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On quantum-mechanical automata
Volume 101A, number 5,6
PHYSICS LETTERS
2 April 1984
ON QUANTUM-MECHANICAL AUTOMATA Asher PERES 1 Center for the Philosophy and History of Science, Boston University, Boston, MA 02215, USA Received 14 December 1983
An automaton which can "measure" or "know" or "predict" the values of physical quantities cannot be described by quantum mechanics.
In a letter with the same title as above [1] Albert considers an automated device capable of performing measurements of physical quantities. This automaton can observe a microscopic system S and, moreover, it can perform measurements upon itself. For example, it may consist of several interconnected instruments, such that one instrument observes S, while another observes the composite system consisting of S and of the first instrument. Up to this point, there is nothing unusual. Albert then describes this automaton in accordance with the laws of quantum mechanics. Its state is represented by a vector in a Hilbert space. This leads to curious paradoxes. The description which a quantum automaton can produce of itself appears to be different, not just in content but in nature, from its description of an external object. As the physical situation is in principle well defined, this remarkable result deserves careful scrutiny. I have no quarrel with the calculations of ref. [1] which are impeccable. On the other hand, I intend to show that the automaton described by these calculations is not a measuring instrument (although it can be part of one). What is called, abusively, a "measurement" in ref. [1] is the yon Neumann dictum [2] q ~ = - - ~ C i ~ i -~ ~
Ci~i~li ,
(1)
where ~bis the initial state of an apparatus, ~k that o f I Permanent address: Department of Physics, Technion Israel Institute of Technology, 32000 Haifa, Israel. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
an observed system, and ¢i and tbi are two sets of orthonormal states of that apparatus and that system, respectively. The evolution (1) is unitary, and therefore reversible [3,4]. It is not a measurement, but may be called a "premeasurement" [5]. For example, it is well known that collisions, which create correlations like (1), are not equivalent to observations [6]. What is a measurement? It is an irreversible event [7] involving the interaction of a macroscopic apparatus with a microscopic object. To describe a measurement, we construct a theoretical model where the behavior of the macroscopic equipment is idealized and reduced to that of a few degrees of freedom, which interact with those of the microscopic system under observation. By means of this model, the behavior of the apparatus is related to the assumed properties of the microscopic system, via an equation like (1). The latter, however, can remain true only for a very short time, because the few macroscopic degrees of freedom which we had in mind cannot be perfectly isolated from the many other degrees of freedom of the macroscopic apparatus [5]. The ~i in eq. (1) will continue to evolve and, after a short time, the reversed evolution will become "impossible" to achieve. (Of course, "impossible" only means "extremely difficult" [8,9], e.g., think of the difficulty of resuscitating Schr6dinger's cat [10] !) Phenomenologically, this irreversible evolution can be expressed as a randomization of the phases of the ci, so that the superposition (1) is replaced by a mixture [5]. This irreversible behavior was completely ignored 249
Volume 101A, number 5,6
PHYSICS LETTERS
by Albert [ 1], although it is essential for the proper functioning of his computerize d automaton. In particular, the process of computing is inherently irreversible [11-13], as no computer can function in thermodynamic equilibrium [14]. There are, however, some distinguished theorists who claim that irreversibility, despite its great practical importance, is not fundamental, that things usually claimed impossible are really only very difficult, that new computing devices may be invented whereby a computer would function reversibly, in brief, that human ignorance or weakness have no role in physics.. Although I may disagree on this point [15] I shall tentatively adopt this approach, just for the sake of the argument, to see how far Albert's analysis can carry us. Let us assume that eq. (1) indeed represents a measurement. Then, we have the familiar hiatus of quantum measurement theory. The external observer, who has prepared the experiment, can only infer from eq. (1) that various outcomes are possible, with probabilities Ici[2. For him, quantum theory is only a mathematical formalism allowing to compute probabilities for the occurrence of events of a specified kind, following a specified preparation [16]. But in his consciousness (or in the records supplied by the automaton) there is only a single world. Each event is unique [17]. Therefore the automaton, or any measuring instrument, must have a special, metaphysical status. It "knows" which is the true outcome (e.g., it may know of additional "hidden" variables, not included in the wavefunction). Yet, as pointed out by Albert [1], it may not communicate this knowledge to the outside world without causing a further change in the wavefunction (1). The novelty in Albert's paper is that it shows how the automaton can make use of this hidden knowledge to create additional hidden knowledge which blatantly violates the limitations imposed on an external observer. The lesson of all this is that one should not attempt to include the measuring apparatus in the wavefunction, because if we do so, we get a unitary evolution which is not the description of a measurement. We
250
2 April 1984
may, of course, describe any apparatus (designed to function as an observer) by quantum theory and have it in a superposition of states such as in eq. (1). However, this precludes the apparatus from being the observer. It is just another quantum system, which then has to be observed by something or someone else [17] There is here a kind of classical-quantum duality [18] whereby a macroscopic apparatus can be treated either classically or by quantum theory. In the classical treatment, the position, momentum, and other properties of the apparatus are c-numbers and have at each instant precise numerical values. Their evolution, however, is subject to probabilistic laws: It is unpredictable for individual cases; only averages can be predicted. On the other hand, a quantum mechanical treatment of the apparatus leaves it in a superposition of macroscopically different states, precluding its use as a measuring apparatus. This classicalquantum duality is not a flaw of the theory. As shown elsewhere [17] it is a logical necessity which is analogous to G6del's undecidability theorem. Any attempt to have a closed theory must lead to inconsistencies, as found in ref. [1].
