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Quantum information and accounting
J. Eng. Technol. Manage. 23 (2006) 33–53 www.elsevier.com/locate/jengtecman
Quantum information and accounting John Fellingham *, Doug Schroeder Ohio State University, OH, USA
Abstract The paper employs quantum probabilities to demonstrate that double entry information processing is efficient in a two-agent control setting. Quantum probabilities differ from classical probabilities as a result of quantum interference. Double entry information processing emerges naturally when interference is combined with quantum correlation or entanglement to produce a reduced set of potential performance measures. That is, both agents’ performances are evaluated based on the same information signal, and this compact information system is more incentive-efficient than evaluating each agent on his or her own signal. # 2006 Published by Elsevier B.V. JEL classification: M41; M52; D82 Keywords: Double entry accounting; Quantum information; Principal-agent
1. Introduction Double entry accounting has played a central role in the communication of financial information for over five centuries. Changes in commerce and technology have been profound, but double entry accounting remains recognizable. There have been a number of papers describing the inherent elegance of double entry (Ijiri, 1971, 1975; Mattessich, 1964; Arya et al., 2000a,b). However, there does not appear to be a clear understanding of how double entry effectively conveys information. The gap between double entry and information is particularly striking in the teaching of accounting. Introductory financial accounting textbooks cover balance sheet and income statement fundamentals, but rarely confront information issues (or even uncertainty) in any formal way. Advanced courses in accounting curricula may analyze information issues, but at that time, double entry mechanics are typically no longer in view. (see discussion in Demski et al., 2002).
* Correspondence to: 2100 Neil Avenue, Fisher College of Business, Columbus, OH 43210, USA. Tel.: +1 614 292 2488; fax: +1 614 292 2118. E-mail address: [email protected] (J. Fellingham). 0923-4748/$ – see front matter # 2006 Published by Elsevier B.V. doi:10.1016/j.jengtecman.2006.02.004
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The purpose of this paper is to develop formally the connection between double entry and information processing in an economic context. The main result is that double entry arises naturally and effectively in a two-person control problem. It is effective: the cost of the twoperson control problem is reduced relative to two single person problems.1 The information structure is double entry: both people in the control problem are assigned the same signal (or number). In particular, if the information structure reports a signal that one agent is successful, then it must also report ‘‘success’’ for the other agent. It is not possible to report success for one and failure for the other. In addition, once the double entry system is committed to, any attempt to report different signals for the different agents yields no benefit in the control problem. In other words, linking the two agents conveys all the useful information. There is no additional information available by relaxing the same signal (double entry) restriction. The reported results depend on the specific probability representation used in the paper, in particular, we use the probability representation used by Nature at the sub atomic level—called quantum probabilities. Quantum probabilities differ from classical probabilities in fundamental ways, the most important of which is the concept of interference: when there are two apparently disjoint ways an event may occur (in a sense discussed later in the paper), the probability of the union of the two ways an event can occur differs from the sum of the two probabilities. The amount of the difference is due to the interference of the two ways, and may be either positive or negative (Feynman et al., 1963; Zuric, 1991; Tegmark and Wheeler, 2001). The purpose of the paper is not to argue that quantum probabilities are more appropriate than classical probabilities for the analysis of economic activities. Rather, the purpose is to derive logical consequences of quantum probabilities if they occur. The consequence of interest in this paper is the natural occurrence of a double entry information structure as an efficient information processor. Nonetheless, some speculation is offered within the discussion as to how quantum probabilities might arise in economic settings. One possible example is the concept of synergy. In an uncertain world synergy could be thought of as increasing the probability of the union of two disjoint events beyond the sum of the probabilities of the two events. Similar manifestations of positive interference appear to be commonplace in the realm of quantum physics. The remainder of the paper is organized as follows. In Section 2 quantum probabilities, and the distinction between classical probabilities, are introduced. The quantum concepts of interference and superposition are discussed. The interferometer, a quantum device used as the basis for the control problem, is introduced in Section 3. Section 4 contains the benchmark oneagent control problem utilizing the interferometer. Section 5 develops the quantum mechanics necessary to construct a two-agent control problem. Multiple quantum objects are subject to the phenomenon termed ‘‘entanglement,’’ another counter intuitive, but valuable, implication of quantum probabilities (Aczel, 2001; Zeilinger, 2000). A consequence of entanglement is presented in Section 6 wherein the benchmark control problem is revisited, this time with two agents. The main results are that the two agents are more efficiently controlled by a double entry information structure and, further, that any attempt to break the double entry connection has no information value. Section 7 revisits the double entry interpretation of the information structure.
1
It is important to note that the double entry and single entry information structures are not generally comparable in the sense of Blackwell (1951). The ranking is obtained in a particular economic setting—a standard principal agent problem.
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2. Quantum probabilities For reasons apparently not well understood, quantum probabilities are described by the mathematics of waves. The mathematics of waves is, fortunately, well understood. Imagine two waves moving on the surface of a liquid. When they meet, interference results, in the sense that, if they meet in phase, that is, trough-to-trough and peak-to-peak, then the amplitude (height) of the resulting wave is the sum of the heights of the two original waves. If they meet out of phase, then the amplitude is determined by subtracting the trough of one from the peak of the other. In other words, wave interference is the adding and subtracting of amplitudes. When waves interfere, amplitudes are added. The average energy of the resulting wave is proportional to the square of the combined amplitudes. In other words, to find the average energy of a wave, first add the amplitudes, then square the result. Quantum probabilities work in parallel fashion. The probability of a particular quantum event – position, spin, polarization, etc. – is equal to the square of the amplitude. If an event can occur in two different ways, the probability of the union is the square of the sum of the amplitudes. In other words, first sum the amplitudes, then square to get the probability. Here’s how Feynman et al. (1963, pp. 37–10) summarizes the procedure. (1) The probability of an event in an ideal experiment is given by the absolute value of a complex number f which is called the probability amplitude: P ¼ probability;
f ¼ probability amplitude;
P ¼ jfj2 :
(2) When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference: f ¼ f1 þ f2 ;
P ¼ jf1 þ f2 j2
notice that Feynman says the amplitude may be a complex number. This eventuality arises from the following train of thought. Sine and cosine functions are quite useful for describing the behavior of waves, but can be cumbersome to manipulate. As operations on exponents are much easier, it is convenient to exploit Euler’s Formula:2 eiu ¼ cos u þ i sin u some care must be taken that the probabilities are real numbers, but the absolute value of a complex number squared (or a complex number times its adjoint) is real. For example, jij2 ¼ iðiÞ ¼ i2 ¼ 1. A striking laboratory example of quantum phenomena is the famous double slit experiment. Send a beam of electrons through a barrier that has two slits, or openings through which the electrons can pass. The pattern of electron arrival on the other side of the barrier looks like a wave formed from the combination of two waves emanating from each of the slits. In particular, the electrons going through one slit interfere with the electrons going through the other. Surprisingly, when the electron beam is reduced to one electron at a time, the interference pattern persists. Hence, the individual electron interferes with itself, and, in a very real sense, 2
Such thinking apparently prompted a remark attributed to French mathematician Jacques Hadamard (1865–1963), ‘‘The shortest path between two truths in the real domain passes through the complex domain’’ (Nahin, 1998, p. 70).
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goes through both slits simultaneously. This is a most counter intuitive state of affairs. Nonetheless, it is descriptive and has important implications. One implication is that a stark distinction between classical and quantum probabilities can be drawn. In classical probability theory it is sensible to define the sample space in such a way that, if an event occurs, then no other event will occur simultaneously. As Feller (1950, p. 14) states, ‘‘The sample space provides a model of an ideal experiment in the sense that, by definition, every thinkable outcome of the experiment is described by one, and only one, sample point.’’ (Italics are Feller’s.) In contrast, quantum probabilities allow for different experimental outcomes to occur simultaneously. For example, in order for the electron to interfere with itself, it must be allowed, in principle, to go through both slits simultaneously. Another implication is that the description of the subatomic world relies, in a fundamental way, on probability statements. There is an inherent uncertainty in the behavior of electrons, protons, photons, and so forth, which require probability statements such as the Heisenberg Uncertainty Principle. The inherent uncertainty prohibits the theory from making deterministic statements. For example, the same experiment, run under identical conditions, may not yield identical results; the theory can only predict the distribution of the results. This uncertainty was the reason for Einstein’s often quoted complaint that ‘‘I don’t believe God plays dice with the Universe.’’ A physical manifestation of this inherent uncertainty is the action of a beam splitter, typically thought of as acting on a photon as illustrated in Fig. 1. For the duration of the paper, instead of speaking of particular subatomic units such as electrons and photons as information-carrying units, we will refer to a generic ‘‘qubit’’ or quantum bit. When a qubit (photon in this case) is
Fig. 1. (a) Beam splitter and (b) beam splitter quantum representation.
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directed to a beam splitter, it may be absorbed or reflected with some probability, say one-half each, pictured in Fig. 1 (Bouwmeester and Zeilinger, 2000). What has been found to be a fruitful way to model the situation is to suppose the qubit possesses, simultaneously, two different components; this is called ‘‘superposition.’’ A qubit is a represented by a two-element vector, say , where the elements a and b are referred to as the b amplitudes. The length of the qubit is standardized so that jaj2 þ jbj2 ¼ 1, hence the square of the amplitude can be interpreted as a probability measure. In terms of the beam splitter, the two elements could represent the polarization of the photon, so the squared amplitude represents the probability the photon is polarized so as to be reflected (or absorbed). The beam splitter alters the qubit. In the vector representation altering a qubit is accomplished by a linear transformation: multiplication by a 2 2 matrix, which does not change the length of the qubit (called a unitary transformation). A useful unitary transformation matrix, and one used to model the behavior of the beam splitter, is a Hadamard transformation: 1 1 1 H ¼ pffiffiffi 2 1 1 pffiffiffi pffiffiffi 1 0 1=p2ffiffiffi 1= p2ffiffiffi ¼ and H ¼ . Now the behavior of a beam splitter can be 0 1 1= 2 1= 2 1 ; the beam splitter is H. So modeled in quantum superposition notation. The entering qubit is 0 pffiffiffi 1 1=p2ffiffiffi after the qubit is acted on by the beam splitter: H ¼ , and the squared amplitudes 0 1= 2 imply probabilities of one-half for reflection and absorption. Dirac notation is an elegant way to keep track of quantum probabilities, and will be 1 0 particularly useful for multiple qubit systems (Dirac, 1958). Define j0i and j1i 0 1 and the beam splitter in Fig. 1 can be redrawn as in Fig. 1b.
note H
3. Interferometer In this section a quantum device called an interferometer is described. The interferometer provides the basis for the economic control problem described in the following section. The basic interferometer, pictured in Fig. 2, is a simple extension of the beam splitter.
Fig. 2. Single qubit interferometer without phase shifter.
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Fig. 3. Single qubit interferometer with phase shifter.
