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Journal of Physics A Mathematical and General

On state-dependent implication in quantum-mechanical distant correlations

Fedor Herbut

Published May 21, 1996• 4 citations
Publisher

Abstract

Let be an arbitrary given composite-system state (statistical operator). It is shown that same-subsystem events imply each other state-dependently according to if and only if they act equally in the range of the corresponding subsystem state (reduced statistical operator). Opposite-subsystem events imply each other in the same way if and only if they are twin events, i.e. . If is a pure state, it is shown that the anti-unitary correlation operator also plays a decisive role in the latter implication.

DOI
10.1088/0305-4470/29/10/015

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Let be an arbitrary given composite-system state (statistical operator). It is shown that same-subsystem events imply each other state-dependently according to if and only if they act equally in the range of the corresponding subsystem state (reduced statistical operator). Opposite-subsystem events imply each other in the same way if and only if they are twin events, i.e. . If is a pure state, it is shown that the anti-unitary correlation operator also plays a decisive role in the latter implication.
A composite quantum system consisting of two distant subsystems and described by a correlated state vector φ12 is considered. It was shown in a previous work by the authors [Ann. Phys. 96, 382 (1976)] that such a system can be equivalently described in terms of the reduced statistical operators ρ1 and ρ2 of φ12 applying to the subsystems and a correlation operator Ua between them. It is argued that this description has a firm physical foundation for the system considered in view of the fact that, on account of the subsystems being distant, one can only measure pairs of subsystem observables A1, B2 in coincidence. The direct measurement of A1 such that [A1,ρ1]=0 on the ensemble of first subsystems performs distantly (without interaction) an orthogonal decomposition of the ensemble of second subsystems ρ2, that amounts to the measurement of the twin observable A2 (A2≡UaA1U−1aQ2, Q2 being the range projector of ρ2). A number of coincidence experiments have confirmed this claim, and have disproved all attempts (on the quantum and on the subquantum levels) to view this decomposition of ρ2 as being present also before the measurement of A1. Hence, this decomposition into subensembles comes about in the very measurement of A1, and Ua determines them in a simple way. It is demonstrated that Ua is essential for twin observables and twin symmetry operators. A detailed study of these operators is presented from a unified point of view. Puzzling features of quantum correlations described by Ua show up in composite states when the mentioned distant decompositions of ρ2 into subensembles can be incompatible with one another. A general definition of such φEPR12 states (called Einstein–Podolsky–Rosen states) is given in a few equivalent forms, and the nonuniqueness of the Schmidt canonical form of φEPR12 is investigated in order to encourage further theoretical and experimental exploration of distant quantum correlations.
Twin observables are opposite subsystem observables A + and A - if they are indistinguishable in measurement in a given mixed or pure state ρ in the sense that the measurement of one amounts to the same as the measurement of the other (direct measurement and so-called distant measurement, respectively). It is pointed out that twin observables may reveal quantum correlations that give rise to the disappearance of interference in the two-slit experiment. Twin observables in general states are investigated in detail algebraically and geometrically. It is shown that there is a far-reaching correspondence between the detectable (in ρ) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and the direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires a singularity of ρ. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing ρ.
Opposite-subsystem twin events and twin observables, studied previously in the context of distant correlations, are first generalized to pure states of not-necessarily-composite systems, and afterwards they are further generalized to delayed twins that are due to unitary evolution of the quantum system. The versatile aspects of delayed twin observables are studied in terms of necessary and sufficient conditions to make possible various applications. Three of these are sketched: Preparation of some quantum experiments, easy solution of a puzzle in an important Scully et al. real experiment, and, finally, it is shown that exact measurement in quantum mechanics is an example of opposite-subsystem delayed twins in bipartite pure states.
As is well known, every mixed or pure state of a bipartite quantum system is given by a statistical operator, which determines, in terms of its two reduced statistical operators, the subsystem states. Necessary and sufficient conditions for the existence of a composite-system state, and, separately, for the possibility of its being correlated or uncorrelated in terms of the range dimensions of the three mentioned statistical operators are derived. As a corollary, it is shown that it cannot happen that two of the mentioned dimensions are finite and the third is infinite.
It is assumed that an arbitrary composite bipartite pure state in which the two subsystems are entangled is given, and it is investigated how the entanglement transmits the influence of measurement on only one of the subsystems to the state of the opposite subsystem. It is shown that any exact subsystem measurement has the same influence as ideal measurement on the opposite subsystem. In particular, the distant effect of subsystem measurement of a twin observable, i. e., so-called 'distant measurement', is always ideal measurement on the distant subsystem no matter how intricate the direct exact measurement on the opposite subsystem is.
Having the quantum correlations in a general mixed or pure bipartite state in mind, the part of information accessible by simultaneous measurement on both subsystems is shown never to exceed the part accessible by measurement on one subsystem, which, in turn is proved not to exceed the von Neumann mutual information. A particular pair of (opposite-subsystem) observables is shown to be responsible both for the amount of quasiclassical correlations and for that of the purely quantum entanglement in the pure-state case: the former via simultaneous subsystem measurements, and the latter through the entropy of coherence or of incompatibility, which is defined for the general case. The observables at issue are so-called twin observables. A general definition of the latter is given in terms of their detailed properties.