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Journal of Mathematical Physics

Knowledge about noncommuting quantum observables by means of Einstein–Podolsky–Rosen correlations

Giuseppe Nisticò, Maria Concetta Romania

Published Sep 1, 1994• 10 citations
Publisher

Abstract

It is well known that quantum physics forbids the simultaneous measurement of two noncommuting projection operators. However, sometimes there exists a third projection (mirror projection) which may be measured together with the second one and whose outcomes are completely correlated, in the Einstein–Podolsky–Rosen sense, with the outcomes of the first projection. In this article the states for which such a mirror projection exists are characterized and a procedure for constructing the mirror is given. In particular, it is shown that for two noncommuting projections there are always states for which the mirror projection exists, but also states for which it does not exist.

DOI
10.1063/1.530802

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It is well known that quantum physics forbids the simultaneous measurement of two noncommuting projection operators. However, sometimes there exists a third projection (mirror projection) which may be measured together with the second one and whose outcomes are completely correlated, in the Einstein–Podolsky–Rosen sense, with the outcomes of the first projection. In this article the states for which such a mirror projection exists are characterized and a procedure for constructing the mirror is given. In particular, it is shown that for two noncommuting projections there are always states for which the mirror projection exists, but also states for which it does not exist.
Over fifty years ago Einstein, Podolsky, and Rosen published a striking argument that quantum theory provides only an incomplete description of physical reality – that is, that some elements of physical reality fail to have a counterpart in quantum-theoretical description. What made this a challenge to the quantum-theoretical Weltanschauung was the very mild character of the sufficient condition for the reality of a physical quantity on which their argument hinged: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” The argument consists in pointing out that it is possible to construct a quantum-mechanical state ψ with the following properties: The state ψ describes two noninteracting systems (I and II). In almost all discussions the two systems are taken to be two particles, and the absence of relevant interaction is built in by taking ψ to assign negligible probability to finding the particles closer together than some macroscopically large distance. By measuring an observable A of system I, one can predict with certainty the result of a subsequent measurement of a corresponding observable P of system II. If, instead, one chooses to measure a different observable B of system I, one can predict with certainty the result of a subsequent measurement of a corresponding observable Q of system II. The observables P and Q are represented in quantum theory by noncommuting operators, which means that they cannot both have definite values.
Based on (i) the physical principle of local causality and (ii) a certain notion of elements of reality, Einstein, Podolsky, and Rosen put forward an argument showing that physical instances may exist in which two noncommuting observables can be joint elements of the physical reality. Here, we introduce a new criterion of joint reality. We demonstrate that according to this criterion, quantum mechanics generally prevents noncommuting observables from having joint elements of reality. In addition, we introduce a measure to quantify the extent to which the criterion is violated and explore the implications of such a measure in connection with incompatibility and correlations. Our findings suggest new manners of interpreting quantum phenomena.
Einstein-Podolsky-Rosen (EPR) steering refers to the ability of one observer to convince a distant observer that they share entanglement by making local measurements. Determining which states allow a demonstration of EPR steering remains an open problem in general. Here, we outline and demonstrate a method of analytically constructing new classes of two-qubit states which are nonsteerable by arbitrary projective measurements, from consideration of local operations performed by the steering party on states known to be nonsteerable.
The nonlocality of Einstein–Podolsky–Rosen (EPR) states is discussed in the framework of Weyl algebras. To include all possible limits of expectation values, the associated canonical cyclic representations of EPR states are considered. The momentum and position operators of individual particles do not exist in such representations, which explains the uncertainty principle in these cases. However, there are still one-to-one correspondences between the local Weyl operators associated with individual particles. The local observable algebras of individual particles are Type II1 von Neumann algebras and have numerous projection operators. The states obtained by applying local projection operations to the EPR states have perfect correlations between local subalgebras. Bell’s inequality is maximally violated by these states. The lack of atomic projections of Type II1 von Neumann algebras implies that the EPR states cannot be disentangled by local projection operations.