Ring 2 — Canonical Grounding

Ring 3 — Framework Connections

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arXiv:0905.4398v1 [quant-ph] 27 May 2009 Von Neumann and Luders postulates and quantum information theory Andrei Khrennikov School of Mathematics and Systems Engineering International Center of Mathematical Modeling in Physics and Cognitive Sciences V¨axj¨o University, S-35195, Sweden May 29, 2018 Abstract This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI) theory, especially quantum teleportation and “one way quantum computing.” We emphasize the role of the projection postulate (determining post-measurement states) in QI and the diﬀerence between its L¨uders and von Neumann versions. These projection postulates diﬀer crucially in the case of observables with degenerate spectra. Such observables play the fundamental role in operations with entangled states: any measurement on one subsystem is represented by an observable with degenerate spectrum in the Hilbert space of a composite system. If von Neumann was right and L¨uders was wrong the canonical schemes of quantum teleportation and “one way quantum computing” would not work. Surprisingly, we found that, in fact, von Neumann’s description of measurements via reﬁnement implies (under natural assumptions) L¨uders projection postulate. It seems that this important observation was missed during last 70 years. This result closed the problem of the proper use of the projection postulate in quantum information theory. One can proceed with L¨uders postulate (as people in quantum information really do). 1 Introduction Although the QI project approached the stage of technological (at least experimental) realizations, research on foundational problems related to quantum information processing1 did not become less important. Moreover, many problems in foundations of QM which were considered as of pure theoretical (or even philosophical) value nowadays play an important role in (expensive) technological projects. Thus such problems could not be simply ignored. Development of QI also induces new approaches which foundational basis should be carefully analyzed. Among such novel approaches I would like to mention quantum teleportation and “one way 1See, e.g., recent book of G. Jaeger [1] and paper of M. Asano, M. Ohya, and Y. Tanaka [2]. 1

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quantum computing”, see, e.g., [3]–[5] – an exciting alternative to the conventional scheme of quantum computing. In a recent series of papers [6]–[9] the author paid attention on crucial diﬀerence of consequences of von Neumann [10] and L¨uders [11] projection postulates for QI, staring with EPR-argument [12]. These postulates coincide for observables with nondegenerate spectra, but they diﬀer in the case of degenerate spectra. We remark that the latter case is the most important for quantum information theory, since measurement on one of systems in a pair of entangled systems is represented by an operator with degenerate spectrum. While L¨uders [11] projection postulate is ﬁne for QI, the appeal to von Neumann postulate induces serious problems []. In the ﬁrst case measurement on a subsystem produces a pure state for another subsystem and it is good for quantum teleportation and computing. However, in the second case even starting with a pure state for a composite system, one obtains in general a statistical mixture. Moreover, by von Neumann the formalism of QM does not predict the post measurement state in the case of degenerate spectrum. Thus even mentioned statistical mixture is unknown. In [10] it was emphasized that measurementd of observables represented by operators with degenerate spectra are ambiguous. This problem can be solved (due to von Neumann) only via reﬁnement measurements. One should ﬁnd an observable, say B, represented by an operator bB with nondegenerate spectrum which commutes with the original operator b A with degenerate spectrum. Then results of A-measurement are obtained as A = f(B), where f is the function coupling the operators: b A = f( bB). Since B can be chosen in various ways, one can select various measurement procedures for A-measurement. It is crucial for foundations of QI that for composite systems reﬁnement of measurement on one of subsystems can be approached only via combined measurement on both subsystems. If it is really the case and von Neumann was right, then foundations of QI should be carefully reconsidered, since a number of important procedures in QI processing is based on L¨uders postulate. First of all we mention quantum teleportation. It were impossible to teleport an unknown quantum state in von Neumann’s framework, see [10]. Alice evidently uses L¨uders postulate to be sure that her measurement would produce the corresponding pure state for Bob (then Bob needs only to perform a local unitary evolution to get the proper state). The situation in quantum computing is not completely clear. It seems that the post-measurement state does not play any role in the conventional scheme of quantum computation: unitary evolution and, ﬁnally, measurement of a proper observable. It seems that only probabilities of results are important. Probabilities are calculated in the same way both in L¨uders and von Neumann’s approach. The situation is completely diﬀerent in the case of so called “one way quantum computing”, see, e.g., [3]–[5]. This scheme (based on measurements, instead of unitary evolution) depends crucially on the possibility to use L¨uders postulate. It would not work if von Neumann was right and L¨uders was wrong. To my surprise, recently I found that, in fact, von Neumann’s description of measurements via reﬁnement2 implies (under natural assumptions) L¨uders projection postulate. It seems that this important observation was missed during last 70 years. This result closed the problem of the proper 2By using an observable represented by an operator with nondegenerate spectrum commuting with operator with degenerate spectrum representing the original observable. 2

