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Conditional quantum state reduction in joint positive operator-valued measures.

Ignacio Pérez, Alfredo Luis

Published Jul 13, 2021• 0 citations
OpenAlex

Abstract

We investigate conditional quantum state reduction associated with positive operator-valued measures describing joint measurements of incompatible observables. To this end we invoke consistency with very basic rules of conditional probabilities, as formulated in the standard Kolmogorovian-Bayesian formulation of statistics. We find an interesting relation with the nonclassical character of the statistics.

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We investigate conditional quantum state reduction associated with positive operator-valued measures describing joint measurements of incompatible observables. To this end we invoke consistency with very basic rules of conditional probabilities, as formulated in the standard Kolmogorovian-Bayesian formulation of statistics. We find an interesting relation with the nonclassical character of the statistics.
We examine conditional probabilities derived from the joint noisy measurement of two complementary observables. We show that conditioning on one variable can lead to reduction of uncertainty in the other one, even completely eliminating uncertainty. We show that the conditional distribution cannot, in general, be represented in Born form using the marginal positive operator valued measure and any physical system state, nor can it be derived from typical rules of state reduction. We examine these issues in two basic quantum-optical schemes. These are double homodyne detection and a qubit measurement.
Abstract We investigate the consistency of conditional quantum probabilities. This is whether there is compatibility between the Kolmogorov–Bayes conditional probabilities and the Born rule. We show that they are not compatible in the sense that there are situations where there is no legitimate density matrix that may reproduce the conditional statistics of the other observable via the Born rule. This is to say that the Gleason theorem does not apply to conditional probabilities. Moreover, we show that when this occurs the joint statistics is nonclassical. We show that conditional probabilities are not equivalent to state reduction, so these results do not affect the validity of the Lüders expression.
A concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a posteriori states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.
Based on a recent proof of free choices in linking equations to the experiments they describe, I clarify relations among some purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and operator-valued measures), thereby allowing applications of these entities to the modeling of a wider variety of physical situations. Conditional probabilities associated with projection-valued measures are expressed by introducing conditional density operators, identical in some but not all cases to the usual reduced density operators. By lifting density operators to the extended Hilbert space featured in Neumark's theorem, I show an obstacle to extending conditional density operators to arbitrary positive operator-valued measures (POVMs); however, tensor products of POVMs are compatible with conditional density operators. By way of application, conditional density operators together with the free choice of probe particles allow the so-called postulate of state reductions to be replaced by a theorem. A second application demonstrates an equivalence between one form of quantum key distribution and another, allowing a formulation of individual eavesdropping attacks against transmitted-state BB84 to work also for entangled-state BB84.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.
Measurement incompatibility and the negativity of quasi-probability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill(MH) quasi-probability distribution associated with two dichotomic observables. { We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension ddd.} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension ddd, the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for nnn dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.