A j0i qubit enters from the left. In between two beam splitters (Hadamard matrices) are reflecting barriers, which do not affect the qubit other than to redirect it. The math is straightforward: two H matrices operate in sequence on a j0i qubit. HHj0i ¼ H
j0i þ j1i j0i þ j1i þ j0i j1i pffiffiffi ¼ ¼ j0i 2 2
For the simple interferometer in Fig. 2 a j0i qubit will emerge with probability one, and a j1i qubit will never appear (interference requires that amplitudes be summed, then squared, to identify quantum probabilities). To allow for a non-trivial stochastic outcome from the interferometer, install a phase shifter along the upper path: a phase shift of angle u can be accomplished by operator Q where: Qj0i ¼ eiu j0i
and
Qj1i ¼ j1i:
eiu 0 Or, in matrix form Q ¼ . 0 1 Now the math is slightly more complicated (Fig. 3): eiu j0i þ j1i eiu ðj0i þ j1iÞ þ j0i j1i pffiffiffi ¼ 2 2 notice there are two ways that a j0i qubit can emerge from the interferometer. It can proceed along the upper path denoted by the term eiu =2j0i. The lower path is captured by j0i/2. A classical probability can be calculated by first squaring the amplitudes and then adding: jeiu =2j2 þ j1=2j2 ¼ ðeiu Þðeiu Þ=4 þ 1=4 ¼ 1=2. In other words, classically the probability of j0i is independent of the size of the angle u; i.e., there is no interference. On the other hand, with quantum probabilities the amplitudes are added together prior to squaring. HQHj0i ¼ H
HQHj0i ¼
eiu þ 1 eiu 1 j0i þ j1i 2 2
To calculate the probability of a j0i emerging from the interferometer: iu e þ 1 2 ðeiu þ 1Þðeiu þ 1Þ 1 þ cos u 2 u ¼ ¼ ¼ cos 2 4 2 2
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The third expression uses Euler’s Formula; the fourth uses the half-angle trigonometric identity cosðuÞ ¼ cos2 ðu=2Þ sin2 ðu=2Þ combined with 1 ¼ cos2 ðu=2Þ þ sin2 ðu=2Þ. Now interference occurs and the probability of j0i depends on u; when the angle is zero, all the probability weight is on j0i. As the phase shift angle increases, more weight is transferred to j1i. The above argument is summarized in the following observation. Observation 1.
The phase shift angle determines the probabilities of the qubit emerging from the interferometer. The next step is to interpret the phase shift as an action taken by an economic agent, embedding quantum probabilities in an economic context. The simplest economic context in which information plays a central role is the standard principal-agent problem, wherein the principal wishes to induce the agent to take an appropriate, but unobservable, action by basing compensation on an informative signal. 4. Control problem formulation In subsequent sections we wish to compare the information carrying capacity of double entry versus a single entry system. The two information structures are not comparable in the Blackwell sense; that is, one does not dominate the other for all decision contexts. Therefore, we will make the comparison in a particular economic setting—a standard principal-agent control problem. Consider quantum performance information processing first in a single-agent setting. Let the agent (who takes the unobservable act) be risk averse with the following domain additive CARA preferences defined over wealth (W) and personal cost of effort (c): UðW; cÞ ¼ erðWcÞ information about the agent’s binary action choice is represented by two angles in the interferometer: uH (working) and uL (shirking). The probability of a success signal, then, is ð1 cosui =2Þ for i = H, L. Let IS be the payment for a success, and IF otherwise (failure). Finally, let RW denote the agent’s reservation wage of employment. For a risk neutral principal the program is written as follows with the standard individual rationality and incentive compatibility constraints, under the assumption the principal wishes to induce the high act. (The structure, and parsimony, of the agency problem here are the same as in Christensen and Demski, 2003, Chapter 11).
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Principal’s problem: min IH ;IF
s:t:
1-cos uH 1 þ cos uH IS þ IF 2 2
1 cos uH 1 þ cos uH UðIS ; cH Þ þ UðIF ; cH Þ UðRWÞ 2 2
1 cos uH 1 þ cos uH 1 cos uL UðIS ; cH Þ þ UðIF ; cH Þ UðIS ; cL Þ 2 2 2 1 þ cos uL þ UðIF ; cL Þ 2
Since the constraints must hold as equalities, and for CARA domain additive U(I, c) = (1)(U(I)/ U(c)), the problem can be rewritten as a linear system. 1 1 cos uH 2 1 cos uL
1 þ cos uH 1 þ cos uL
UðIS Þ UðIF Þ
¼
UðRW þ cH Þ UðRW þ cL Þ
The general solution is
UðIS Þ UðIF Þ
1 1 þ cosuL ¼ cosuL cosuH cosuL 1
1 cosuH 1 cosuH
UðRW þ cH Þ UðRW þ cL Þ
A benchmark numerical example will prove useful for subsequent comparisons. Let the risk aversion parameter, r, be .001. Let the two possible actions be uH = 458 and uL = 08. The two acts have associated personal cost to the agent of cH = 100 and cL = 0. The agent’s reservation wage is 200. Solving pffiffiffi pffiffiffi UðIS Þ ¼ ð4 þ 2 2ÞUðRW þ cH Þ þ ð2 2 3ÞUðRW þ cL Þ ¼ :286711 ) IS ¼ 1249:28 and UðIF Þ ¼ UðRW þ cL Þ ) IF ¼ 200 The expected payment is 353.66. The efficiency loss is 53.66, since for an observable action the agent needs to be compensated for the reservation wage (200) plus the cost of the high act (100). 5. Two qubits—two activities In order to expand the control problem to accommodate two activities performed by two separate agents, this section introduces some rudimentary two qubit quantum operations.
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Two qubits are combined into one system according to tensor multiplication of vectors. That is,
7 6 7 6 6 a1 a2 7 6 a 1 b2 7 a1 a2 7 ¼6 4 b1 a2 5 b1 b2 b1 b2
there are a couple of things to do in the way of setup so multiple qubit states can be analyzed. The first is to note there are multiple qubit transformations, the most important of which is the ‘‘controlled not’’ transformation: 6 61 6 60 CNOT ¼ 6 40 0
0 1 0 0
0 0 0 1
7 07 7 07 7 15 0
Dirac 6 notation simplifies tensor multiplication. For example, j0ij1i 6 ¼ j01i, where 7 7 6a7 607 6 7 6 7 607 617 a 1 7 7, Also, and jcij0i ¼ aj00i þ bj10i, since ¼6 j01i ¼ 6 4 b 5. 405 b 0 0 0 CNOTj00i ¼ j00i, as well as CNOTj10i ¼ j11i. The first qubit is called the control qubit; if it is j1i, then the second qubit (the target qubit) flips from a j0i to a j1i, or vice-versa. If the control qubit is a j0i, then the target is unchanged. An important two qubit state is called an ‘‘entangled’’ pair of qubits, so important it has its own conventional notation, denoted jb00i (Nielsen and Chuang, 2003, p. 25). Start with j00i and perform, in order, an H transformation on the first qubit (denote the operation H1), and then a CNOT transformation on the pair. jb00 i ¼ CNOT H1 j00i ¼ CNOT
j00i þ j10i j00i þ j11i pffiffiffi pffiffiffi ¼ 2 2
The resulting two-qubit system is referred to as an entangled state; note that it cannot be created by the tensor combination of any two individual qubits. Furthermore, when the state of one of the qubits is determined, the state of the other is also determined. That is, if the first qubit is measured as j0i, then the other must be j0i, as well. In other words, there is uncertainty in the system, but if the uncertainty is resolved for one qubit, it is resolved pffiffiffifor the other, also. jb00i is a Bell or EPR state and its orthogonal complements are jb i ¼ 1= 2ðj01i þ j10iÞ; jb10 i ¼ 1= 01 pffiffiffi pffiffiffi 2ðj00i j11iÞ, and jb11 i ¼ 1= 2ðj01i j10iÞ: Together, the four are said to form the Bell basis. In theory Bell states can be used to generate some apparently strange, or at least non-intuitive, results. See Bell (1964). A further implication is the particularly non-intuitive process of quantum teleportation (Zeilinger, 2003, and Nielsen and Chuang, 2000, pp. 26–28). For quantum information processing and computation tasks, entanglement is considered an important resource; the next section demonstrates that, not only is entanglement a resource in the control problem, it induces a double entry property in the information structure.
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6. Two-qubit control In this section the control problem of interest is one involving two agents or two activities. As an example, suppose the principal wishes to provide incentives both for a production process and also a marketing process. Two-phase shift angles are required to model the effort levels of the two activities, say Q1 and Q2. A two-qubit interferometer performs a phase shift for each qubit: algebraically, H1Q1H1H2Q2H2 where the subscripts refer to the qubit operated on. 6.1. Tensor qubits j00i as interferometer input The first two-qubit problem analyzed is where the two qubits entering the interferometer are the tensor combination j00i. There are four possible outputs of the interferometer: j00i, j00i, j10i, and j11i. The amplitudes and probabilities for each are presented in Observation 2; the derivation is in Appendix A. Observation 2.
Comparing Observation 2 with Observation 1, it can be seen that there is no gain in the control problem. That is, the best that can be done in the two-activity problem is the same as two applications of the one-qubit problem. From Observation 1 the unconditional probability of j0i is cos2 ðu1 =2Þ. In the two-qubit problem of Observation 2 the conditional probability of a j0i in the first position given a j0i in position two is probj00i=ðprobj00iþ probj10iÞ ¼ cos2 ðu1 =2Þcos2 ðu2 =2Þ=ðcos2 ðu2 =2Þ ¼ cos2 ðu1 =2Þ. Therefore, the second qubit is not conditionally informative about the first (Christensen and Demski, 2003 or Holmstrom, 1979). 6.2. Entangled qubits jb00i as interferometer input The next setting differs from Observation 2 only in the preparation of the two-qubit state entering the interferometer. Instead of tensor combinationp j00i, ffiffiffi the entering state is an entangled one, jb00i. Recall jb00 i ¼ CNOTH1 j00i ¼ ðj00i þ j11i= 2Þ. With this change in the experiment Observation 3 reports the amplitudes and probabilities of possible outcomes. (The derivation is in Appendix A). Note that informativeness in the control setting depends critically on the difference between classical and quantum probabilities; that is, adding amplitudes and then squaring gives a distinctly different result than simply adding probabilities.
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Observation 3.
The notable result is that, when classified in the Bell basis, only two of the four possible pffiffiffi Bell states appear as outcomes. The probability of the outcome jb01 i ¼ ðj01i þ j10i= 2Þ increases in both u1 and u2, the effort angles for the two agents (retaining the convention that u1 + u2 p), so it is labeled as a ‘‘success’’ signal. A success for one agent must necessarily be accompanied by a success for the other; there is no possibility of a success for one and a failure for the other. This linkage of the signals is interpreted as double entry. Both agents must receive the same number, whatever it is. Hence, the number is entered twice. The primary question is whether the arrangement in Observation 3 (double entry) yields an efficiency gain relative to Observation 2. The two information structures are not comparable for all decision problems—that is, one is not a subpartition of the other. However, for the control problem under consideration, the double entry system strictly dominates the other. First reconsider the numerical example. Assume the ‘‘other’’ agent chooses u2 = 458, and solve the problem for the u1-agent. Collusion issues will be addressed later. Let all the other parameters be the same as in the previous example. The solution is as follows: UðIS Þ ¼ ð1 þ
pffiffiffi pffiffiffi 2ÞUð300Þ 2Uð200Þ
UðIS Þ ¼ :630633260 ) IS ¼ 461:0307898 pffiffiffi pffiffiffi UðIF Þ ¼ ð1 2ÞUð300Þ þ 2Uð200Þ UðIF Þ ¼ :851003181 ) IF ¼ 161:3394125 the expected payment is 311.185 implying an efficiency loss of 11.185, and a substantial efficiency gain relative to the Observation 2 result (efficiency loss equal to 53.66). The probability of a success in the entangled problem is :5 ¼ sin2 ðu1H þ u2H =2Þ. For the tensor problem, the corresponding probability is :146 ¼ sin2 ðu1H =2Þ. The increase in probability of a success, however, is not accompanied by increased payments or efficiency loss. In the entangled qubit problem, the efficiency loss declines from 53.66 to 11.185. Furthermore, that decline is accomplished as both payments decline. Payment for a success, Is, goes from 1,249.28 to 461.03. Likewise, the low payment declines from 200 to 161.34. It is not merely that the payments become closer together, thereby reducing the risk premium, they also both decline. The two-qubit control example is summarized in Table 1.