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use of the projection postulate in quantum information theory. One can proceed with L¨uders postulate (as people in quantum information really do). 2 Von Neumann’s and L¨uders’ postulates for pure states 2.1 Nondegenerate (discrete) spectrum Everywhere below H denotes complex Hilbert space. Let ψ ∈H be a pure state, i.e., ∥ψ∥2 = 1. We remark that any pure state induces density operator: bρψ = ψ ⊗ψ = bPψ where bPψ denotes the orthogonal projector on the vector ψ. This operator describes an ensemble of identically prepared systems each of them in the same state ψ. For an observable A represented by the operator b A with nondegenerate spectrum von Neumann’s and L¨uders projection postulates coincide. For simplicity we restrict our considerations to operators with purely discrete spectra. In this case spectrum consists of eigenvalues αk of b A : b Aek = αkek. Nondegeneracy of spectrum means that subspaces consisting of eigenvectors corresponding to diﬀerent eigenvalues are one dimensional. PP: Let A be an observable described by the self-adjoint operator b A having purely discrete nondegenerate spectrum. Measurement of observable A on a system in the (pure) quantum state ψ producing the result A = αk induces transition from the state ψ into the corresponding eigenvector ek of the operator b A. If we select only systems with the ﬁxed measurement result A = αk, then we obtain an ensemble described by the density operator bqk = ek⊗ek. Any system in this ensemble is in the same state ek. If we do not perform selections, we obtain obtain an ensemble described by the density operator bqψ = X k |⟨ψ, ek⟩|2 bPek = X k ⟨bρψek, ek⟩bPek = X k bPek bρψ bPek. where bPek is projector on the eigenvector ek. 2.2 Degenerate (discrete) spectrum: L¨uders viewpoint L¨uders generalized this postulate to the case of operators having degenerate spectra. Let us consider spectral decomposition for a self-adjoint operator b A having purely discrete spectrum: b A = X i αi bPi, where αi ∈R are diﬀerent eigenvalues of b A (so αi ̸= αj) and bPi, i = 1, 2, …, is projector onto subspace Hi of eigenvectors corresponding to αi. 3

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By L¨uders’ postulate after measurement of an observable A represented by the operator b A that gives the result αi the initial pure state ψ is transformed again into a pure state, namely, ψi = bPiψ ∥bPiψ∥ . Thus for corresponding density operator we have bQi = ψi ⊗ψi = bPiψ ⊗bPiψ ∥bPiψ∥2

bPibρψ bPi ∥bPiψ∥2 . If one does not make selections corresponding to values αi the ﬁnal postmeasurement state is given by bqψ = X i pi bQi, pi = ∥bPiψ∥2, (1) or simply bqψ = X i bqi, bqi = bPiρψ bPi. (2) This is the statistical mixture of pure states ψi. Thus by L¨uders there is no essential diﬀerence between measurements of observables with degenerate and nondegenerate spectra. 2.3 Degenerate (discrete) spectrum: von Neumann’s viewpoint Von Neumann had the completely diﬀerent viewpoint on the post-measurement state [10]. Even for a pure state ψ the post-measurement state (for measurement with selection with respect to a ﬁxed result A = αk) will not be a pure state again. If b A has degenerate (discrete) spectrum, then according to von Neumann [10] A measurement of an observable A giving the value A = αi does not induce projection of ψ on the subspace Hi. The result will not be the ﬁxed pure state, in particular, not L¨uders’ state ψi. Moreover, the post-measurement state, say bgψ, is not determined by the formalism of QM! Only a subsequent measurement of an observable D such that A = f(D) and bD is an operator with nondegenerate spectrum (“reﬁnement measurement”) will determine the ﬁnal state. Following von Neumann, we choose in each Hi an orthonormal basis {ein}. Let us take sequence of real numbers {γin} such that all numbers are distinct. We deﬁne the corresponding self-adjoint operator bD having eigenvectors {ein} and eigenvalues {γin} : bD = X i X n γin bPein. A measurement of the observable D represented by the operator bD can be considered as measurement of the observable A, because A = f(D), where f is some function such that f(γin) = αi. The D-measurement (without post-measurement selection with respect to eigenvalues) produces the statistical mixture bOD;ψ = X i X n |⟨ψ, ein⟩|2 bPein. (3) 4

