- Email: [email protected]
Controlling quantum state reductions
30 April 2001
Physics Letters A 282 (2001) 336–342 www.elsevier.nl/locate/pla
Controlling quantum state reductions Masanao Ozawa CREST, Japan Science and Technology, Graduate School of Human Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan Received 30 January 2001; accepted 7 March 2001 Communicated by P.R. Holland
Abstract The ensemble after a measurement of an observable can be controlled to be an arbitrary family of states by choosing the object–probe interaction properly. In particular, there is a position measuring apparatus that does not disturb the object prepared in a momentum eigenstate just before the measurement. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ta; 03.67.-a; 04.80.Nn Keywords: Measurement; State reduction; Completely positive; Operation; Indirect measurement; Position measurement; Uncertainty principle
1. Introduction Every quantum measurement is specified by the output distribution, the probability distribution of the outcome, and the quantum state reduction, the object state change from the state just before measurement to the state just after measurement conditional upon the outcome. Conventional measurement theory has restricted its scope to measurements such that output distributions are described by projection valued measures [1] and that quantum state reductions satisfy the projection postulate [2]. Modern measurement theory extends the scope to more general measurements such that output distributions are described by probability operator valued measures (POVMs) [3,4] and quantum state reductions by normalized completely positive operation valued measures [5].
E-mail address: [email protected] (M. Ozawa).
Since no measurements of continuous observables satisfy the repeatability hypothesis [5], the projection postulate, which implies the repeatability hypothesis, has confined the conventional theory to measurements of discrete observables. In the modern theory, the projection postulate is abandoned so that the notion of a measurement of an observable is defined by the sole requirement that the corresponding POVM value the spectral projections of the observable to be measured. This gives us an interesting problem of determining all the possible quantum state reductions caused by measurements of the same observable. In the recent paper [6], the case of nondegenerate discrete observables was investigated in detail. In this Letter, we examine continuous observables and show that for any (Borel) family of states there is a measuring apparatus that measures a given observable and leaves the object in that family of states. We also construct a potentially realizable object–probe interaction that measures position and leaves the object in the states which are translations of a given wave function. In the limit where the given wave function tends to a
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 7 4 - 8
M. Ozawa / Physics Letters A 282 (2001) 336–342
momentum eigenstate, this model demonstrates a position measuring apparatus that does not disturb the object prepared in a momentum eigenstate just before the measurement. In order to determine the possible quantum state reductions, we need to consider measuring processes described by indirect measurement models. An indirect measurement model consists of the following two stages: in the first stage the measured observable is transduced to the probe observable by a quantum mechanical interaction between the object and the probe, and in the second stage the probe observable is detected by amplifying it to a directly-sensible, macroscopic output variable. In the literature, we have often found the following argument to determine the quantum state reduction: to apply the unitary evolution to the object–probe composite system for describing the state change during the first stage, to apply the projection postulate to the composite system, and to trace out the probe system [7]. The validity of the projection postulate in the above argument is, however, limited or questionable, because of the following reasons (see Ref. [8] for discussions): 1. The probe detection, such as photon counting, in some measuring apparatus does not satisfy the projection postulate [9]. 2. When the probe observable has continuous spectrum, the projection postulate cannot be formulated properly [5]. 3. The projection postulate determines the state after the second stage. However, in the repeated measurement experiment, the second measurement on the same object can follow immediately after the first stage of the first measurement [10]. Thus, we need to determine the state after the first stage conditional upon the outcome to be obtained by the second stage. A rigorous and consistent derivation of the quantum state reduction without appealing to the projection postulate has been established in [5]. Based on this approach, the statistical equivalence classes of all the possible quantum measurements have been characterized as the normalized completely positive map valued measures in [5], where two measurements are statistically equivalent, if they have the same output distribution and the same quantum state reduction.
337
The above result shows quite generally that the quantum state reductions of measurements of the same observable can occur in surprisingly rich variety of ways. In order to exploit such richness for precision measurements and quantum information, this Letter investigates how quantum state reductions can be controlled by measuring interactions. It will be shown that for any Borel function x → ρx from the real line to the density operators there is a model of an indirect measurement of any observable such that the measured object is left in the state ρx with outcome x regardless of the prior object state. A potentially realizable interaction will be found also for a position measurement such that the object is left in ρx = e−ixP /h¯ |φφ|eixP /h¯ , where P is the momentum observable and φ is an arbitrarily given wave function. The implication of this model to the uncertainty principle will conclude this Letter.