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Table 1 Two-agent quantum control example Parameter values r = 0.001 RW = 200
cH = 100 cL = 0
u1H = 458 u1L = 08
u2 = 458
Signal j11i (S) Independent performance evaluation (j00i) sin2(u1H/2)0.1464 Probability(signal ju1H) Probability(signal ju1L) sin2(u1L/2) = 0.0 Payments (I) 1249.28
E[I] j01i (F) cos2(u1H/2)/2 0.8536 cos2(u1L/2) = 1.0 200
353.66
Signal jb01i (S) Coordinated performance evaluation (jb00i) sin2(u1H + u2)/2 = 0.5 Probability(signalju1H) Probability(signalju1L) sin2(u1L + u2)/20.1464 Payments (I) 461.03
E[I] jb00i (F) cos2(u1H + u2)/2 = 0.5 cos2(u1L + u2)/20.8536 161.34
311.19
That these results generalize beyond the numerical example is stated in the Proposition.3 The Proposition establishes that, in the presence of entanglement, the efficiency of the contract with agent one increases with the effort level of agent two. Therefore, for any non-zero agent two effort, the entangled (double entry based) contract dominates a tensor (single entry) contract. (For agent two effort equal to zero, the two contracts are equivalent.) Proposition (The proof is in Appendix B). In the entangled qubit control problem, for 0 u1L < u1H p=2 and marginal increase in u2 (i) increases the probability of a success signal; (ii) reduces both payments; (iii) decreases the efficiency loss.
0 u2 p=2, a
Finally, it is also worth noting there is no collusion problem in the entangled control problem even though the constraints are written so that high effort is a Nash best response to high effort from the other agent.4 In a standard correlated state two-agent control problem, writing the constraints as best response may cause the existence of another equilibrium that is preferred by the agents and dispreferred by the principal. Roughly speaking, this occurs 3 To focus on accounting implications the quantum control experiment is subject to the following restrictions: (1) initial states are limited to j00i or jb00i; (2) information processing employs the standard two-qubit interferometer H1Q1H1H2Q2H2; and (3) probabilities are based on projections into computational eigenstates (summarizing as compactly as possible without compromising likelihood ratios effectively allows measurement in the Bell eigenstates for initial state jb00i). 4 Collusion does not arise from initial Bell states with positive interference (amplitudes of the same sign). However, jb10i and jb11i have negative interference and produce severe collusion concerns. We only discuss the initial state jb00i but jb01i consistently utilized produces the same incentive-efficiency (with some relabeling). Again, this is a commitment to double entry accounting—a convention is adopted and once adopted it is critical that it be consistently applied.
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because the Nash incentive contract rewards the agents for similar signals and punishes dissimilar. This, in turn, induces the agents to collude, providing similar signals with high probability by shirking. One reason the dominated equilibrium is not a problem in the entangled control problem is that there is no possibility of dissimilar signals. That is, there is only one signal that applies to both agents. Additionally, a way to see that there is only one equilibrium which satisfies both agents’ individual rationality constraints is to present the agent payoffs in normal form.
Agent one chooses the row. Note that shirking by agent one does not increase agent one’s expected utility—the first number in each cell. Agent one’s shirking does, however, decrease agent two’s expected utility. Furthermore, the shirk/shirk strategy pair is clearly dominated by work/work. (When both agents shirk, IF = 161.3 is guaranteed to both agents, as the probability of a success is zero.). The question arises as to whether there is any benefit to disentangling the signal in order to obtain a signal particular to each individual. The answer is, in the presence of entangled input, there is no benefit to individual signals. The first step is to disentangle the output so each individual is recognizable, that is, the output is in tensor form—j00i, j01i, and so forth. The amplitudes and probabilities for the first agent (qubit) of disentangled signals are presented in Observation 4. (The derivation is in Appendix A). Observation 4.
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Consider the benefit of individual measurements. To calculate the probability of obtaining a j0i qubit in position one, for example, note that there are two different ways for this to occur and add the amplitudes together. eiðu1 þu2 Þ þ 1 þ eiðu1 þu2 Þ 1 eiðu1 þu2 Þ pffiffiffi ¼ pffiffiffi 2 2 2 pffiffiffi The probability is ðjeiðu1 þu2 Þ = 2j2 Þ ¼ ð1=2Þ. That is, the probability of the first output qubit being j0i is one-half regardless of the effort angle chosen by agent 1. Hence, measuring the first qubit only is useless for controlling the first activity. A similar argument applies to the second agent (qubit). 7. Some remarks about accounting and double entry Broadly speaking two main lessons of the previous section are: 1 Entanglement yields an efficiency gain in the control problem, as in the proposition, and 2 The information system associated with entanglement is restricted in the sense that it uses only two of the possible four signals. That is, jb00i and jb01i are used, but the other two states in a Bell basis, jb10i and jb11i are not. In this section we briefly speculate on parallels between the double entry accounting system and the information system, which shows up in the entangled qubit control problem. Double entry is also restricted in that it only uses two of four possible signals. The restriction is a consequence of the balancing property: assets must equal equities. Of four possible signals, accounting can only produce two, as follows.
Not allowed Not allowed
Assets
Equities
Positive Negative Positive Negative
Positive Negative Negative Positive
A variation on the restricted signal theme is that asset recognition consists of two requirements: control of the item and a future economic benefit. As an example, inventory may be recognized only if the item exists – has been produced – and a market exists for its sale. In terms of the control problem, we can think of two agents supplying inputs: a production manager and a marketing manager. Because of the restricted information system, a success for one cannot occur in the absence of a success for the other. A restricted information system yields efficiency gains only in the presence of an entangled entry qubit. How the agents’ performance evaluation might be entangled is a difficult problem. The concepts of superposition and interference are deep and non-intuitive in the domain of physics. How they might be operationalized in an economic context is an open question.
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We will, however, offer some brief speculations. In quantum physics entanglement occurs with the following sequence of operations: j00i þ j11i pffiffiffi 2 CNOT ties the two qubits together. While in quantum physics the intuition for how this occurs is not strong, it may be easier to see if the qubits act as rational economic agents. The agents can coordinate. For example, the marketing manager can agree to market the design chosen by the production manager regardless of the state of the market, or vice-versa. It is not implausible that coordination of this type occurs on a regular basis in economic organizations. The speculative conclusion is that double entry accounting is appropriate in organizations where coordination of activities is a primary concern. The beam splitter embodied by the H transformation is critical to the generation of interference necessary for quantum probabilities. Here the economic analog may be synergy (positive or negative) among activities. A common example in the academic domain is the synergy between teaching and research activities. Some might argue (or act as if) there is negative interference between the two activities—more time or energy devoted to one necessarily diminishes the amount available for the other. On the other hand, cranking up the phase shift angle devoted to one of the activities may, indeed, increase the efficiency of the other, possibly in terms of creativity, insight, or general organization. jb00 i ¼ CNOT H1 j00i ¼
7.1. Concluding remarks The paper has argued, and attempted to demonstrate, that classical probabilities and quantum probabilities are fundamentally different due to quantum interference. Interference has the potential to enhance the efficiency of organizations. For instance, universities face potential interference (positive or negative) between research and teaching, and their survivorship is enhanced if this interference is, on average, positive. Similarly, efficient coordination of activities in industrial organizations and hence their survivorship are enhanced by exploiting positive interference. Employing one performance measurement for the evaluation of two agents or two activities (a double entry characteristic) is more efficient than independent evaluation of the two agents in a setting with quantum interference. This is accomplished via quantum entanglement of the agents’ performance evaluation. Not only is independent evaluation of the agents inefficient but also it may lead to serious compromise of the cause and effect relationship of double entry and increase the likelihood of transgressions such as experienced by Enron. This work on accounting implications of quantum probabilities is in its infancy and much remains to be explored. One of the big challenges in quantum information processing is maintaining coherent quantum states. Two related issues warrant further exploration. First, there exists a demand for error correction in the processing of quantum information. Aside from the (complex) diagonal matrix Q to represent the agent’s impact on performance, our control experiment restricts quantum operations to Hadamard transformations.5 This and similar 0 1 1 0 and Z ¼ , known as Pauli-X and Pauli-Z. It is noteworthy that 1 0 0 1 0 i if one admits complex, non-diagonal operators such as Pauli- Y ¼ then initial state j00i is sometimes more i 0 efficient than initial state jb00i in the control setting. For example, initial state j00i is more efficient pffiffiffi than is initial state jb00i when processed via U1Q1H1U2Q2H2 for u1H = p/4, u1L = 0, and u2 = p/4 where U ¼ 1= 2ðZ þ YÞ. 5
pffiffiffi Note H ¼ 1= 2ðZ þ XÞ where X ¼
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transformations are the operators employed by most of the known quantum error correction algorithms (Nielsen and Chuang, 2000, Chapter 10). Second, quantum interference is destroyed by interaction with the (unknown and unobservable) environment. This is referred to as decoherence (Zurek, 1991). Of course, decoherence makes the potential efficiency of quantum probabilities inaccessible. More work on quantum coherency and its relation to double entry accounting and internal control system design seems desirable. Acknowledgment Helpful comments from Anil Arya, John Christensen, Peter Christensen, Joel Demski, Hans Frimor, Jonathan Glover, Thorbjørn Knudsen, Scott Liao, Florin Sabac, Eric Spires, two anonymous referees, and workshop participants at the University of Southern Denmark are gratefully acknowledged. Appendix A. Derivation of observations A.1. Derivation of Observation 2 ðeiu2 þ 1Þj00i þ ðeiu2 1Þjj01i 2 iu1 iu1 iu2 þ 1Þðe j00i þ e j10iÞ þ ðe 1Þðeiu1 j01i þ eiu1 j11iÞ pffiffiffi 2 2
H1 Q1 H1 H2 Q2 H2 j00i ¼ H1 Q1 H1 ¼ H1
ðeiu2
ðeiu2 þ 1Þðeiu1 þ 1Þj00i þ ðeiu2 1Þðeiu1 þ 1Þj01i þ ðeiu2 þ 1Þðeiu1 1Þj10i þ ðeiu2 1Þðeiu1 1Þj11i ¼ 4 The probabilities are calculated by recalling jeiu1 þ 1=2j2 ¼ cos2 ðu1 =2Þ and jeiu1 1=2j2 ¼ sin2 ðu1 =2Þ.