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By selection for the value αi of A (its measurements realized via measurements of a reﬁnement observable D) we obtain the statistical mixture described by normalization of the operator bOi,D;ψ = X n |⟨ψ, ein⟩|2 bPein. (4) Von Neumann emphasized that the mathematical formalism of QM could not describe the post-measurement state for measurements (without reﬁnement) of degenerate observables. He did not discuss directly properties of such a state, he described them only indirectly via reﬁnement measurements.3 We would like to proceed by considering this (“hidden”) state under assumption that it can be described by a density operator, say bgψ. We formalize a list of properties of this hidden (post-measurement) state which can be extracted from von Neumann’s considerations on re- ﬁnement measurements. Finally, we prove, see Theorem 1, that bgψ should coincide with the post-measurement state postulated by L¨uders, (2). Consider the A-measurement without reﬁnement. By von Neumann, for each quantum system s in the initial pure state ψ, the A-measurement with the αi-selection transforms the ψ in one of states φ = φ(s) belonging to the eigensubspace Hi. Unlike L¨uders’ approach, it implies that, instead of one ﬁxed state, namely, ψi ∈Hi, such an experiment produces a probability distribution of states on the unit sphere of the subspace Hi. We postulate DO For any value αi such that bPiψ ̸= 0, the post-measurement probability distribution on Hi can be described by a density operator, say bΓi. Here bΓi : Hi →Hi is such that bΓi ≥0 and TrbΓi = 1. Consider now the corresponding density operator bGi in H. Its restriction on Hi coincides with bΓi. In particular this implies its property: bGi(Hi) ⊂Hi. (5) We remark that bGi is determined by ψ, so bGi ≡bGi;ψ. We would like to present the list of other properties of bGi induced by von Neumann’s considerations on reﬁnement. Since, for each reﬁnement measurement D, the operators b A and bD commute, the measurement of A with reﬁnement can be performed in two ways. First we perform the D- measurement and then we get A as A = f(D). However, we also can ﬁrst perform the A-measurement, obtain the post-measurement state described by the density operator bGi, then measure D and, ﬁnally, we again ﬁnd A = f(D). Take an arbitrary φ ∈Hi and consider a reﬁnement measurement D such that φ is an eigenvector of bD. Thus bDφ = γφφ. Then for the cases – [direct measurement of D] and [ﬁrst A and then D] – we get probabilities which are coupled in a simple way. In the ﬁrst case (by Born’s rule) P(D = γφ|bρψ) = | < ψ, φ > |2. (6) In the second case, after the A-measurement, we obtain the state bGi with probability P(A = αi|bρψ) = ∥bPiψ∥2. 3For him this state was a kind of hidden variable. It might even be that he had in mind that it “does not exist at all”, i.e., it could not be described by a density operator. 5

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Performing the D-measurement for the state bGi we get the value γφ with probability: P(D = γφ| bGi) = Tr bGibPφ. (7) By (classical) Bayes’ rule P(D = γφ|bρψ) = P(A = αi|bρψ)P(D = γφ| bGi). (8) Finally, we obtain P(D = γφ| bGi) = Tr bGibPφ = | < ψ, φ > |2 ∥bPiψ∥2 . (9) Thus TrbGibPφ = | < ψ, φ > |2 ∥bPiψ∥2 . (10) This is one of the basic features of the post-measurement state bGi (for the A-measurement with the αi-selection, but without any reﬁnement). Another basic equality we obtain in the following way. Take an arbitrary φ′ ∈H⊥ i , and consider a measurement of the observable described by the orthogonal projector bPφ′ under the state bGi. Since the later describes a probability distribution concentrated on Hi, we have: P(Pφ′ = 1| bGi) = 0. (11) Thus Tr; bGibPφ′ = 0. (12) This is the second basic feature of the post-measurement state. Our aim is to show that (10) and (12) imply that, in fact, bGi = bPibρψ bPi/∥bPiψ∥2 ≡bPiψ ⊗bPiψ/∥bPiψ∥2, (13) i.e., to derive L¨uders postulate which is a theorem in our approach. Lemma. The post-measurement density operator bGi maps H into Hi. Proof. By (5) it is suﬃcient to show that bGi(H⊥ i ) ⊂Hi. By (12) we obtain < bGiφ′, φ′ >= 0 (14) for any φ′ ∈H⊥ i . This immediately implies that < bGiφ′ 1, φ′ 2 >= 0 for any pair of vectors from H⊥ i . The latter implies that bGiφ′ ∈Hi for any φ′ ∈H⊥ i . Consider now the A-measurement without reﬁnement and selection. The post-measurement state bgψ can be represented as bgψ = X m pm bGm, p=∥bPmψ∥2, (15) Proposition 1. For any pure state ψ and self-adjoint operator b A (with purely discrete degenerate) spectrum the post-measurement state (in the absence of reﬁnement measurement) can be represented as bgψ = X m bgm, (16) where bgm : H →Hm, bgm ≥0, and, for any φ ∈Hm, < bgmφ, φ >= | < ψ, φ > |2. (17) 6