2. Indirect measurement models Let S be a quantum system to be measured. Let A(a) be an apparatus to measure an observable A of S with output variable a [6]. The probe P is a microscopic subsystem of the apparatus A(a) that actually interacts with S, or more precisely the smallest subsystem of A(a) such that the composite system S + P is closed during the measuring interaction [6]. The Hilbert spaces of S and P are denoted by H and K, respectively. The process of measurement of A at time t using A(a) is described as follows. The interaction between S and P is turned on from time t to a later time t + t so that after t + t the object is free from the apparatus. The outcome of measurement is obtained by detecting the probe observable, denoted by M, in P at time t + t. This detection process takes time τ so that at time t + t + τ the observer can read out directly the output variable. The time t is called the time of measurement, the time t + t is called the time just after measurement, and the time t + t + τ is called the time of readout. For any Borel set ∆ in the real line R, the spectral projection of A corresponding to ∆ is denoted by E A (∆). Suppose that the time evolution of the composite system S + P from t to t + t is represented by a unitary operator U on the Hilbert space H ⊗ K. Suppose that at time t the object S is in an arbitrary
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M. Ozawa / Physics Letters A 282 (2001) 336–342
state ρ(t) and that the probe P is prepared in a fixed state σ . Then, S + P is in the state U (ρ(t) ⊗ σ )U † at time t + t. Since the outcome is obtained by the probe detection at t + t, according to the Born statistical formula the output distribution is given by Pr a(t) ∈ ∆ = Tr I ⊗ E M (∆) U ρ(t) ⊗ σ U † (1) for any Borel set ∆, where “a(t) ∈ ∆” denotes the probabilistic event that the outcome of the measurement using the apparatus A(a) at time t is in ∆. Thus, any apparatus is modeled generally by the quadruple (K, σ, U, M) of a Hilbert space K, a density operator σ on K, a unitary operator U on K ⊗ H, and a self-adjoint operator M on K, where K represents the state space of the probe, σ the preparation of the probe, U the interaction between the object and the probe, and M the probe observable to be detected. A model apparatus (K, σ, U, M) is said to measure an observable A, if the output distribution is identical with the probability distribution of A in the state ρ(t), i.e., Pr a(t) ∈ ∆ = Tr E A (∆)ρ(t) (2)
for the simultaneous measurement of I ⊗ M and B ⊗ I in the state U (ρ(t) ⊗ σ )U † [11]. It follows that the joint probability distribution of the output from the apparatus A(a) and the output from the apparatus A(b) is given by Pr a(t) ∈ ∆, b(t + t) ∈ ∆ = Tr E B (∆ ) ⊗ E M (∆) U ρ(t) ⊗ σ U † . (4)
for any Borel set ∆. For any Borel set ∆, let ρ(t + t | a(t) ∈ ∆) be the state at t + t of S conditional upon a(t) ∈ ∆; if the object S is sampled randomly from the subensemble of the similar systems that yield the outcome of the Ameasurement in the Borel set ∆, then S is in the state ρ(t + t | a(t) ∈ ∆) at time t + t. Since the condition a(t) ∈ R makes no selection, the state change ρ(t) → ρ(t + t | a(t) ∈ R) is called the nonselective state change and when ∆ = R the state change ρ(t) → ρ(t + t | a(t) ∈ ∆) is called the selective state change. According to the standard argument, the nonselective state change is determined by ρ t + t | a(t) ∈ R = TrK U ρ(t) ⊗ σ U † , (3)
for every Borel set ∆ with Pr{a(t) ∈ ∆} > 0. The above formula was obtained first in [5]. It should be noted that Eq. (6) does not assume that the composite system S + P with the outcome a(t) ∈ ∆ is in the state ρS+P t + t | a(t) ∈ ∆
where TrK denotes the partial trace over K. In order to determine the selective state change caused by the apparatus A(a), suppose that at time t + t the observer were to measure an arbitrary observable B of the same object S. Let A(b) be the apparatus to measure B with output variable b. Since the M-measurement at time t + t does not disturb the object S, the joint probability distribution of the outcome of the M-measurement and the outcome of the B-measurement satisfies the joint probability formula
On the other hand, using the state ρ(t + t | a(t) ∈ ∆) the same joint probability distribution can be represented by Pr a(t) ∈ ∆, b(t + t) ∈ ∆ = Tr E B (∆ )ρ t + t | a(t) ∈ ∆ × Pr a(t) ∈ ∆ . (5) From Eqs. (1), (4), and (5), the state ρ(t + t | a(t) ∈ ∆) is uniquely determined as ρ t + t | a(t) ∈ ∆ =
=
TrK [(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † ] Tr[(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † ]
(6)
(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † (I ⊗ E M (∆)) Tr[(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † ]
(7) just after the measurement. In fact, such assumption is not correct, since for any partition ∆ = ∆ ∪ ∆
the state ρS+P (t + t | a(t) ∈ ∆) should be a mixture of ρS+P (t + t | a(t) ∈ ∆ ) and ρS+P (t + t | a(t) ∈ ∆
) but this is not the case for Eq. (7). It is a significant merit of our derivation of Eq. (6) to make no assumptions on the state of the composite system after the measurement.
3. Operational distributions In order to examine the mathematical properties of the selective state change, for any Borel set ∆ define
M. Ozawa / Physics Letters A 282 (2001) 336–342
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the transformation X(∆) on the space τ c(H) of trace class operators on H by X(∆)ρ = TrK I ⊗ E M (∆) U (ρ ⊗ σ )U † (8)
Lewis [3] and were shown to be realized by completely positive map valued measures later in [5].
for any ρ ∈ τ c(H) and we shall call the family {X(∆) | ∆ ∈ B(R)} the operational distribution of the apparatus A(a), where B(R) is the collection of Borel sets in R. Then, it is easy to check that the family {X(∆) | ∆ ∈ B(R)} satisfies the following conditions:
4. Quantum state reductions
(i) For any Borel set ∆, the map ρ → X(∆)ρ is completely positive [12], i.e., we have n
ξi |X(∆) ρi† ρj |ξj 0
(9)
i,j =1
for any finite sequences ξ1 , . . . , ξn ∈ H and ρ1 , . . . , ρn ∈ τ c(H). (ii) For any Borel set ∆ and disjoint Borel sets ∆n such that ∆ = n ∆n , we have X(∆)ρ = (10) X(∆n )ρ n
for any ρ ∈ τ c(H). (iii) For any Borel set ∆ and any ρ ∈ τ c(H), Tr X(∆)ρ = Tr E A (∆)ρ .