A.2. Derivation of Observation 3 j00iþj01iþj10ij11i eiu2 j00iþj01iþeiu2 j10ij11i ¼H2 2 2 eiu2 ðj00iþj01iÞþj00ij01iþeiu2 ðj10ij11iÞj10iþj11i ðeiu2 þ1Þ ðeiu2 1Þ pffiffiffi jb00 iþ jb01 i ¼ ¼ 2 2 2 2
H2 Q2 H2 jb00 i¼H2 Q2
Likewise, ðeiu1 þ 1Þ ðeiu1 1Þ jb00 i þ jb01 i and 2 2 ðeiu1 1Þ ðeiu1 þ 1Þ jb00 i þ jb01 i: H1 Q1 H1 jb01 i ¼ 2 2
H1 Q1 H1 jb00 i ¼
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Hence,
ðeiu1 þ 1Þðeiu2 þ 1Þ þ ðeiu1 1Þðeiu2 1Þ H 1 Q1 H 1 H 2 Q2 H 2 jb00 i ¼ jb00 i 4 iu2 ðe þ 1Þðeiu1 1Þ þ ðeiu2 1Þðeiu1 þ 1Þ þ jb01 i 4
¼ jb00 iðeiðu1 þu2 Þ þ 1=2Þ þ jb01 iðeiðu1 þu2 Þ 1=2Þ A.3. Derivation of Observation 4 Start with the expression in Observation 3. eiðu1 þu2 Þ þ 1 eiðu1 þu2 Þ 1 H1 Q1 H1 H2 Q2 H2 jb00 i ¼ jb00 i þ jb01 i 2 2 eiðu1 þu2 Þ þ 1 j00i þ j11i eiðu1 þu2 Þ 1 j01i þ j10i eiðu1 þu2 Þ þ 1 pffiffiffi pffiffiffi pffiffiffi ¼ þ ¼ j00i 2 2 2 2 2 2 þ
eiðu1 þu2 Þ 1 eiðu1 þu2 Þ 1 eiðu1 þu2 Þ 1 pffiffiffi pffiffiffi pffiffiffi j01i þ j10i þ j11i 2 2 2 2 2 2
Appendix B. Proof of proposition First, for reference purposes, we state some relationships derived from the statement of the control problem. 1 cosðu1H þ u2 Þ 1 þ cosðu1H þ u2 Þ EðpaymentÞ ¼ E½I ¼ IS þ IF 2 2 1 1 IS ¼ lnðUðIS ÞÞ IF ¼ lnðUðIF ÞÞ r r UðIS Þ ¼
ð1 þ cosðu1L þ u2 ÞÞUðRW þ cH Þ ð1 þ cosðu1H þ u2 ÞÞUðRW þ cL Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
UðIF Þ ¼
ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
dIS 1 dUðIS Þ=du2 ¼ r du2 UðIS Þ
dIL 1 dUðIL Þ=du2 ¼ r du2 UðIL Þ
dUðIS Þ ½sinðu1H u1L Þ þ sinðu1H þ u2 Þ sinðu1L þ u2 Þ ¼ ½UðRW þ cL Þ UðRW þ cH Þ du2 ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 dUðIF Þ ½sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ ¼ ½UðRW þ cL Þ du2 ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 UðRW þ cH Þ
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It is feasible to motivate uH if probabilityðfailurejuL Þ=probabilityðfailurejuH Þ > erðcH cL Þ (see Christensen and Demski, 2003, pp. 250–251). Since 1 þ cosðu1L þ u2 Þ=1 þ cosðu1H þ u2 Þ 1 þ cosðu1L Þ=1 þ cosðu1H Þ for 0 uj p/2 and u1L < u1H (equal iff u2 = 0), there exist values of u2 for an entangled state when it is feasible to motivate uH while it is infeasible to motivate uH when the state is not entangled. Hence, the results proven below are monotonic in u2. (i) Probability of success increases with u2.Probability of success is 1 cosðu1H þ u2 Þ=2. The first derivative with respect to u2 is sinðu1H þ u2 Þ=2 > 0. (ii) IS and IF both decrease in u2. As U(I) is monotonic in I, need to check dUðIS Þ=du2 and dUðIF Þ=du2 . dUðIS Þ=du2 is clearly negative. The sign of dUðIF Þ=du2 depends on sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ, which is positive by the lemma in Appendix C. Hence, dUðIF Þ=du2 < 0. (iii) The change in the expected payment is negative as u2 increases. dE½I 1 cosðu1H þ u2 Þ dIS 1 þ cosðu1H þ u2 Þ dIF sinðu1H þ u2 Þ ðIS IF Þ ¼ þ þ du2 2 2 2 du2 du2 The first two terms are negative by (ii) above; the third term is positive. Rewrite the third term: sinðu1H þ u2 Þ=2r lnðUðIF Þ=UðIS ÞÞ where UðIF Þ=UðIS Þ > 1. Now expand the first two terms, which is a little bit more work. ð1 cosðu1H þ u2 ÞÞ dIS ð1 þ cosðu1H þ u2 ÞÞ dIL þ 2 2 du2 du2 ð1 cosðu1H þ u2 ÞÞ=2ð1=rÞðUðRW þ cL Þ UðRW þ cH ÞÞ½sinðu1H u1L Þ þ sinðu1H þ u2 Þ sinðu1L þ u2 Þ=½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 ¼ ½ð1 þ cosðu1L þ u2 ÞÞUðRW þ cH Þ ð1 þ cosðu1H þ u2 ÞÞUðRW þ cL Þ= ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ ð1 þ cosðu1H þ u2 ÞÞ=2ð1=rÞðUðRW þ cL Þ UðRW þ cH ÞÞ½sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ=½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 þ ½ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ= ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ Numerator: ¼
1 ½UðRW þ cL Þ UðRW þ cH Þ 2r fð1 cosðu1H þ u2 ÞÞ½sinðu1H u1L Þ þ sinðu1H þ u2 Þ sinðu1L þ u2 Þ ½ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ þ ð1 þ cosðu1H þ u2 ÞÞ½sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ ½ð1 þ cosðu1L þ u2 ÞÞUðRW þ cH Þ ð1 þ cosðu1H þ u2 ÞÞUðRW þ cL Þg
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Denominator: ðcosðu1L þ u2 Þ cosðu1H þ u2 ÞÞ ½ð1 þ cosðu1L þ u2 ÞUðRW þ cH Þ ð1 þ cosðu1L þ u2 ÞÞUðRW þ cL ÞÞ ½ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ Expand and combine terms. Numerator:
1 ½UðRW þ cL Þ UðRW þ cH Þ 2r f2½UðRW þ cL Þ UðRW þ cH Þ ½sinðu1H þ u2 Þ cosðu1H þ u2 Þsinðu1H u1L Þ sinðu1L þ u2 Þ þ ½2UðRW þ cH Þcosðu1L þ u2 Þ 2UðRW þ cL Þcosðu1H þ u2 Þ ½cosðu1H þ u2 Þsinðu1H þ u2 Þ þ sinðu1H u1L Þ þ cosðu1H þ u2 Þsinðu1L þ u2 Þg
Denominator: UðIF ÞUðIS Þ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ3 Substitute UðIS Þ UðIF Þ ¼ 2
UðRW þ cH Þ UðRW þ cL Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
UðIS Þ þ UðIF Þ ¼ 2
UðRW þ cH Þcosðu1L þ u2 Þ UðRW þ cL Þcosðu1H þ u2 Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
Numerator:
1 UðIF Þ UðIS Þ ½n1 ðUðIF Þ UðIS ÞÞ þ n2 ðUðIF Þ þ UðIS ÞÞ 2r 2
where n1 ¼ sinðu1H þ u2 Þ þ cosðu1H þ u2 Þsinðu1H u1L Þ þ sinðu1L þ u2 Þ n2 ¼ cosðu1H þ u2 Þsinðu1H þ u2 Þ þ sinðu1H u1L Þ þ cosðu1H þ u2 Þsinðu1L þ u2 Þ Denominator: UðIF ÞUðIS Þ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ
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Recombining to get all the terms in dE½I=du2 , we can write 4r
dE½I sinðu1H þ u2 Þ du2
UðIF Þ UðIS Þ UðIF Þ n1 n2 ½UðIS Þ þ UðIF Þ ¼ 2ln ½UðIS Þ UðIF Þ UðIS Þ UðIS ÞUðIF Þ den den where den ¼ sinðu1H þ u2 Þ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ Since n1 =den ¼ sinðu1H u1L =2Þ=sinðu1H þ u1L =2 þ u2 Þ > 0 and n2 =den ¼ 1, we can write 4r dE½I UðIF Þ UðIS Þ UðIF Þ < 2ln þ ½UðIS Þ þ UðIF Þ sinðu1H þ u2 Þ du2 UðIS Þ UðIS ÞUðIF Þ UðIF Þ UðIS Þ2 UðIF Þ2 UðIF Þ UðIF Þ UðIS Þ ¼ 2ln þ <0 þ ¼ 2ln UðIS Þ UðIS Þ UðIS Þ UðIF Þ UðIF ÞUðIS Þ Recall UðIF Þ=UðIS Þ > 1; the expression above is equal to zero at UðIF Þ=UðIS Þ ¼ 1, and the derivative with respect to UðIF Þ=UðIS Þ is everywhere negative. Hence, the inequality follows, and dE½I <0 du2 Appendix C. Lemma
Lemma. Let u1H þ u2 equal ffAOE and u1L þ u2 equal ffBOE in the above diagram. For 0 u1L < u1H
p 2
and
0 u2
p ; 2
sinðu1H u1L Þ sinðu1H þ u2 Þ þ sinðu1L þ u2 Þ 0
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Proof. ðu1H u1L Þ ¼ ðu1H þ u2 Þ ðu1H þ u2 Þ or ffAOB then sinðu1H u1L Þ ¼ AD; sinðu1H þ u2 Þ ¼ AC;
sinðu1L þ u2 Þ ¼ BF
Since AD is perpendicular to OB, it is clear AC < AD + BF. One or more of the angles may be in the second quadrant. If sinðu1L þ u2 Þ sinðu1H þ u2 Þ, then the lemma is immediate. If not, a similar diagram can be drawn and the logic is similar. References Aczel, A., 2001. Entanglement: The Greatest Mystery in Physics. Four Walls Eight Windows, New York, NY. Arya, A., Fellingham, J., Schroeder, D., 2000a. Accounting information, aggregation, and discriminant analysis. Management Science 46 (6). Arya, A., Fellingham, J., Glover, J., Schroeder, D.A., Strang, G., 2000b. Inferring transactions from financial statements. Contemporary Accounting Research 17 (3). Bell, J., 1964. On the Einstein-Podolsky-Rosen paradox. Physics 1 . Blackwell, D., 1951. Comparison of experiments. In: Proceedings of Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA,. University of California Press. Bouwmeester, D., Zeilinger, A., 2000. The physics of quantum information: basic concepts. Dirk Bouwmeester. In: Artur, Ekert, Anton, Zeilinger (Eds.), The Physics of Quantum Information. Springer, Berlin. Christensen, J., Demski, J., 2003. Accounting Theory. McGraw Hill-Irwin, New York, NY. Dirac, P., 1958. The Principles of Quantum Mechanics. Oxford University Press, Oxford, UK. Feller, W., 1950. An Introduction to Probability Theory and Its Applictions. John Wiley & Sons, New York. Feynman, R., Leighton, R., Sands, M., 1963. The Feynman Lectures on Physics, vol. 1. Addison-Wesley Publishing Co., Reading MA. Holmstrom, B., 1979. Moral hazard and observability. Bell Journal of Economics. Ijiri, Y., 1971. Fundamental queries in aggregation theory. Journal of the American Statistical Association 66 . Ijiri, Y., 1975. Theory of Accounting Measurement. Studies in Accounting Research No. 10. American Accounting Association, Sarasota, FL. Mattessich, R., 1964. Accounting and Analytical Methods. Richard D. Irwin, Homewood, IL. pffiffiffiffiffiffiffi Nahin, P., 1998. An Imaginary Tale: The Story of 1. Princeton, NJ, Princeton University Press. Nielsen, M., Chuang, I., 2000. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK. Tegmark, M., Wheeler, J., 2001. As quantum theory celebrates its 100th birthday, spectacular successes are mixed with persistant puzzles. Scientific American. Zeilinger, A., 2000. Quantum teleportation. Scientific American 282 4. Zuric, W., 1991. Decoherence and the transition from quantum to classical. Physics Today.
Further reading Nielsen, M., 2003. Simple rules for a complex quantum world. Scientific America 288, 5.