References [1] D.Z. Albert, Phys. Lett. 98A (1983) 249. [2] J. yon Neumann, Mathematical foundations of quantum mechanics (Princeton Univ. Press, Princeton, 1955). [3] P.A. Moldauer, Phys. Rev. D5 (1972) 1028. [4] A. Peres, Am. J. Phys. 42 (1974) 886. [5] A. Peres, Plays. Rev. D22 (1980) 879. [6] J. Wheather and R. Peierls, Phys. Rev. Lett. 51 (1983) 1601.
[7] A. Daneri, A. Loinq~,erand G.M. Prosperi, Nucl. Phys. 33 (1962) 297; Nuovo Cimento B44 (1966) 119.
[8] J.S. Bell, Helv. Phys. Acta 48 (1975) 93. [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
P. Jasselette, Int. J. Quantum Chem. 17 (1980) 83. E. Schr6dinger, Naturwiss. 23 (1935) 807,824,844. M.S. Gupta, Int. J. Theor. Phys. 21 (1982) 275. R. Landauer, Int. J. Theor. Phys. 21 (1982) 283. K.K. Likhaxev, Int. J. Theor. Phys. 21 (1982) 311. I. Prigogine, Science 201 (1978) 777. A. Peres, Found. Phys. 10 (1980) 631. H.P. Stapp, Am. J. Phys. 40 (1972) 1098. A. Peres and W.H. Zurek, Am. J. Phys. 50 (1982) 807. A. Peres, Am. J. Phys. 52 (1984), to be published.
PHYSICS LETTERS
2 April 1984
ON QUANTUM-MECHANICAL AUTOMATA Asher PERES 1 Center for the Philosophy and History of Science, Boston University, Boston, MA 02215, USA Received 14 December 1983
An automaton which can "measure" or "know" or "predict" the values of physical quantities cannot be described by quantum mechanics.
In a letter with the same title as above [1] Albert considers an automated device capable of performing measurements of physical quantities. This automaton can observe a microscopic system S and, moreover, it can perform measurements upon itself. For example, it may consist of several interconnected instruments, such that one instrument observes S, while another observes the composite system consisting of S and of the first instrument. Up to this point, there is nothing unusual. Albert then describes this automaton in accordance with the laws of quantum mechanics. Its state is represented by a vector in a Hilbert space. This leads to curious paradoxes. The description which a quantum automaton can produce of itself appears to be different, not just in content but in nature, from its description of an external object. As the physical situation is in principle well defined, this remarkable result deserves careful scrutiny. I have no quarrel with the calculations of ref. [1] which are impeccable. On the other hand, I intend to show that the automaton described by these calculations is not a measuring instrument (although it can be part of one). What is called, abusively, a "measurement" in ref. [1] is the yon Neumann dictum [2] q ~ = - - ~ C i ~ i -~ ~
Ci~i~li ,
(1)
where ~bis the initial state of an apparatus, ~k that o f I Permanent address: Department of Physics, Technion Israel Institute of Technology, 32000 Haifa, Israel. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
an observed system, and ¢i and tbi are two sets of orthonormal states of that apparatus and that system, respectively. The evolution (1) is unitary, and therefore reversible [3,4]. It is not a measurement, but may be called a "premeasurement" [5]. For example, it is well known that collisions, which create correlations like (1), are not equivalent to observations [6]. What is a measurement? It is an irreversible event [7] involving the interaction of a macroscopic apparatus with a microscopic object. To describe a measurement, we construct a theoretical model where the behavior of the macroscopic equipment is idealized and reduced to that of a few degrees of freedom, which interact with those of the microscopic system under observation. By means of this model, the behavior of the apparatus is related to the assumed properties of the microscopic system, via an equation like (1). The latter, however, can remain true only for a very short time, because the few macroscopic degrees of freedom which we had in mind cannot be perfectly isolated from the many other degrees of freedom of the macroscopic apparatus [5]. The ~i in eq. (1) will continue to evolve and, after a short time, the reversed evolution will become "impossible" to achieve. (Of course, "impossible" only means "extremely difficult" [8,9], e.g., think of the difficulty of resuscitating Schr6dinger's cat [10] !) Phenomenologically, this irreversible evolution can be expressed as a randomization of the phases of the ci, so that the superposition (1) is replaced by a mixture [5]. This irreversible behavior was completely ignored 249