∆
(11)
Mathematically, we shall call any map ∆ → X(∆) satisfying conditions (i) and (ii) a completely positive (CP) map valued measure and it is said to be A-compatible if condition (iii) holds. We have shown that the operational distribution of any model apparatus (K, σ, U, M) measuring A is an A-compatible CP map valued measure. The converse of this assertion was proved in [5] so that for any A-compatible CP map valued measure ∆ → X(∆) there is a model apparatus (K, σ, U, M) measuring A such that its operational distribution is {X(∆) | ∆ ∈ B(R)}. From Eqs. (2) and (11) we have Pr a(t) ∈ ∆ = Tr X(∆)ρ(t) , (12) and from Eqs. (6) and (8), if Pr{a(t) ∈ ∆} > 0, we have ρ t + t | a(t) ∈ ∆ =
X(∆)ρ(t) . Tr[X(∆)ρ(t)]
For any real number a, let ρ(t + t | a(t) = a) be the state at t + t of S conditional upon a(t) = a; if the object leads to the outcome a(t) = a, it has been in the state ρ(t + t | a(t) = a) at t + t. The quantum state reduction caused by the apparatus A(a) is the state change ρ(t) → ρ(t + t | a(t) = a) for all real number a. The family {ρ(t + t | a(t) = a) | a ∈ R} of states is called the posterior states for the prior state ρ(t). Together with the probability distribution Pr{a(t) ∈ ∆}, it is also called the posterior ensemble. According to the above definition, the quantum state reduction and the selective state change are related by Pr a(t) ∈ ∆ ρ t + t | a(t) ∈ ∆ = ρ t + t | a(t) = a Pr a(t) ∈ da , (14)
(13)
Thus, both the output distribution and the selective state change are determined by the operational distribution; relations (12) and (13) were postulated for general positive map valued measures first by Davies and
or equivalently, X(∆)ρ(t) = ρ t + t | a(t) = a ∆
× Tr E A (da)ρ(t)
(15)
for any Borel set ∆. It was shown in [5] that the above formula determines the posterior states as a Borel function a → ρ(t + t | a(t) = a) from R to the space of density operators uniquely up to probability one. To disregard irrelevant difference, two families of posterior states are considered to be identical if they are identical on a set of outcomes with probability one. Then, the quantum state reduction and the selective state change are equivalent under relation (14). We conclude, therefore, that two model apparatuses are statistically equivalent if and only if they have the same operational distribution. Let a → a be an arbitrary Borel function from R to the density operators. We shall consider the following problem: Is there any model apparatus measuring a given observable A such that the measured object is left in the state a with outcome a regardless of the prior state ρ(t). The affirmative answer to this problem is obtained as follows. For any Borel set ∆, let X(∆)
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M. Ozawa / Physics Letters A 282 (2001) 336–342
be the transformation of τ c(H) defined by X(∆)ρ = a Tr E A (da)ρ
(16)
∆
for any ρ ∈ τ c(H). In [13], it was shown that the map ∆ → X(∆) is an A-compatible completely positive map valued measure. It follows that there is a model apparatus (K, σ, U, M) measuring A such that its operational distribution is given by Eq. (16). By comparing Eqs. (15) and (16), we have [14] ρ t + t | a(t) = a = a . Thus, we have shown that for any observable A and any Borel function a → a there is a model apparatus measuring A such that the posterior states is { a | a ∈ R} regardless of the prior state of the object.
By the Schrödinger equation, the time evolution of S + P during the coupling is described by the unitary evolution operators
i pˆx ⊗ yˆ , U (t + t/2, t) = exp (21) h¯
i U (t + t, t + t/2) = exp − xˆ ⊗ pˆ y . (22) h¯ Then, in the position basis we have x, y|U (t + t/2, t)|x , y = x + y, y|x , y , (23) x, y|U (t + t, t + t/2)|x , y = x, y − x|x , y , and hence x, y|U (t + t, t + t/2)U (t + t/2, t)|x , y = y, y − x|x , y .
5. Position measuring apparatus Now, we shall consider the following model of position measurement, which was found first in [15] as the time independent Hamiltonian model. The object S is a one-dimensional mass with position x, ˆ momentum pˆx , and Hamiltonian Hˆ S . The probe P of the apparatus A(a) to measure the object position xˆ is another one-dimensional mass with position y, ˆ momentum pˆ y , and Hamiltonian Hˆ P . The Hilbert space of S is H = L2 (Rx ), the L2 space of the x-coordinate, and the Hilbert space of P is K = L2 (Ry ). The object–probe coupling is turned on from time t to t + t. Suppose that the time dependent total Hamiltonian Hˆ S+P (T ) of S + P is taken to be Hˆ S+P (T ) = Hˆ S ⊗ I + I ⊗ Hˆ P − K1 (T )pˆ x ⊗ yˆ + K2 (T )xˆ ⊗ pˆ y ,
(17)
where the strengths of couplings, K1 (T ) and K2 (T ), satisfy / (t, t + t/2), K1 (T ) = 0 if T ∈
(18)
/ (t + t/2, t + t), K2 (T ) = 0 if T ∈ (19) t + t /2 t + t K1 (T ) dT = 1, K2 (T ) dT = 1. t
(25)
Let ξ be an arbitrary normalized wave function of the apparatus. We shall show that the model M = (K, σ, U, M) measures xˆ with K = L2 (Ry ), σ = |ξ ξ |, U = U (t + t, t + t/2)U (t + t/2, t), M = y. ˆ Suppose that S is in the state ρ(t) = |ψ(t)ψ(t)| at time t, where ψ(t) is a normalized wave function of S. Then, S + P has the wave function Ψ (t) = ψ(t) ⊗ ξ at t and the wave function Ψ (t + t) = U Ψ (t) at t + t. In this case, we have
U ρ(t) ⊗ σ U † = Ψ (t + t) Ψ (t + t) , (26) and by Eq. (25) we have
x, y Ψ (t + t) = y|ψ(t)y − x|ξ .