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3 Derivation of Luders’ postulate from von Neumann’s postulate Theorem. Let bg ≡bgψ be a density operator described by Proposition 1. Then bgm = bPmψ ⊗bPmψ. (18) Proof. Let {emk} be an orthonormal basis in Hm and let u ∈H. We represent it as u = um + u⊥ m, where um ∈Hm and u⊥ m ∈H⊥ m. Then < bgmu, u >=< bgmum, um > + < bgmum, u⊥ m > + < bgmu⊥ m, um > + < bgmu⊥ m, u⊥ m > . The second and last terms equals to zero, since bgm : H 7→Hm. To show that the third term also equals to zero, we should use self-adjointness of bgm. Thus < bgmu, u >= X k,k′ < u, eku >< emk′, u >< bgmeku, emk′ > . For each emn, we have < bgmemn, emn >= | < ψ, emn > |2. Thus the diagonal elements of the matrix of operator bgm coincide with diagonal elements of operator bPmψ ⊗bPmψ. Take now another basis in Hm which is constructed in the following way. We ﬁx two indexes, say n and j, and choose two new basis vectors: fmn = (emn + emj)/ √ 2, fmj = (emn −emj)/ √ 2. Then we have < bgmfmn, fmn >= | < ψ, fmn > |2, or < bgmemn, emn > + < bgmemj, emj > + < bgmemn, emj > + < bgmemj, emn > = | < ψ, emn > |2+| < ψ, emj > |2+ < ψ, emn >< emj, ψ > + < ψ, emj >< emn, ψ > . Thus < bgmemn, emj > +< bgmemn, emj > =< ψ, emn >< emj, ψ > +< ψ, emn >< emj, ψ >. Thus we proved that Re [< bgmemn, emj >] = Re [< ψ, emn >< emj, ψ >]. Let us now choose two new basis vectors ¯fmn = (emn + iemj)/ √ 2, , ¯fmj = (emn + iemj)/ √ 2. Then we have: < bgm ¯fmn, ¯fmn >=< bgmemn, emn > + < bgmemj, emj > +i < bgmemj, emn > −i < bgmemn, emj >= | < ψ, emn > |2+ < ψ, emj > |2+i < ψ, emn >< emj, ψ > −i < ψ, emj >< emn, ψ > . Thus: < qmemj, emn > −< bgmemn, emj >=< ψ, emn >< emj, ψ > −< ψ, emj >< emn, ψ > . Thus < qmemn, emj >=< ψ, emj >< emn, ψ > . We obtained the following representation for the quadratic form of the operator bgm < bgmu, u >= X k,k′ < u, emk >< emk′, u >< ψ, emk′ >< emk, ψ >= | < ψ, u > |2. Hence bgm = bPmψ ⊗bP ψ m. 7

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Conclusion: The general scheme of measurement of observables with degenerate spectra provided by von Neumann [10] implies, in fact, the L¨uders projection postulate. This postulate is a theorem (missed for 70 years) in von Neumann’s framework. Thus (in the canonical formalism of QM) the post-measurement state is always a pure state. This supports existing schemes of quantum teleportation and computing. References [1] G. Jaeger, Quantum information. An overview. Springer, Berlin, 2007. [2] M. Asano, M. Ohya, and Y. Tanaka, Complete m-level teleportation based on Kossakowski-Ohya scheme. Proceedings of QBIC-2, Quantum Probability and White Noise Analysis, 24, 19-29 (2009). [3] R. Rausendorf and J. Briegel, A one way quantum computer. Phys.Rev. Lett 86, 5188(2001). [4] G. Vallone, E. Pomarico, F. De Martini, and P. Mataloni, One way quantum computation with two-photon multiqubit cluster state, arhiv.0807.3887. [5] N. C. Menicucci, S. T. Flammia, O. Pﬁster, One way quantum computing in optical frequency comb, arxiv 0804.4468. [6] A. Yu. Khrennikov, The role of von Neumann and Luders postulates in the Einstein, Podolsky, and Rosen considerations: Comparing measurements with degenerate and nondegenerate spectra, J. Math. Phys., 49, N 5, art. no. 052102 (2008). [7] A. Yu. Khrennikov, Analysis of the role of von Neumann’s projection postulate in the canonical scheme of quantum teleportation, J. Russian Laser Research, 29, N 3, 296-301 (2008). [8] Khennikov, Analysis of explicit and implicit assumptions in the theorems of J. Von Neumann and J. Bell. J. Russian Laser Research 28, 244 (2007). [9] A. Yu. Khrennikov, EPR ”Paradox”, projection postulate, time synchronization “nonlocality”. Int. J. Quantum Information (IJQI), 7, N 1, 71 - 8 (2009), [10] J. von Neumann, Matematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932). [11] G. L¨uders, Ann. Phys., Lpz 8 322 (1951). [12] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935). 8

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