Quantum information and accounting John Fellingham *, Doug Schroeder Ohio State University, OH, USA
Abstract The paper employs quantum probabilities to demonstrate that double entry information processing is efficient in a two-agent control setting. Quantum probabilities differ from classical probabilities as a result of quantum interference. Double entry information processing emerges naturally when interference is combined with quantum correlation or entanglement to produce a reduced set of potential performance measures. That is, both agents’ performances are evaluated based on the same information signal, and this compact information system is more incentive-efficient than evaluating each agent on his or her own signal. # 2006 Published by Elsevier B.V. JEL classification: M41; M52; D82 Keywords: Double entry accounting; Quantum information; Principal-agent
1. Introduction Double entry accounting has played a central role in the communication of financial information for over five centuries. Changes in commerce and technology have been profound, but double entry accounting remains recognizable. There have been a number of papers describing the inherent elegance of double entry (Ijiri, 1971, 1975; Mattessich, 1964; Arya et al., 2000a,b). However, there does not appear to be a clear understanding of how double entry effectively conveys information. The gap between double entry and information is particularly striking in the teaching of accounting. Introductory financial accounting textbooks cover balance sheet and income statement fundamentals, but rarely confront information issues (or even uncertainty) in any formal way. Advanced courses in accounting curricula may analyze information issues, but at that time, double entry mechanics are typically no longer in view. (see discussion in Demski et al., 2002).
* Correspondence to: 2100 Neil Avenue, Fisher College of Business, Columbus, OH 43210, USA. Tel.: +1 614 292 2488; fax: +1 614 292 2118. E-mail address: [email protected] (J. Fellingham). 0923-4748/$ – see front matter # 2006 Published by Elsevier B.V. doi:10.1016/j.jengtecman.2006.02.004
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The purpose of this paper is to develop formally the connection between double entry and information processing in an economic context. The main result is that double entry arises naturally and effectively in a two-person control problem. It is effective: the cost of the twoperson control problem is reduced relative to two single person problems.1 The information structure is double entry: both people in the control problem are assigned the same signal (or number). In particular, if the information structure reports a signal that one agent is successful, then it must also report ‘‘success’’ for the other agent. It is not possible to report success for one and failure for the other. In addition, once the double entry system is committed to, any attempt to report different signals for the different agents yields no benefit in the control problem. In other words, linking the two agents conveys all the useful information. There is no additional information available by relaxing the same signal (double entry) restriction. The reported results depend on the specific probability representation used in the paper, in particular, we use the probability representation used by Nature at the sub atomic level—called quantum probabilities. Quantum probabilities differ from classical probabilities in fundamental ways, the most important of which is the concept of interference: when there are two apparently disjoint ways an event may occur (in a sense discussed later in the paper), the probability of the union of the two ways an event can occur differs from the sum of the two probabilities. The amount of the difference is due to the interference of the two ways, and may be either positive or negative (Feynman et al., 1963; Zuric, 1991; Tegmark and Wheeler, 2001). The purpose of the paper is not to argue that quantum probabilities are more appropriate than classical probabilities for the analysis of economic activities. Rather, the purpose is to derive logical consequences of quantum probabilities if they occur. The consequence of interest in this paper is the natural occurrence of a double entry information structure as an efficient information processor. Nonetheless, some speculation is offered within the discussion as to how quantum probabilities might arise in economic settings. One possible example is the concept of synergy. In an uncertain world synergy could be thought of as increasing the probability of the union of two disjoint events beyond the sum of the probabilities of the two events. Similar manifestations of positive interference appear to be commonplace in the realm of quantum physics. The remainder of the paper is organized as follows. In Section 2 quantum probabilities, and the distinction between classical probabilities, are introduced. The quantum concepts of interference and superposition are discussed. The interferometer, a quantum device used as the basis for the control problem, is introduced in Section 3. Section 4 contains the benchmark oneagent control problem utilizing the interferometer. Section 5 develops the quantum mechanics necessary to construct a two-agent control problem. Multiple quantum objects are subject to the phenomenon termed ‘‘entanglement,’’ another counter intuitive, but valuable, implication of quantum probabilities (Aczel, 2001; Zeilinger, 2000). A consequence of entanglement is presented in Section 6 wherein the benchmark control problem is revisited, this time with two agents. The main results are that the two agents are more efficiently controlled by a double entry information structure and, further, that any attempt to break the double entry connection has no information value. Section 7 revisits the double entry interpretation of the information structure.
1
It is important to note that the double entry and single entry information structures are not generally comparable in the sense of Blackwell (1951). The ranking is obtained in a particular economic setting—a standard principal agent problem.
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2. Quantum probabilities For reasons apparently not well understood, quantum probabilities are described by the mathematics of waves. The mathematics of waves is, fortunately, well understood. Imagine two waves moving on the surface of a liquid. When they meet, interference results, in the sense that, if they meet in phase, that is, trough-to-trough and peak-to-peak, then the amplitude (height) of the resulting wave is the sum of the heights of the two original waves. If they meet out of phase, then the amplitude is determined by subtracting the trough of one from the peak of the other. In other words, wave interference is the adding and subtracting of amplitudes. When waves interfere, amplitudes are added. The average energy of the resulting wave is proportional to the square of the combined amplitudes. In other words, to find the average energy of a wave, first add the amplitudes, then square the result. Quantum probabilities work in parallel fashion. The probability of a particular quantum event – position, spin, polarization, etc. – is equal to the square of the amplitude. If an event can occur in two different ways, the probability of the union is the square of the sum of the amplitudes. In other words, first sum the amplitudes, then square to get the probability. Here’s how Feynman et al. (1963, pp. 37–10) summarizes the procedure. (1) The probability of an event in an ideal experiment is given by the absolute value of a complex number f which is called the probability amplitude: P ¼ probability;
f ¼ probability amplitude;
P ¼ jfj2 :
(2) When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference: f ¼ f1 þ f2 ;
P ¼ jf1 þ f2 j2
notice that Feynman says the amplitude may be a complex number. This eventuality arises from the following train of thought. Sine and cosine functions are quite useful for describing the behavior of waves, but can be cumbersome to manipulate. As operations on exponents are much easier, it is convenient to exploit Euler’s Formula:2 eiu ¼ cos u þ i sin u some care must be taken that the probabilities are real numbers, but the absolute value of a complex number squared (or a complex number times its adjoint) is real. For example, jij2 ¼ iðiÞ ¼ i2 ¼ 1. A striking laboratory example of quantum phenomena is the famous double slit experiment. Send a beam of electrons through a barrier that has two slits, or openings through which the electrons can pass. The pattern of electron arrival on the other side of the barrier looks like a wave formed from the combination of two waves emanating from each of the slits. In particular, the electrons going through one slit interfere with the electrons going through the other. Surprisingly, when the electron beam is reduced to one electron at a time, the interference pattern persists. Hence, the individual electron interferes with itself, and, in a very real sense, 2
Such thinking apparently prompted a remark attributed to French mathematician Jacques Hadamard (1865–1963), ‘‘The shortest path between two truths in the real domain passes through the complex domain’’ (Nahin, 1998, p. 70).
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goes through both slits simultaneously. This is a most counter intuitive state of affairs. Nonetheless, it is descriptive and has important implications. One implication is that a stark distinction between classical and quantum probabilities can be drawn. In classical probability theory it is sensible to define the sample space in such a way that, if an event occurs, then no other event will occur simultaneously. As Feller (1950, p. 14) states, ‘‘The sample space provides a model of an ideal experiment in the sense that, by definition, every thinkable outcome of the experiment is described by one, and only one, sample point.’’ (Italics are Feller’s.) In contrast, quantum probabilities allow for different experimental outcomes to occur simultaneously. For example, in order for the electron to interfere with itself, it must be allowed, in principle, to go through both slits simultaneously. Another implication is that the description of the subatomic world relies, in a fundamental way, on probability statements. There is an inherent uncertainty in the behavior of electrons, protons, photons, and so forth, which require probability statements such as the Heisenberg Uncertainty Principle. The inherent uncertainty prohibits the theory from making deterministic statements. For example, the same experiment, run under identical conditions, may not yield identical results; the theory can only predict the distribution of the results. This uncertainty was the reason for Einstein’s often quoted complaint that ‘‘I don’t believe God plays dice with the Universe.’’ A physical manifestation of this inherent uncertainty is the action of a beam splitter, typically thought of as acting on a photon as illustrated in Fig. 1. For the duration of the paper, instead of speaking of particular subatomic units such as electrons and photons as information-carrying units, we will refer to a generic ‘‘qubit’’ or quantum bit. When a qubit (photon in this case) is
Fig. 1. (a) Beam splitter and (b) beam splitter quantum representation.
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directed to a beam splitter, it may be absorbed or reflected with some probability, say one-half each, pictured in Fig. 1 (Bouwmeester and Zeilinger, 2000). What has been found to be a fruitful way to model the situation is to suppose the qubit possesses, simultaneously, two different components; this is called ‘‘superposition.’’ A qubit is a represented by a two-element vector, say , where the elements a and b are referred to as the b amplitudes. The length of the qubit is standardized so that jaj2 þ jbj2 ¼ 1, hence the square of the amplitude can be interpreted as a probability measure. In terms of the beam splitter, the two elements could represent the polarization of the photon, so the squared amplitude represents the probability the photon is polarized so as to be reflected (or absorbed). The beam splitter alters the qubit. In the vector representation altering a qubit is accomplished by a linear transformation: multiplication by a 2 2 matrix, which does not change the length of the qubit (called a unitary transformation). A useful unitary transformation matrix, and one used to model the behavior of the beam splitter, is a Hadamard transformation: 1 1 1 H ¼ pffiffiffi 2 1 1 pffiffiffi pffiffiffi 1 0 1=p2ffiffiffi 1= p2ffiffiffi ¼ and H ¼ . Now the behavior of a beam splitter can be 0 1 1= 2 1= 2 1 ; the beam splitter is H. So modeled in quantum superposition notation. The entering qubit is 0 pffiffiffi 1 1=p2ffiffiffi after the qubit is acted on by the beam splitter: H ¼ , and the squared amplitudes 0 1= 2 imply probabilities of one-half for reflection and absorption. Dirac notation is an elegant way to keep track of quantum probabilities, and will be 1 0 particularly useful for multiple qubit systems (Dirac, 1958). Define j0i and j1i 0 1 and the beam splitter in Fig. 1 can be redrawn as in Fig. 1b.
note H
3. Interferometer In this section a quantum device called an interferometer is described. The interferometer provides the basis for the economic control problem described in the following section. The basic interferometer, pictured in Fig. 2, is a simple extension of the beam splitter.
Fig. 2. Single qubit interferometer without phase shifter.
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Fig. 3. Single qubit interferometer with phase shifter.