(27)
From Eq. (1) the output distribution is given by Pr a(t) ∈ ∆
= Ψ (t + t) I ⊗ E yˆ (∆) Ψ (t + t) 2 2 = dy y|ψ(t) y − x|ξ dx
t + t /2
(20) We assume that t is so small that the system Hamiltonians Hˆ S and Hˆ P can be neglected from t to t + t.
(24)
∆
= ∆
R
y|ψ(t) 2 dy.
(28)
M. Ozawa / Physics Letters A 282 (2001) 336–342
Thus, the output distribution coincides with the xˆ distribution at time t, so that the model M measures position x. ˆ Now, let φ be an arbitrary normalized wave function and suppose that the probe preparation ξ is such that y|ξ = −y|φ. Then, we shall prove that the quantum state reduction caused by the model M is given by ρ t + t | a(t) = a = e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯ (29) for arbitrary prior state ρ(t) of S. By Eqs. (8), (26), and (27), we have x|X(∆) |ψ(t)ψ(t)| |x
= x, a Ψ (t + t) Ψ (t + t) x , a da ∆
=
2 x − a|φφ|x − a ψ(t)|a da
∆
= x|
e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯
∆
× ψ(t)|E xˆ (da)|ψ(t)|x .
Since x and x are arbitrary, we have X(∆) |ψ(t)ψ(t)| = e−ia pˆ y /h¯ |φφ|eiy pˆ y /h¯ ∆
× ψ(t)|E xˆ (da)|ψ(t).
341
posterior states is controlled to be an arbitrary family of states. This disproves the following interpretation of the uncertainty principle: if the position is measured with noise x then the back action disturbs the momentum so that the momentum uncertainty p just after measurement is not less than the order of h¯ / x. In [11], it was proved that any measuring apparatus disturbs every observable not commuting with the measured observable. The present model suggests, however, that the assertion cannot be strengthened so that the apparatus disturbs every state with density operator not commuting with the measured observable. Suppose the limit case where the probe is prepared in a momentum eigenstate |p. Then, the position measurement using M leaves the object in the momentum eigenstate |−p regardless of the outcome. Thus, if the object is prepared in the momentum eigenstate |−p just before the measurement, the posterior state is the same state. Thus, there is a position measuring apparatus that does not disturb the object prepared in a momentum eigenstate just before the measurement. The object–probe coupling U of the model M can be considered to be potentially realizable at least in the equivalent optical setting with orthogonal quadraˆ as well ture components, since the interaction pˆx ⊗ y, as xˆ ⊗ pˆ y , has been known as the backaction-evading (BAE) measurement [16] and has been optically realized by [17]. The detailed optical implementation of the model M will be discussed in a forthcoming paper.
(30)
By the linearity of X(∆), we have X(∆)ρ = e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯ Tr E xˆ (da)ρ ∆
(31) for any trace class operator ρ. This shows that the model M has the operational distribution of the form Eq. (16) with a = e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯ , and hence the quantum state reduction caused by M is given by Eq. (29).
6. Uncertainty principle — concluding remarks In this Letter, it is demonstrated that the measuring apparatus can be designed, in principle, so that the
References [1] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955. [2] G. Lüders, Ann. Phys. (Leipzig) 6 (8) (1951) 322. [3] E.B. Davies, J.T. Lewis, Comm. Math. Phys. 17 (1970) 239. [4] E.B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976; C.W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 1976; A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982. [5] M. Ozawa, in: K. Itô, J.V. Prohorov (Eds.), Probability Theory and Mathematical Statistics, Lecture Notes in Mathematics, Vol. 1021, Springer, Berlin, 1983, pp. 518–525; M. Ozawa, J. Math. Phys. 25 (1984) 79; M. Ozawa, Publ. Res. Inst. Math. Sci. Kyoto Univ. 21 (1985) 279. [6] M. Ozawa, Phys. Rev. A 62 (2000) 062101. [7] K. Kraus, Ann. Phys. (NY) 64 (1971) 311;
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[8] [9] [10] [11] [12]
[13]
M. Ozawa / Physics Letters A 282 (2001) 336–342 K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, Berlin, 1983; C.M. Caves, Phys. Rev. D 33 (1986) 1643; V.B. Braginsky, F.Ya. Khalili, Quantum Measurement, Cambridge University Press, Cambridge, 1992; P. Busch, M. Grabowski, P.J. Lahti, Operational Quantum Physics, Springer, Berlin, 1995. M. Ozawa, Phys. Rev. A 63 (2001) 032109. N. Imoto, M. Ueda, T. Ogawa, Phys. Rev. A 41 (1990) 4127. M. Ozawa, quant-ph/9802022. M. Ozawa, Phys. Lett. A 275 (2000) 5. An alternative but equivalent condition is that there is a sequence {An } of bounded operators on H such that X(∆)ρ = † n An ρAn for all ρ ∈ τ c(H). M. Ozawa, J. Math. Phys. 26 (1985) 1948.
