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Physical Review A

Spin-measurement retrodiction

Shahar Ben‐Menahem

Published Feb 1, 1989• 17 citations
Publisher

Abstract

We address the problem of choosing an initial state and final measurement that enable maximal retrodiction of the result of an unknown intermediate spin measurement of a spin-(1/2 particle, along any one of a given set of directions. Details are given of the derivation of the most general possible solution of the problem, which we reported recently. The maximal retrodiction is shown to be the determination of the result of the intermediate spin measurement, along any of four different directions that span three-dimensional space and satisfy a specific linear relation.

DOI
10.1103/physreva.39.1621

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We address the problem of choosing an initial state and final measurement that enable maximal retrodiction of the result of an unknown intermediate spin measurement of a spin-(1/2 particle, along any one of a given set of directions. Details are given of the derivation of the most general possible solution of the problem, which we reported recently. The maximal retrodiction is shown to be the determination of the result of the intermediate spin measurement, along any of four different directions that span three-dimensional space and satisfy a specific linear relation.
We study how well we can retrodict results of measurements made on a quantum system if we can make measurements on its final state. We know what measurements were made, but not their results. An initial examination shows that we can gain anywhere from no information to perfect information about the results of previous measurements, depending on the measurements and the initial state of the system. The case of two two-outcome measurements, the second of which is a projective measurement, is examined in some detail. We then look at a model of a qubit interferometer in which measurements are made in order to determine the path the qubit followed. The measurement made on the final state of the qubit depends on the information about previous measurement results that we are trying to determine. One can attempt to find the result of just one of the measurements or all of them or to find a measurement sequence that was not realized. We study all three possibilities.
If a generalized measurement is performed on a quantum system and we do not know the outcome, are we able to retrodict it with a second measurement? We obtain a necessary and sufficient condition for perfect retrodiction of the outcome of a known generalized measurement, given the final state, for an arbitrary initial state. From this, we deduce that, when the input and output Hilbert spaces have equal (finite) dimension, it is impossible to perfectly retrodict the outcome of any fine-grained measurement [where each positive, operator-valued measure (POVM) element corresponds to a single Kraus operator] for all initial states unless the measurement is unitarily equivalent to a projective measurement. It also enables us to show that every POVM can be realized in such a way that perfect outcome retrodiction is possible for an arbitrary initial state when the number of outcomes does not exceed the output Hilbert space dimension. We then consider the situation where the initial state is not arbitrary, though it may be entangled, and describe the conditions under which unambiguous outcome retrodiction is possible for a fine-grained generalized measurement. We find that this is possible for some state if the Kraus operators are linearly independent. This condition is also necessary when the Kraus operators are nonsingular. From this, we deduce that every trace-preserving quantum operation is associated with a generalized measurement whose outcome is unambiguously retrodictable for some initial state, and also that a set of unitary operators can be unambiguously discriminated iff they are linearly independent. We then examine the issue of unambiguous outcome retrodiction without entanglement. This has important connections with the theory of locally linearly dependent and locally linearly independent operators.
A measurement procedure on a spin 1/2 system is described, whose effect on the (possibly unknown) state of the system can be reversed, by means of other similar measurements, with a sizable probability of success Prev{P}^{\mathrm{rev}}Prev (e.g., 12<Prev<1\frac{1}{2}<{P}^{\mathrm{rev}}<121​<Prev<1, depending on the state). Repeated such measurements followed by reversal, on a single system, allow us to determine its (a priori unknown) state operator with a probability of success which is nonzero, though extremely small (because Prev<1{P}^{\mathrm{rev}}<1Prev<1).
The standard Bloch sphere representation has been recently generalized to describe not only systems of arbitrary dimension, but also their measurements, in what has been called the extended Bloch representation of quantum mechanics . This model, which offers a solution to the longstanding measurement problem, is based on the hidden-measurement interpretation of quantum mechanics , according to which the Born rule results from our lack of knowledge of the measurement interaction that each time is actualized between the measuring apparatus and the measured entity. In this article, we present the extended Bloch model and use it to investigate, more specifically, the nature of the quantum spin entities and of their relation to our three-dimensional Euclidean theater. Our analysis shows that spin eigenstates cannot generally be associated with directions in the Euclidean space, but only with generalized directions in the Blochean space, which apart from the special case of spin one-half entities, is a space of higher dimensionality. Accordingly, spin entities have to be considered as genuine non-spatial entities. We also show, however, that specific vectors can be identified in the Blochean theater that are isomorphic to the Euclidean space directions, and therefore representative of them, and that spin eigenstates always have a predetermined orientation with respect to them. We use the details of our results to put forward a new view of realism, that we call multiplex realism , providing a specific framework with which to interpret the human observations and understanding of the component parts of the world. Elements of reality can be represented in different theaters, one being our customary Euclidean space, and another one the quantum realm, revealed to us through our sophisticated experiments, whose elements of reality, in the quantum jargon, are the eigenvalues and eigenstates. Our understanding of the component parts of the world can then be guided by looking for the possible connections, in the form of partial morphisms, between the different representations, which is precisely what we do in this article with regard to spin entities.