A j0i qubit enters from the left. In between two beam splitters (Hadamard matrices) are reflecting barriers, which do not affect the qubit other than to redirect it. The math is straightforward: two H matrices operate in sequence on a j0i qubit. HHj0i ¼ H
j0i þ j1i j0i þ j1i þ j0i j1i pffiffiffi ¼ ¼ j0i 2 2
For the simple interferometer in Fig. 2 a j0i qubit will emerge with probability one, and a j1i qubit will never appear (interference requires that amplitudes be summed, then squared, to identify quantum probabilities). To allow for a non-trivial stochastic outcome from the interferometer, install a phase shifter along the upper path: a phase shift of angle u can be accomplished by operator Q where: Qj0i ¼ eiu j0i
and
Qj1i ¼ j1i:
eiu 0 Or, in matrix form Q ¼ . 0 1 Now the math is slightly more complicated (Fig. 3): eiu j0i þ j1i eiu ðj0i þ j1iÞ þ j0i j1i pffiffiffi ¼ 2 2 notice there are two ways that a j0i qubit can emerge from the interferometer. It can proceed along the upper path denoted by the term eiu =2j0i. The lower path is captured by j0i/2. A classical probability can be calculated by first squaring the amplitudes and then adding: jeiu =2j2 þ j1=2j2 ¼ ðeiu Þðeiu Þ=4 þ 1=4 ¼ 1=2. In other words, classically the probability of j0i is independent of the size of the angle u; i.e., there is no interference. On the other hand, with quantum probabilities the amplitudes are added together prior to squaring. HQHj0i ¼ H
HQHj0i ¼
eiu þ 1 eiu 1 j0i þ j1i 2 2
To calculate the probability of a j0i emerging from the interferometer: iu e þ 1 2 ðeiu þ 1Þðeiu þ 1Þ 1 þ cos u 2 u ¼ ¼ ¼ cos 2 4 2 2
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The third expression uses Euler’s Formula; the fourth uses the half-angle trigonometric identity cosðuÞ ¼ cos2 ðu=2Þ sin2 ðu=2Þ combined with 1 ¼ cos2 ðu=2Þ þ sin2 ðu=2Þ. Now interference occurs and the probability of j0i depends on u; when the angle is zero, all the probability weight is on j0i. As the phase shift angle increases, more weight is transferred to j1i. The above argument is summarized in the following observation. Observation 1.
The phase shift angle determines the probabilities of the qubit emerging from the interferometer. The next step is to interpret the phase shift as an action taken by an economic agent, embedding quantum probabilities in an economic context. The simplest economic context in which information plays a central role is the standard principal-agent problem, wherein the principal wishes to induce the agent to take an appropriate, but unobservable, action by basing compensation on an informative signal. 4. Control problem formulation In subsequent sections we wish to compare the information carrying capacity of double entry versus a single entry system. The two information structures are not comparable in the Blackwell sense; that is, one does not dominate the other for all decision contexts. Therefore, we will make the comparison in a particular economic setting—a standard principal-agent control problem. Consider quantum performance information processing first in a single-agent setting. Let the agent (who takes the unobservable act) be risk averse with the following domain additive CARA preferences defined over wealth (W) and personal cost of effort (c): UðW; cÞ ¼ erðWcÞ information about the agent’s binary action choice is represented by two angles in the interferometer: uH (working) and uL (shirking). The probability of a success signal, then, is ð1 cosui =2Þ for i = H, L. Let IS be the payment for a success, and IF otherwise (failure). Finally, let RW denote the agent’s reservation wage of employment. For a risk neutral principal the program is written as follows with the standard individual rationality and incentive compatibility constraints, under the assumption the principal wishes to induce the high act. (The structure, and parsimony, of the agency problem here are the same as in Christensen and Demski, 2003, Chapter 11).
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Principal’s problem: min IH ;IF
s:t:
1-cos uH 1 þ cos uH IS þ IF 2 2
1 cos uH 1 þ cos uH UðIS ; cH Þ þ UðIF ; cH Þ UðRWÞ 2 2
1 cos uH 1 þ cos uH 1 cos uL UðIS ; cH Þ þ UðIF ; cH Þ UðIS ; cL Þ 2 2 2 1 þ cos uL þ UðIF ; cL Þ 2
Since the constraints must hold as equalities, and for CARA domain additive U(I, c) = (1)(U(I)/ U(c)), the problem can be rewritten as a linear system. 1 1 cos uH 2 1 cos uL
1 þ cos uH 1 þ cos uL
UðIS Þ UðIF Þ
¼
UðRW þ cH Þ UðRW þ cL Þ
The general solution is
UðIS Þ UðIF Þ
1 1 þ cosuL ¼ cosuL cosuH cosuL 1
1 cosuH 1 cosuH
UðRW þ cH Þ UðRW þ cL Þ
A benchmark numerical example will prove useful for subsequent comparisons. Let the risk aversion parameter, r, be .001. Let the two possible actions be uH = 458 and uL = 08. The two acts have associated personal cost to the agent of cH = 100 and cL = 0. The agent’s reservation wage is 200. Solving pffiffiffi pffiffiffi UðIS Þ ¼ ð4 þ 2 2ÞUðRW þ cH Þ þ ð2 2 3ÞUðRW þ cL Þ ¼ :286711 ) IS ¼ 1249:28 and UðIF Þ ¼ UðRW þ cL Þ ) IF ¼ 200 The expected payment is 353.66. The efficiency loss is 53.66, since for an observable action the agent needs to be compensated for the reservation wage (200) plus the cost of the high act (100). 5. Two qubits—two activities In order to expand the control problem to accommodate two activities performed by two separate agents, this section introduces some rudimentary two qubit quantum operations.
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Two qubits are combined into one system according to tensor multiplication of vectors. That is,
7 6 7 6 6 a1 a2 7 6 a 1 b2 7 a1 a2 7 ¼6 4 b1 a2 5 b1 b2 b1 b2
there are a couple of things to do in the way of setup so multiple qubit states can be analyzed. The first is to note there are multiple qubit transformations, the most important of which is the ‘‘controlled not’’ transformation: 6 61 6 60 CNOT ¼ 6 40 0
0 1 0 0
0 0 0 1
7 07 7 07 7 15 0
Dirac 6 notation simplifies tensor multiplication. For example, j0ij1i 6 ¼ j01i, where 7 7 6a7 607 6 7 6 7 607 617 a 1 7 7, Also, and jcij0i ¼ aj00i þ bj10i, since ¼6 j01i ¼ 6 4 b 5. 405 b 0 0 0 CNOTj00i ¼ j00i, as well as CNOTj10i ¼ j11i. The first qubit is called the control qubit; if it is j1i, then the second qubit (the target qubit) flips from a j0i to a j1i, or vice-versa. If the control qubit is a j0i, then the target is unchanged. An important two qubit state is called an ‘‘entangled’’ pair of qubits, so important it has its own conventional notation, denoted jb00i (Nielsen and Chuang, 2003, p. 25). Start with j00i and perform, in order, an H transformation on the first qubit (denote the operation H1), and then a CNOT transformation on the pair. jb00 i ¼ CNOT H1 j00i ¼ CNOT
j00i þ j10i j00i þ j11i pffiffiffi pffiffiffi ¼ 2 2
The resulting two-qubit system is referred to as an entangled state; note that it cannot be created by the tensor combination of any two individual qubits. Furthermore, when the state of one of the qubits is determined, the state of the other is also determined. That is, if the first qubit is measured as j0i, then the other must be j0i, as well. In other words, there is uncertainty in the system, but if the uncertainty is resolved for one qubit, it is resolved pffiffiffifor the other, also. jb00i is a Bell or EPR state and its orthogonal complements are jb i ¼ 1= 2ðj01i þ j10iÞ; jb10 i ¼ 1= 01 pffiffiffi pffiffiffi 2ðj00i j11iÞ, and jb11 i ¼ 1= 2ðj01i j10iÞ: Together, the four are said to form the Bell basis. In theory Bell states can be used to generate some apparently strange, or at least non-intuitive, results. See Bell (1964). A further implication is the particularly non-intuitive process of quantum teleportation (Zeilinger, 2003, and Nielsen and Chuang, 2000, pp. 26–28). For quantum information processing and computation tasks, entanglement is considered an important resource; the next section demonstrates that, not only is entanglement a resource in the control problem, it induces a double entry property in the information structure.
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6. Two-qubit control In this section the control problem of interest is one involving two agents or two activities. As an example, suppose the principal wishes to provide incentives both for a production process and also a marketing process. Two-phase shift angles are required to model the effort levels of the two activities, say Q1 and Q2. A two-qubit interferometer performs a phase shift for each qubit: algebraically, H1Q1H1H2Q2H2 where the subscripts refer to the qubit operated on. 6.1. Tensor qubits j00i as interferometer input The first two-qubit problem analyzed is where the two qubits entering the interferometer are the tensor combination j00i. There are four possible outputs of the interferometer: j00i, j00i, j10i, and j11i. The amplitudes and probabilities for each are presented in Observation 2; the derivation is in Appendix A. Observation 2.
Comparing Observation 2 with Observation 1, it can be seen that there is no gain in the control problem. That is, the best that can be done in the two-activity problem is the same as two applications of the one-qubit problem. From Observation 1 the unconditional probability of j0i is cos2 ðu1 =2Þ. In the two-qubit problem of Observation 2 the conditional probability of a j0i in the first position given a j0i in position two is probj00i=ðprobj00iþ probj10iÞ ¼ cos2 ðu1 =2Þcos2 ðu2 =2Þ=ðcos2 ðu2 =2Þ ¼ cos2 ðu1 =2Þ. Therefore, the second qubit is not conditionally informative about the first (Christensen and Demski, 2003 or Holmstrom, 1979). 6.2. Entangled qubits jb00i as interferometer input The next setting differs from Observation 2 only in the preparation of the two-qubit state entering the interferometer. Instead of tensor combinationp j00i, ffiffiffi the entering state is an entangled one, jb00i. Recall jb00 i ¼ CNOTH1 j00i ¼ ðj00i þ j11i= 2Þ. With this change in the experiment Observation 3 reports the amplitudes and probabilities of possible outcomes. (The derivation is in Appendix A). Note that informativeness in the control setting depends critically on the difference between classical and quantum probabilities; that is, adding amplitudes and then squaring gives a distinctly different result than simply adding probabilities.
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Observation 3.
The notable result is that, when classified in the Bell basis, only two of the four possible pffiffiffi Bell states appear as outcomes. The probability of the outcome jb01 i ¼ ðj01i þ j10i= 2Þ increases in both u1 and u2, the effort angles for the two agents (retaining the convention that u1 + u2 p), so it is labeled as a ‘‘success’’ signal. A success for one agent must necessarily be accompanied by a success for the other; there is no possibility of a success for one and a failure for the other. This linkage of the signals is interpreted as double entry. Both agents must receive the same number, whatever it is. Hence, the number is entered twice. The primary question is whether the arrangement in Observation 3 (double entry) yields an efficiency gain relative to Observation 2. The two information structures are not comparable for all decision problems—that is, one is not a subpartition of the other. However, for the control problem under consideration, the double entry system strictly dominates the other. First reconsider the numerical example. Assume the ‘‘other’’ agent chooses u2 = 458, and solve the problem for the u1-agent. Collusion issues will be addressed later. Let all the other parameters be the same as in the previous example. The solution is as follows: UðIS Þ ¼ ð1 þ
pffiffiffi pffiffiffi 2ÞUð300Þ 2Uð200Þ
UðIS Þ ¼ :630633260 ) IS ¼ 461:0307898 pffiffiffi pffiffiffi UðIF Þ ¼ ð1 2ÞUð300Þ þ 2Uð200Þ UðIF Þ ¼ :851003181 ) IF ¼ 161:3394125 the expected payment is 311.185 implying an efficiency loss of 11.185, and a substantial efficiency gain relative to the Observation 2 result (efficiency loss equal to 53.66). The probability of a success in the entangled problem is :5 ¼ sin2 ðu1H þ u2H =2Þ. For the tensor problem, the corresponding probability is :146 ¼ sin2 ðu1H =2Þ. The increase in probability of a success, however, is not accompanied by increased payments or efficiency loss. In the entangled qubit problem, the efficiency loss declines from 53.66 to 11.185. Furthermore, that decline is accomplished as both payments decline. Payment for a success, Is, goes from 1,249.28 to 461.03. Likewise, the low payment declines from 200 to 161.34. It is not merely that the payments become closer together, thereby reducing the risk premium, they also both decline. The two-qubit control example is summarized in Table 1.