[14] Note that ρ(t + t | a(t) = a) depends on ρ(t) in general, but a is given independent of ρ(t). [15] M. Ozawa, Phys. Rev. Lett. 60 (1988) 385; See also, M. Ozawa, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light, Plenum, New York, 1989, pp. 263– 286; M. Ozawa, Phys. Rev. A 41 (1990) 1735. [16] C.M. Caves, K.S. Thorne, R.W.P. Drever, V.D. Sandberg, M. Zimmermann, Rev. Mod. Phys. 52 (1980) 341; B. Yurke, J. Opt. Soc. Am. B 2 (1985) 732. [17] S.F. Pereira, Z.Y. Ou, H.J. Kimble, Phys. Rev. Lett. 72 (1994) 214; K. Bencheikh, J.A. Levenson, Ph. Grangier, O. Lopez, Phys. Rev. Lett. 75 (1995) 3422.
Physics Letters A 282 (2001) 336–342 www.elsevier.nl/locate/pla
Controlling quantum state reductions Masanao Ozawa CREST, Japan Science and Technology, Graduate School of Human Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan Received 30 January 2001; accepted 7 March 2001 Communicated by P.R. Holland
Abstract The ensemble after a measurement of an observable can be controlled to be an arbitrary family of states by choosing the object–probe interaction properly. In particular, there is a position measuring apparatus that does not disturb the object prepared in a momentum eigenstate just before the measurement. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ta; 03.67.-a; 04.80.Nn Keywords: Measurement; State reduction; Completely positive; Operation; Indirect measurement; Position measurement; Uncertainty principle
1. Introduction Every quantum measurement is specified by the output distribution, the probability distribution of the outcome, and the quantum state reduction, the object state change from the state just before measurement to the state just after measurement conditional upon the outcome. Conventional measurement theory has restricted its scope to measurements such that output distributions are described by projection valued measures [1] and that quantum state reductions satisfy the projection postulate [2]. Modern measurement theory extends the scope to more general measurements such that output distributions are described by probability operator valued measures (POVMs) [3,4] and quantum state reductions by normalized completely positive operation valued measures [5].
E-mail address: [email protected] (M. Ozawa).
Since no measurements of continuous observables satisfy the repeatability hypothesis [5], the projection postulate, which implies the repeatability hypothesis, has confined the conventional theory to measurements of discrete observables. In the modern theory, the projection postulate is abandoned so that the notion of a measurement of an observable is defined by the sole requirement that the corresponding POVM value the spectral projections of the observable to be measured. This gives us an interesting problem of determining all the possible quantum state reductions caused by measurements of the same observable. In the recent paper [6], the case of nondegenerate discrete observables was investigated in detail. In this Letter, we examine continuous observables and show that for any (Borel) family of states there is a measuring apparatus that measures a given observable and leaves the object in that family of states. We also construct a potentially realizable object–probe interaction that measures position and leaves the object in the states which are translations of a given wave function. In the limit where the given wave function tends to a
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 7 4 - 8
M. Ozawa / Physics Letters A 282 (2001) 336–342
momentum eigenstate, this model demonstrates a position measuring apparatus that does not disturb the object prepared in a momentum eigenstate just before the measurement. In order to determine the possible quantum state reductions, we need to consider measuring processes described by indirect measurement models. An indirect measurement model consists of the following two stages: in the first stage the measured observable is transduced to the probe observable by a quantum mechanical interaction between the object and the probe, and in the second stage the probe observable is detected by amplifying it to a directly-sensible, macroscopic output variable. In the literature, we have often found the following argument to determine the quantum state reduction: to apply the unitary evolution to the object–probe composite system for describing the state change during the first stage, to apply the projection postulate to the composite system, and to trace out the probe system [7]. The validity of the projection postulate in the above argument is, however, limited or questionable, because of the following reasons (see Ref. [8] for discussions): 1. The probe detection, such as photon counting, in some measuring apparatus does not satisfy the projection postulate [9]. 2. When the probe observable has continuous spectrum, the projection postulate cannot be formulated properly [5]. 3. The projection postulate determines the state after the second stage. However, in the repeated measurement experiment, the second measurement on the same object can follow immediately after the first stage of the first measurement [10]. Thus, we need to determine the state after the first stage conditional upon the outcome to be obtained by the second stage. A rigorous and consistent derivation of the quantum state reduction without appealing to the projection postulate has been established in [5]. Based on this approach, the statistical equivalence classes of all the possible quantum measurements have been characterized as the normalized completely positive map valued measures in [5], where two measurements are statistically equivalent, if they have the same output distribution and the same quantum state reduction.