Volume 101A, number 5,6
PHYSICS LETTERS
by Albert [ 1], although it is essential for the proper functioning of his computerize d automaton. In particular, the process of computing is inherently irreversible [11-13], as no computer can function in thermodynamic equilibrium [14]. There are, however, some distinguished theorists who claim that irreversibility, despite its great practical importance, is not fundamental, that things usually claimed impossible are really only very difficult, that new computing devices may be invented whereby a computer would function reversibly, in brief, that human ignorance or weakness have no role in physics.. Although I may disagree on this point [15] I shall tentatively adopt this approach, just for the sake of the argument, to see how far Albert's analysis can carry us. Let us assume that eq. (1) indeed represents a measurement. Then, we have the familiar hiatus of quantum measurement theory. The external observer, who has prepared the experiment, can only infer from eq. (1) that various outcomes are possible, with probabilities Ici[2. For him, quantum theory is only a mathematical formalism allowing to compute probabilities for the occurrence of events of a specified kind, following a specified preparation [16]. But in his consciousness (or in the records supplied by the automaton) there is only a single world. Each event is unique [17]. Therefore the automaton, or any measuring instrument, must have a special, metaphysical status. It "knows" which is the true outcome (e.g., it may know of additional "hidden" variables, not included in the wavefunction). Yet, as pointed out by Albert [1], it may not communicate this knowledge to the outside world without causing a further change in the wavefunction (1). The novelty in Albert's paper is that it shows how the automaton can make use of this hidden knowledge to create additional hidden knowledge which blatantly violates the limitations imposed on an external observer. The lesson of all this is that one should not attempt to include the measuring apparatus in the wavefunction, because if we do so, we get a unitary evolution which is not the description of a measurement. We
250
2 April 1984
may, of course, describe any apparatus (designed to function as an observer) by quantum theory and have it in a superposition of states such as in eq. (1). However, this precludes the apparatus from being the observer. It is just another quantum system, which then has to be observed by something or someone else [17] There is here a kind of classical-quantum duality [18] whereby a macroscopic apparatus can be treated either classically or by quantum theory. In the classical treatment, the position, momentum, and other properties of the apparatus are c-numbers and have at each instant precise numerical values. Their evolution, however, is subject to probabilistic laws: It is unpredictable for individual cases; only averages can be predicted. On the other hand, a quantum mechanical treatment of the apparatus leaves it in a superposition of macroscopically different states, precluding its use as a measuring apparatus. This classicalquantum duality is not a flaw of the theory. As shown elsewhere [17] it is a logical necessity which is analogous to G6del's undecidability theorem. Any attempt to have a closed theory must lead to inconsistencies, as found in ref. [1].
References [1] D.Z. Albert, Phys. Lett. 98A (1983) 249. [2] J. yon Neumann, Mathematical foundations of quantum mechanics (Princeton Univ. Press, Princeton, 1955). [3] P.A. Moldauer, Phys. Rev. D5 (1972) 1028. [4] A. Peres, Am. J. Phys. 42 (1974) 886. [5] A. Peres, Plays. Rev. D22 (1980) 879. [6] J. Wheather and R. Peierls, Phys. Rev. Lett. 51 (1983) 1601.
[7] A. Daneri, A. Loinq~,erand G.M. Prosperi, Nucl. Phys. 33 (1962) 297; Nuovo Cimento B44 (1966) 119.
[8] J.S. Bell, Helv. Phys. Acta 48 (1975) 93. [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
P. Jasselette, Int. J. Quantum Chem. 17 (1980) 83. E. Schr6dinger, Naturwiss. 23 (1935) 807,824,844. M.S. Gupta, Int. J. Theor. Phys. 21 (1982) 275. R. Landauer, Int. J. Theor. Phys. 21 (1982) 283. K.K. Likhaxev, Int. J. Theor. Phys. 21 (1982) 311. I. Prigogine, Science 201 (1978) 777. A. Peres, Found. Phys. 10 (1980) 631. H.P. Stapp, Am. J. Phys. 40 (1972) 1098. A. Peres and W.H. Zurek, Am. J. Phys. 50 (1982) 807. A. Peres, Am. J. Phys. 52 (1984), to be published.