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Table 1 Two-agent quantum control example Parameter values r = 0.001 RW = 200
cH = 100 cL = 0
u1H = 458 u1L = 08
u2 = 458
Signal j11i (S) Independent performance evaluation (j00i) sin2(u1H/2)0.1464 Probability(signal ju1H) Probability(signal ju1L) sin2(u1L/2) = 0.0 Payments (I) 1249.28
E[I] j01i (F) cos2(u1H/2)/2 0.8536 cos2(u1L/2) = 1.0 200
353.66
Signal jb01i (S) Coordinated performance evaluation (jb00i) sin2(u1H + u2)/2 = 0.5 Probability(signalju1H) Probability(signalju1L) sin2(u1L + u2)/20.1464 Payments (I) 461.03
E[I] jb00i (F) cos2(u1H + u2)/2 = 0.5 cos2(u1L + u2)/20.8536 161.34
311.19
That these results generalize beyond the numerical example is stated in the Proposition.3 The Proposition establishes that, in the presence of entanglement, the efficiency of the contract with agent one increases with the effort level of agent two. Therefore, for any non-zero agent two effort, the entangled (double entry based) contract dominates a tensor (single entry) contract. (For agent two effort equal to zero, the two contracts are equivalent.) Proposition (The proof is in Appendix B). In the entangled qubit control problem, for 0 u1L < u1H p=2 and marginal increase in u2 (i) increases the probability of a success signal; (ii) reduces both payments; (iii) decreases the efficiency loss.
0 u2 p=2, a
Finally, it is also worth noting there is no collusion problem in the entangled control problem even though the constraints are written so that high effort is a Nash best response to high effort from the other agent.4 In a standard correlated state two-agent control problem, writing the constraints as best response may cause the existence of another equilibrium that is preferred by the agents and dispreferred by the principal. Roughly speaking, this occurs 3 To focus on accounting implications the quantum control experiment is subject to the following restrictions: (1) initial states are limited to j00i or jb00i; (2) information processing employs the standard two-qubit interferometer H1Q1H1H2Q2H2; and (3) probabilities are based on projections into computational eigenstates (summarizing as compactly as possible without compromising likelihood ratios effectively allows measurement in the Bell eigenstates for initial state jb00i). 4 Collusion does not arise from initial Bell states with positive interference (amplitudes of the same sign). However, jb10i and jb11i have negative interference and produce severe collusion concerns. We only discuss the initial state jb00i but jb01i consistently utilized produces the same incentive-efficiency (with some relabeling). Again, this is a commitment to double entry accounting—a convention is adopted and once adopted it is critical that it be consistently applied.
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because the Nash incentive contract rewards the agents for similar signals and punishes dissimilar. This, in turn, induces the agents to collude, providing similar signals with high probability by shirking. One reason the dominated equilibrium is not a problem in the entangled control problem is that there is no possibility of dissimilar signals. That is, there is only one signal that applies to both agents. Additionally, a way to see that there is only one equilibrium which satisfies both agents’ individual rationality constraints is to present the agent payoffs in normal form.
Agent one chooses the row. Note that shirking by agent one does not increase agent one’s expected utility—the first number in each cell. Agent one’s shirking does, however, decrease agent two’s expected utility. Furthermore, the shirk/shirk strategy pair is clearly dominated by work/work. (When both agents shirk, IF = 161.3 is guaranteed to both agents, as the probability of a success is zero.). The question arises as to whether there is any benefit to disentangling the signal in order to obtain a signal particular to each individual. The answer is, in the presence of entangled input, there is no benefit to individual signals. The first step is to disentangle the output so each individual is recognizable, that is, the output is in tensor form—j00i, j01i, and so forth. The amplitudes and probabilities for the first agent (qubit) of disentangled signals are presented in Observation 4. (The derivation is in Appendix A). Observation 4.
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Consider the benefit of individual measurements. To calculate the probability of obtaining a j0i qubit in position one, for example, note that there are two different ways for this to occur and add the amplitudes together. eiðu1 þu2 Þ þ 1 þ eiðu1 þu2 Þ 1 eiðu1 þu2 Þ pffiffiffi ¼ pffiffiffi 2 2 2 pffiffiffi The probability is ðjeiðu1 þu2 Þ = 2j2 Þ ¼ ð1=2Þ. That is, the probability of the first output qubit being j0i is one-half regardless of the effort angle chosen by agent 1. Hence, measuring the first qubit only is useless for controlling the first activity. A similar argument applies to the second agent (qubit). 7. Some remarks about accounting and double entry Broadly speaking two main lessons of the previous section are: 1 Entanglement yields an efficiency gain in the control problem, as in the proposition, and 2 The information system associated with entanglement is restricted in the sense that it uses only two of the possible four signals. That is, jb00i and jb01i are used, but the other two states in a Bell basis, jb10i and jb11i are not. In this section we briefly speculate on parallels between the double entry accounting system and the information system, which shows up in the entangled qubit control problem. Double entry is also restricted in that it only uses two of four possible signals. The restriction is a consequence of the balancing property: assets must equal equities. Of four possible signals, accounting can only produce two, as follows.
Not allowed Not allowed
Assets
Equities
Positive Negative Positive Negative
Positive Negative Negative Positive
A variation on the restricted signal theme is that asset recognition consists of two requirements: control of the item and a future economic benefit. As an example, inventory may be recognized only if the item exists – has been produced – and a market exists for its sale. In terms of the control problem, we can think of two agents supplying inputs: a production manager and a marketing manager. Because of the restricted information system, a success for one cannot occur in the absence of a success for the other. A restricted information system yields efficiency gains only in the presence of an entangled entry qubit. How the agents’ performance evaluation might be entangled is a difficult problem. The concepts of superposition and interference are deep and non-intuitive in the domain of physics. How they might be operationalized in an economic context is an open question.
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We will, however, offer some brief speculations. In quantum physics entanglement occurs with the following sequence of operations: j00i þ j11i pffiffiffi 2 CNOT ties the two qubits together. While in quantum physics the intuition for how this occurs is not strong, it may be easier to see if the qubits act as rational economic agents. The agents can coordinate. For example, the marketing manager can agree to market the design chosen by the production manager regardless of the state of the market, or vice-versa. It is not implausible that coordination of this type occurs on a regular basis in economic organizations. The speculative conclusion is that double entry accounting is appropriate in organizations where coordination of activities is a primary concern. The beam splitter embodied by the H transformation is critical to the generation of interference necessary for quantum probabilities. Here the economic analog may be synergy (positive or negative) among activities. A common example in the academic domain is the synergy between teaching and research activities. Some might argue (or act as if) there is negative interference between the two activities—more time or energy devoted to one necessarily diminishes the amount available for the other. On the other hand, cranking up the phase shift angle devoted to one of the activities may, indeed, increase the efficiency of the other, possibly in terms of creativity, insight, or general organization. jb00 i ¼ CNOT H1 j00i ¼
7.1. Concluding remarks The paper has argued, and attempted to demonstrate, that classical probabilities and quantum probabilities are fundamentally different due to quantum interference. Interference has the potential to enhance the efficiency of organizations. For instance, universities face potential interference (positive or negative) between research and teaching, and their survivorship is enhanced if this interference is, on average, positive. Similarly, efficient coordination of activities in industrial organizations and hence their survivorship are enhanced by exploiting positive interference. Employing one performance measurement for the evaluation of two agents or two activities (a double entry characteristic) is more efficient than independent evaluation of the two agents in a setting with quantum interference. This is accomplished via quantum entanglement of the agents’ performance evaluation. Not only is independent evaluation of the agents inefficient but also it may lead to serious compromise of the cause and effect relationship of double entry and increase the likelihood of transgressions such as experienced by Enron. This work on accounting implications of quantum probabilities is in its infancy and much remains to be explored. One of the big challenges in quantum information processing is maintaining coherent quantum states. Two related issues warrant further exploration. First, there exists a demand for error correction in the processing of quantum information. Aside from the (complex) diagonal matrix Q to represent the agent’s impact on performance, our control experiment restricts quantum operations to Hadamard transformations.5 This and similar 0 1 1 0 and Z ¼ , known as Pauli-X and Pauli-Z. It is noteworthy that 1 0 0 1 0 i if one admits complex, non-diagonal operators such as Pauli- Y ¼ then initial state j00i is sometimes more i 0 efficient than initial state jb00i in the control setting. For example, initial state j00i is more efficient pffiffiffi than is initial state jb00i when processed via U1Q1H1U2Q2H2 for u1H = p/4, u1L = 0, and u2 = p/4 where U ¼ 1= 2ðZ þ YÞ. 5
pffiffiffi Note H ¼ 1= 2ðZ þ XÞ where X ¼
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transformations are the operators employed by most of the known quantum error correction algorithms (Nielsen and Chuang, 2000, Chapter 10). Second, quantum interference is destroyed by interaction with the (unknown and unobservable) environment. This is referred to as decoherence (Zurek, 1991). Of course, decoherence makes the potential efficiency of quantum probabilities inaccessible. More work on quantum coherency and its relation to double entry accounting and internal control system design seems desirable. Acknowledgment Helpful comments from Anil Arya, John Christensen, Peter Christensen, Joel Demski, Hans Frimor, Jonathan Glover, Thorbjørn Knudsen, Scott Liao, Florin Sabac, Eric Spires, two anonymous referees, and workshop participants at the University of Southern Denmark are gratefully acknowledged. Appendix A. Derivation of observations A.1. Derivation of Observation 2 ðeiu2 þ 1Þj00i þ ðeiu2 1Þjj01i 2 iu1 iu1 iu2 þ 1Þðe j00i þ e j10iÞ þ ðe 1Þðeiu1 j01i þ eiu1 j11iÞ pffiffiffi 2 2
H1 Q1 H1 H2 Q2 H2 j00i ¼ H1 Q1 H1 ¼ H1
ðeiu2
ðeiu2 þ 1Þðeiu1 þ 1Þj00i þ ðeiu2 1Þðeiu1 þ 1Þj01i þ ðeiu2 þ 1Þðeiu1 1Þj10i þ ðeiu2 1Þðeiu1 1Þj11i ¼ 4 The probabilities are calculated by recalling jeiu1 þ 1=2j2 ¼ cos2 ðu1 =2Þ and jeiu1 1=2j2 ¼ sin2 ðu1 =2Þ.