337
The above result shows quite generally that the quantum state reductions of measurements of the same observable can occur in surprisingly rich variety of ways. In order to exploit such richness for precision measurements and quantum information, this Letter investigates how quantum state reductions can be controlled by measuring interactions. It will be shown that for any Borel function x → ρx from the real line to the density operators there is a model of an indirect measurement of any observable such that the measured object is left in the state ρx with outcome x regardless of the prior object state. A potentially realizable interaction will be found also for a position measurement such that the object is left in ρx = e−ixP /h¯ |φφ|eixP /h¯ , where P is the momentum observable and φ is an arbitrarily given wave function. The implication of this model to the uncertainty principle will conclude this Letter.
2. Indirect measurement models Let S be a quantum system to be measured. Let A(a) be an apparatus to measure an observable A of S with output variable a [6]. The probe P is a microscopic subsystem of the apparatus A(a) that actually interacts with S, or more precisely the smallest subsystem of A(a) such that the composite system S + P is closed during the measuring interaction [6]. The Hilbert spaces of S and P are denoted by H and K, respectively. The process of measurement of A at time t using A(a) is described as follows. The interaction between S and P is turned on from time t to a later time t + t so that after t + t the object is free from the apparatus. The outcome of measurement is obtained by detecting the probe observable, denoted by M, in P at time t + t. This detection process takes time τ so that at time t + t + τ the observer can read out directly the output variable. The time t is called the time of measurement, the time t + t is called the time just after measurement, and the time t + t + τ is called the time of readout. For any Borel set ∆ in the real line R, the spectral projection of A corresponding to ∆ is denoted by E A (∆). Suppose that the time evolution of the composite system S + P from t to t + t is represented by a unitary operator U on the Hilbert space H ⊗ K. Suppose that at time t the object S is in an arbitrary
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state ρ(t) and that the probe P is prepared in a fixed state σ . Then, S + P is in the state U (ρ(t) ⊗ σ )U † at time t + t. Since the outcome is obtained by the probe detection at t + t, according to the Born statistical formula the output distribution is given by Pr a(t) ∈ ∆ = Tr I ⊗ E M (∆) U ρ(t) ⊗ σ U † (1) for any Borel set ∆, where “a(t) ∈ ∆” denotes the probabilistic event that the outcome of the measurement using the apparatus A(a) at time t is in ∆. Thus, any apparatus is modeled generally by the quadruple (K, σ, U, M) of a Hilbert space K, a density operator σ on K, a unitary operator U on K ⊗ H, and a self-adjoint operator M on K, where K represents the state space of the probe, σ the preparation of the probe, U the interaction between the object and the probe, and M the probe observable to be detected. A model apparatus (K, σ, U, M) is said to measure an observable A, if the output distribution is identical with the probability distribution of A in the state ρ(t), i.e., Pr a(t) ∈ ∆ = Tr E A (∆)ρ(t) (2)
for the simultaneous measurement of I ⊗ M and B ⊗ I in the state U (ρ(t) ⊗ σ )U † [11]. It follows that the joint probability distribution of the output from the apparatus A(a) and the output from the apparatus A(b) is given by Pr a(t) ∈ ∆, b(t + t) ∈ ∆ = Tr E B (∆ ) ⊗ E M (∆) U ρ(t) ⊗ σ U † . (4)
for any Borel set ∆. For any Borel set ∆, let ρ(t + t | a(t) ∈ ∆) be the state at t + t of S conditional upon a(t) ∈ ∆; if the object S is sampled randomly from the subensemble of the similar systems that yield the outcome of the Ameasurement in the Borel set ∆, then S is in the state ρ(t + t | a(t) ∈ ∆) at time t + t. Since the condition a(t) ∈ R makes no selection, the state change ρ(t) → ρ(t + t | a(t) ∈ R) is called the nonselective state change and when ∆ = R the state change ρ(t) → ρ(t + t | a(t) ∈ ∆) is called the selective state change. According to the standard argument, the nonselective state change is determined by ρ t + t | a(t) ∈ R = TrK U ρ(t) ⊗ σ U † , (3)
for every Borel set ∆ with Pr{a(t) ∈ ∆} > 0. The above formula was obtained first in [5]. It should be noted that Eq. (6) does not assume that the composite system S + P with the outcome a(t) ∈ ∆ is in the state ρS+P t + t | a(t) ∈ ∆
where TrK denotes the partial trace over K. In order to determine the selective state change caused by the apparatus A(a), suppose that at time t + t the observer were to measure an arbitrary observable B of the same object S. Let A(b) be the apparatus to measure B with output variable b. Since the M-measurement at time t + t does not disturb the object S, the joint probability distribution of the outcome of the M-measurement and the outcome of the B-measurement satisfies the joint probability formula
On the other hand, using the state ρ(t + t | a(t) ∈ ∆) the same joint probability distribution can be represented by Pr a(t) ∈ ∆, b(t + t) ∈ ∆ = Tr E B (∆ )ρ t + t | a(t) ∈ ∆ × Pr a(t) ∈ ∆ . (5) From Eqs. (1), (4), and (5), the state ρ(t + t | a(t) ∈ ∆) is uniquely determined as ρ t + t | a(t) ∈ ∆ =
=
TrK [(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † ] Tr[(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † ]
(6)
(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † (I ⊗ E M (∆)) Tr[(I ⊗ E M (∆))U (ρ(t) ⊗ σ )U † ]
(7) just after the measurement. In fact, such assumption is not correct, since for any partition ∆ = ∆ ∪ ∆
the state ρS+P (t + t | a(t) ∈ ∆) should be a mixture of ρS+P (t + t | a(t) ∈ ∆ ) and ρS+P (t + t | a(t) ∈ ∆
) but this is not the case for Eq. (7). It is a significant merit of our derivation of Eq. (6) to make no assumptions on the state of the composite system after the measurement.
3. Operational distributions In order to examine the mathematical properties of the selective state change, for any Borel set ∆ define
M. Ozawa / Physics Letters A 282 (2001) 336–342
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the transformation X(∆) on the space τ c(H) of trace class operators on H by X(∆)ρ = TrK I ⊗ E M (∆) U (ρ ⊗ σ )U † (8)
Lewis [3] and were shown to be realized by completely positive map valued measures later in [5].
for any ρ ∈ τ c(H) and we shall call the family {X(∆) | ∆ ∈ B(R)} the operational distribution of the apparatus A(a), where B(R) is the collection of Borel sets in R. Then, it is easy to check that the family {X(∆) | ∆ ∈ B(R)} satisfies the following conditions:
4. Quantum state reductions
(i) For any Borel set ∆, the map ρ → X(∆)ρ is completely positive [12], i.e., we have n
ξi |X(∆) ρi† ρj |ξj 0
(9)
i,j =1
for any finite sequences ξ1 , . . . , ξn ∈ H and ρ1 , . . . , ρn ∈ τ c(H). (ii) For any Borel set ∆ and disjoint Borel sets ∆n such that ∆ = n ∆n , we have X(∆)ρ = (10) X(∆n )ρ n
for any ρ ∈ τ c(H). (iii) For any Borel set ∆ and any ρ ∈ τ c(H), Tr X(∆)ρ = Tr E A (∆)ρ .
∆
(11)
Mathematically, we shall call any map ∆ → X(∆) satisfying conditions (i) and (ii) a completely positive (CP) map valued measure and it is said to be A-compatible if condition (iii) holds. We have shown that the operational distribution of any model apparatus (K, σ, U, M) measuring A is an A-compatible CP map valued measure. The converse of this assertion was proved in [5] so that for any A-compatible CP map valued measure ∆ → X(∆) there is a model apparatus (K, σ, U, M) measuring A such that its operational distribution is {X(∆) | ∆ ∈ B(R)}. From Eqs. (2) and (11) we have Pr a(t) ∈ ∆ = Tr X(∆)ρ(t) , (12) and from Eqs. (6) and (8), if Pr{a(t) ∈ ∆} > 0, we have ρ t + t | a(t) ∈ ∆ =
X(∆)ρ(t) . Tr[X(∆)ρ(t)]
For any real number a, let ρ(t + t | a(t) = a) be the state at t + t of S conditional upon a(t) = a; if the object leads to the outcome a(t) = a, it has been in the state ρ(t + t | a(t) = a) at t + t. The quantum state reduction caused by the apparatus A(a) is the state change ρ(t) → ρ(t + t | a(t) = a) for all real number a. The family {ρ(t + t | a(t) = a) | a ∈ R} of states is called the posterior states for the prior state ρ(t). Together with the probability distribution Pr{a(t) ∈ ∆}, it is also called the posterior ensemble. According to the above definition, the quantum state reduction and the selective state change are related by Pr a(t) ∈ ∆ ρ t + t | a(t) ∈ ∆ = ρ t + t | a(t) = a Pr a(t) ∈ da , (14)
(13)
Thus, both the output distribution and the selective state change are determined by the operational distribution; relations (12) and (13) were postulated for general positive map valued measures first by Davies and
or equivalently, X(∆)ρ(t) = ρ t + t | a(t) = a ∆
× Tr E A (da)ρ(t)
(15)
for any Borel set ∆. It was shown in [5] that the above formula determines the posterior states as a Borel function a → ρ(t + t | a(t) = a) from R to the space of density operators uniquely up to probability one. To disregard irrelevant difference, two families of posterior states are considered to be identical if they are identical on a set of outcomes with probability one. Then, the quantum state reduction and the selective state change are equivalent under relation (14). We conclude, therefore, that two model apparatuses are statistically equivalent if and only if they have the same operational distribution. Let a → a be an arbitrary Borel function from R to the density operators. We shall consider the following problem: Is there any model apparatus measuring a given observable A such that the measured object is left in the state a with outcome a regardless of the prior state ρ(t). The affirmative answer to this problem is obtained as follows. For any Borel set ∆, let X(∆)
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be the transformation of τ c(H) defined by X(∆)ρ = a Tr E A (da)ρ
(16)
∆
for any ρ ∈ τ c(H). In [13], it was shown that the map ∆ → X(∆) is an A-compatible completely positive map valued measure. It follows that there is a model apparatus (K, σ, U, M) measuring A such that its operational distribution is given by Eq. (16). By comparing Eqs. (15) and (16), we have [14] ρ t + t | a(t) = a = a . Thus, we have shown that for any observable A and any Borel function a → a there is a model apparatus measuring A such that the posterior states is { a | a ∈ R} regardless of the prior state of the object.
By the Schrödinger equation, the time evolution of S + P during the coupling is described by the unitary evolution operators
i pˆx ⊗ yˆ , U (t + t/2, t) = exp (21) h¯
i U (t + t, t + t/2) = exp − xˆ ⊗ pˆ y . (22) h¯ Then, in the position basis we have x, y|U (t + t/2, t)|x , y = x + y, y|x , y , (23) x, y|U (t + t, t + t/2)|x , y = x, y − x|x , y , and hence x, y|U (t + t, t + t/2)U (t + t/2, t)|x , y = y, y − x|x , y .