A.2. Derivation of Observation 3 j00iþj01iþj10ij11i eiu2 j00iþj01iþeiu2 j10ij11i ¼H2 2 2 eiu2 ðj00iþj01iÞþj00ij01iþeiu2 ðj10ij11iÞj10iþj11i ðeiu2 þ1Þ ðeiu2 1Þ pffiffiffi jb00 iþ jb01 i ¼ ¼ 2 2 2 2
H2 Q2 H2 jb00 i¼H2 Q2
Likewise, ðeiu1 þ 1Þ ðeiu1 1Þ jb00 i þ jb01 i and 2 2 ðeiu1 1Þ ðeiu1 þ 1Þ jb00 i þ jb01 i: H1 Q1 H1 jb01 i ¼ 2 2
H1 Q1 H1 jb00 i ¼
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Hence,
ðeiu1 þ 1Þðeiu2 þ 1Þ þ ðeiu1 1Þðeiu2 1Þ H 1 Q1 H 1 H 2 Q2 H 2 jb00 i ¼ jb00 i 4 iu2 ðe þ 1Þðeiu1 1Þ þ ðeiu2 1Þðeiu1 þ 1Þ þ jb01 i 4
¼ jb00 iðeiðu1 þu2 Þ þ 1=2Þ þ jb01 iðeiðu1 þu2 Þ 1=2Þ A.3. Derivation of Observation 4 Start with the expression in Observation 3. eiðu1 þu2 Þ þ 1 eiðu1 þu2 Þ 1 H1 Q1 H1 H2 Q2 H2 jb00 i ¼ jb00 i þ jb01 i 2 2 eiðu1 þu2 Þ þ 1 j00i þ j11i eiðu1 þu2 Þ 1 j01i þ j10i eiðu1 þu2 Þ þ 1 pffiffiffi pffiffiffi pffiffiffi ¼ þ ¼ j00i 2 2 2 2 2 2 þ
eiðu1 þu2 Þ 1 eiðu1 þu2 Þ 1 eiðu1 þu2 Þ 1 pffiffiffi pffiffiffi pffiffiffi j01i þ j10i þ j11i 2 2 2 2 2 2
Appendix B. Proof of proposition First, for reference purposes, we state some relationships derived from the statement of the control problem. 1 cosðu1H þ u2 Þ 1 þ cosðu1H þ u2 Þ EðpaymentÞ ¼ E½I ¼ IS þ IF 2 2 1 1 IS ¼ lnðUðIS ÞÞ IF ¼ lnðUðIF ÞÞ r r UðIS Þ ¼
ð1 þ cosðu1L þ u2 ÞÞUðRW þ cH Þ ð1 þ cosðu1H þ u2 ÞÞUðRW þ cL Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
UðIF Þ ¼
ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
dIS 1 dUðIS Þ=du2 ¼ r du2 UðIS Þ
dIL 1 dUðIL Þ=du2 ¼ r du2 UðIL Þ
dUðIS Þ ½sinðu1H u1L Þ þ sinðu1H þ u2 Þ sinðu1L þ u2 Þ ¼ ½UðRW þ cL Þ UðRW þ cH Þ du2 ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 dUðIF Þ ½sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ ¼ ½UðRW þ cL Þ du2 ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 UðRW þ cH Þ
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It is feasible to motivate uH if probabilityðfailurejuL Þ=probabilityðfailurejuH Þ > erðcH cL Þ (see Christensen and Demski, 2003, pp. 250–251). Since 1 þ cosðu1L þ u2 Þ=1 þ cosðu1H þ u2 Þ 1 þ cosðu1L Þ=1 þ cosðu1H Þ for 0 uj p/2 and u1L < u1H (equal iff u2 = 0), there exist values of u2 for an entangled state when it is feasible to motivate uH while it is infeasible to motivate uH when the state is not entangled. Hence, the results proven below are monotonic in u2. (i) Probability of success increases with u2.Probability of success is 1 cosðu1H þ u2 Þ=2. The first derivative with respect to u2 is sinðu1H þ u2 Þ=2 > 0. (ii) IS and IF both decrease in u2. As U(I) is monotonic in I, need to check dUðIS Þ=du2 and dUðIF Þ=du2 . dUðIS Þ=du2 is clearly negative. The sign of dUðIF Þ=du2 depends on sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ, which is positive by the lemma in Appendix C. Hence, dUðIF Þ=du2 < 0. (iii) The change in the expected payment is negative as u2 increases. dE½I 1 cosðu1H þ u2 Þ dIS 1 þ cosðu1H þ u2 Þ dIF sinðu1H þ u2 Þ ðIS IF Þ ¼ þ þ du2 2 2 2 du2 du2 The first two terms are negative by (ii) above; the third term is positive. Rewrite the third term: sinðu1H þ u2 Þ=2r lnðUðIF Þ=UðIS ÞÞ where UðIF Þ=UðIS Þ > 1. Now expand the first two terms, which is a little bit more work. ð1 cosðu1H þ u2 ÞÞ dIS ð1 þ cosðu1H þ u2 ÞÞ dIL þ 2 2 du2 du2 ð1 cosðu1H þ u2 ÞÞ=2ð1=rÞðUðRW þ cL Þ UðRW þ cH ÞÞ½sinðu1H u1L Þ þ sinðu1H þ u2 Þ sinðu1L þ u2 Þ=½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 ¼ ½ð1 þ cosðu1L þ u2 ÞÞUðRW þ cH Þ ð1 þ cosðu1H þ u2 ÞÞUðRW þ cL Þ= ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ ð1 þ cosðu1H þ u2 ÞÞ=2ð1=rÞðUðRW þ cL Þ UðRW þ cH ÞÞ½sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ=½cosðu1L þ u2 Þ cosðu1H þ u2 Þ2 þ ½ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ= ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ Numerator: ¼
1 ½UðRW þ cL Þ UðRW þ cH Þ 2r fð1 cosðu1H þ u2 ÞÞ½sinðu1H u1L Þ þ sinðu1H þ u2 Þ sinðu1L þ u2 Þ ½ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ þ ð1 þ cosðu1H þ u2 ÞÞ½sinðu1H u1L Þ þ sinðu1L þ u2 Þ sinðu1H þ u2 Þ ½ð1 þ cosðu1L þ u2 ÞÞUðRW þ cH Þ ð1 þ cosðu1H þ u2 ÞÞUðRW þ cL Þg
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Denominator: ðcosðu1L þ u2 Þ cosðu1H þ u2 ÞÞ ½ð1 þ cosðu1L þ u2 ÞUðRW þ cH Þ ð1 þ cosðu1L þ u2 ÞÞUðRW þ cL ÞÞ ½ðcosðu1L þ u2 Þ 1ÞUðRW þ cH Þ þ ð1 cosðu1H þ u2 ÞÞUðRW þ cL Þ Expand and combine terms. Numerator:
1 ½UðRW þ cL Þ UðRW þ cH Þ 2r f2½UðRW þ cL Þ UðRW þ cH Þ ½sinðu1H þ u2 Þ cosðu1H þ u2 Þsinðu1H u1L Þ sinðu1L þ u2 Þ þ ½2UðRW þ cH Þcosðu1L þ u2 Þ 2UðRW þ cL Þcosðu1H þ u2 Þ ½cosðu1H þ u2 Þsinðu1H þ u2 Þ þ sinðu1H u1L Þ þ cosðu1H þ u2 Þsinðu1L þ u2 Þg
Denominator: UðIF ÞUðIS Þ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ3 Substitute UðIS Þ UðIF Þ ¼ 2
UðRW þ cH Þ UðRW þ cL Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
UðIS Þ þ UðIF Þ ¼ 2
UðRW þ cH Þcosðu1L þ u2 Þ UðRW þ cL Þcosðu1H þ u2 Þ cosðu1L þ u2 Þ cosðu1H þ u2 Þ
Numerator:
1 UðIF Þ UðIS Þ ½n1 ðUðIF Þ UðIS ÞÞ þ n2 ðUðIF Þ þ UðIS ÞÞ 2r 2
where n1 ¼ sinðu1H þ u2 Þ þ cosðu1H þ u2 Þsinðu1H u1L Þ þ sinðu1L þ u2 Þ n2 ¼ cosðu1H þ u2 Þsinðu1H þ u2 Þ þ sinðu1H u1L Þ þ cosðu1H þ u2 Þsinðu1L þ u2 Þ Denominator: UðIF ÞUðIS Þ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ
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Recombining to get all the terms in dE½I=du2 , we can write 4r
dE½I sinðu1H þ u2 Þ du2
UðIF Þ UðIS Þ UðIF Þ n1 n2 ½UðIS Þ þ UðIF Þ ¼ 2ln ½UðIS Þ UðIF Þ UðIS Þ UðIS ÞUðIF Þ den den where den ¼ sinðu1H þ u2 Þ½cosðu1L þ u2 Þ cosðu1H þ u2 Þ Since n1 =den ¼ sinðu1H u1L =2Þ=sinðu1H þ u1L =2 þ u2 Þ > 0 and n2 =den ¼ 1, we can write 4r dE½I UðIF Þ UðIS Þ UðIF Þ < 2ln þ ½UðIS Þ þ UðIF Þ sinðu1H þ u2 Þ du2 UðIS Þ UðIS ÞUðIF Þ UðIF Þ UðIS Þ2 UðIF Þ2 UðIF Þ UðIF Þ UðIS Þ ¼ 2ln þ <0 þ ¼ 2ln UðIS Þ UðIS Þ UðIS Þ UðIF Þ UðIF ÞUðIS Þ Recall UðIF Þ=UðIS Þ > 1; the expression above is equal to zero at UðIF Þ=UðIS Þ ¼ 1, and the derivative with respect to UðIF Þ=UðIS Þ is everywhere negative. Hence, the inequality follows, and dE½I <0 du2 Appendix C. Lemma
Lemma. Let u1H þ u2 equal ffAOE and u1L þ u2 equal ffBOE in the above diagram. For 0 u1L < u1H
p 2
and
0 u2
p ; 2
sinðu1H u1L Þ sinðu1H þ u2 Þ þ sinðu1L þ u2 Þ 0
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Proof. ðu1H u1L Þ ¼ ðu1H þ u2 Þ ðu1H þ u2 Þ or ffAOB then sinðu1H u1L Þ ¼ AD; sinðu1H þ u2 Þ ¼ AC;
sinðu1L þ u2 Þ ¼ BF
Since AD is perpendicular to OB, it is clear AC < AD + BF. One or more of the angles may be in the second quadrant. If sinðu1L þ u2 Þ sinðu1H þ u2 Þ, then the lemma is immediate. If not, a similar diagram can be drawn and the logic is similar. References Aczel, A., 2001. Entanglement: The Greatest Mystery in Physics. Four Walls Eight Windows, New York, NY. Arya, A., Fellingham, J., Schroeder, D., 2000a. Accounting information, aggregation, and discriminant analysis. Management Science 46 (6). Arya, A., Fellingham, J., Glover, J., Schroeder, D.A., Strang, G., 2000b. Inferring transactions from financial statements. Contemporary Accounting Research 17 (3). Bell, J., 1964. On the Einstein-Podolsky-Rosen paradox. Physics 1 . Blackwell, D., 1951. Comparison of experiments. In: Proceedings of Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA,. University of California Press. Bouwmeester, D., Zeilinger, A., 2000. The physics of quantum information: basic concepts. Dirk Bouwmeester. In: Artur, Ekert, Anton, Zeilinger (Eds.), The Physics of Quantum Information. Springer, Berlin. Christensen, J., Demski, J., 2003. Accounting Theory. McGraw Hill-Irwin, New York, NY. Dirac, P., 1958. The Principles of Quantum Mechanics. Oxford University Press, Oxford, UK. Feller, W., 1950. An Introduction to Probability Theory and Its Applictions. John Wiley & Sons, New York. Feynman, R., Leighton, R., Sands, M., 1963. The Feynman Lectures on Physics, vol. 1. Addison-Wesley Publishing Co., Reading MA. Holmstrom, B., 1979. Moral hazard and observability. Bell Journal of Economics. Ijiri, Y., 1971. Fundamental queries in aggregation theory. Journal of the American Statistical Association 66 . Ijiri, Y., 1975. Theory of Accounting Measurement. Studies in Accounting Research No. 10. American Accounting Association, Sarasota, FL. Mattessich, R., 1964. Accounting and Analytical Methods. Richard D. Irwin, Homewood, IL. pffiffiffiffiffiffiffi Nahin, P., 1998. An Imaginary Tale: The Story of 1. Princeton, NJ, Princeton University Press. Nielsen, M., Chuang, I., 2000. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK. Tegmark, M., Wheeler, J., 2001. As quantum theory celebrates its 100th birthday, spectacular successes are mixed with persistant puzzles. Scientific American. Zeilinger, A., 2000. Quantum teleportation. Scientific American 282 4. Zuric, W., 1991. Decoherence and the transition from quantum to classical. Physics Today.
Further reading Nielsen, M., 2003. Simple rules for a complex quantum world. Scientific America 288, 5.