5. Position measuring apparatus Now, we shall consider the following model of position measurement, which was found first in [15] as the time independent Hamiltonian model. The object S is a one-dimensional mass with position x, ˆ momentum pˆx , and Hamiltonian Hˆ S . The probe P of the apparatus A(a) to measure the object position xˆ is another one-dimensional mass with position y, ˆ momentum pˆ y , and Hamiltonian Hˆ P . The Hilbert space of S is H = L2 (Rx ), the L2 space of the x-coordinate, and the Hilbert space of P is K = L2 (Ry ). The object–probe coupling is turned on from time t to t + t. Suppose that the time dependent total Hamiltonian Hˆ S+P (T ) of S + P is taken to be Hˆ S+P (T ) = Hˆ S ⊗ I + I ⊗ Hˆ P − K1 (T )pˆ x ⊗ yˆ + K2 (T )xˆ ⊗ pˆ y ,
(17)
where the strengths of couplings, K1 (T ) and K2 (T ), satisfy / (t, t + t/2), K1 (T ) = 0 if T ∈
(18)
/ (t + t/2, t + t), K2 (T ) = 0 if T ∈ (19) t + t /2 t + t K1 (T ) dT = 1, K2 (T ) dT = 1. t
(25)
Let ξ be an arbitrary normalized wave function of the apparatus. We shall show that the model M = (K, σ, U, M) measures xˆ with K = L2 (Ry ), σ = |ξ ξ |, U = U (t + t, t + t/2)U (t + t/2, t), M = y. ˆ Suppose that S is in the state ρ(t) = |ψ(t)ψ(t)| at time t, where ψ(t) is a normalized wave function of S. Then, S + P has the wave function Ψ (t) = ψ(t) ⊗ ξ at t and the wave function Ψ (t + t) = U Ψ (t) at t + t. In this case, we have
U ρ(t) ⊗ σ U † = Ψ (t + t) Ψ (t + t) , (26) and by Eq. (25) we have
x, y Ψ (t + t) = y|ψ(t)y − x|ξ .
(27)
From Eq. (1) the output distribution is given by Pr a(t) ∈ ∆
= Ψ (t + t) I ⊗ E yˆ (∆) Ψ (t + t) 2 2 = dy y|ψ(t) y − x|ξ dx
t + t /2
(20) We assume that t is so small that the system Hamiltonians Hˆ S and Hˆ P can be neglected from t to t + t.
(24)
∆
= ∆
R
y|ψ(t) 2 dy.
(28)
M. Ozawa / Physics Letters A 282 (2001) 336–342
Thus, the output distribution coincides with the xˆ distribution at time t, so that the model M measures position x. ˆ Now, let φ be an arbitrary normalized wave function and suppose that the probe preparation ξ is such that y|ξ = −y|φ. Then, we shall prove that the quantum state reduction caused by the model M is given by ρ t + t | a(t) = a = e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯ (29) for arbitrary prior state ρ(t) of S. By Eqs. (8), (26), and (27), we have x|X(∆) |ψ(t)ψ(t)| |x
= x, a Ψ (t + t) Ψ (t + t) x , a da ∆
=
2 x − a|φφ|x − a ψ(t)|a da
∆
= x|
e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯
∆
× ψ(t)|E xˆ (da)|ψ(t)|x .
Since x and x are arbitrary, we have X(∆) |ψ(t)ψ(t)| = e−ia pˆ y /h¯ |φφ|eiy pˆ y /h¯ ∆
× ψ(t)|E xˆ (da)|ψ(t).
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posterior states is controlled to be an arbitrary family of states. This disproves the following interpretation of the uncertainty principle: if the position is measured with noise x then the back action disturbs the momentum so that the momentum uncertainty p just after measurement is not less than the order of h¯ / x. In [11], it was proved that any measuring apparatus disturbs every observable not commuting with the measured observable. The present model suggests, however, that the assertion cannot be strengthened so that the apparatus disturbs every state with density operator not commuting with the measured observable. Suppose the limit case where the probe is prepared in a momentum eigenstate |p. Then, the position measurement using M leaves the object in the momentum eigenstate |−p regardless of the outcome. Thus, if the object is prepared in the momentum eigenstate |−p just before the measurement, the posterior state is the same state. Thus, there is a position measuring apparatus that does not disturb the object prepared in a momentum eigenstate just before the measurement. The object–probe coupling U of the model M can be considered to be potentially realizable at least in the equivalent optical setting with orthogonal quadraˆ as well ture components, since the interaction pˆx ⊗ y, as xˆ ⊗ pˆ y , has been known as the backaction-evading (BAE) measurement [16] and has been optically realized by [17]. The detailed optical implementation of the model M will be discussed in a forthcoming paper.
(30)
By the linearity of X(∆), we have X(∆)ρ = e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯ Tr E xˆ (da)ρ ∆
(31) for any trace class operator ρ. This shows that the model M has the operational distribution of the form Eq. (16) with a = e−ia pˆ y /h¯ |φφ|eia pˆ y /h¯ , and hence the quantum state reduction caused by M is given by Eq. (29).
6. Uncertainty principle — concluding remarks In this Letter, it is demonstrated that the measuring apparatus can be designed, in principle, so that the
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