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Institute of Physics eBooks

Quantum measurements

Paul Skrzypczyk, Daniel Cavalcanti

Published Mar 1, 2023• 0 citations
Publisher

Abstract

In chapter 4, we shift focus to quantum measurements. We will first discuss the problem of estimating which measurement a given measuring device is performing. We then study quantum state discrimination, the problem of determining—by performing a measurement—which state, out of a set of possibilities, a given source is producing. We show how to gain new insight into this problem by using the key concept of semidefinite programming duality, and how the tool of complementary slackness can be used to derive optimality conditions.

DOI
10.1088/978-0-7503-3343-6ch4

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In chapter 4, we shift focus to quantum measurements. We will first discuss the problem of estimating which measurement a given measuring device is performing. We then study quantum state discrimination, the problem of determining—by performing a measurement—which state, out of a set of possibilities, a given source is producing. We show how to gain new insight into this problem by using the key concept of semidefinite programming duality, and how the tool of complementary slackness can be used to derive optimality conditions.
Chapter 3 addresses one of most basic problems in quantum physics, how to determine the properties of quantum states produced by an uncharacterised source. There are numerous variants of this problem, the most well-known being quantum state tomography, that is, estimating exactly which state the source is emitting. In this chapter we review this and other variants of state estimation, including the so-called quantum marginal problem. This will allow us to study various aspects of semidefinite programs, including how to obtain certificates of feasibility and the prevalent notion of a ‘witness’.
This chapter is concerned with the measure of quantum states . This requires one to introduce the subtle notion of quantum measurement , an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, starting with Dirac notation , a formalism that is not only very elegant but is relatively simple to handle. The introduction of Dirac notation makes it possible to become familiar with the inner product for quantum states, as well as different properties for operators and states concerning projection , change of basis , unitary transformations , matrix elements , similarity transformations , eigenvalues and eigenstates , spectral decomposition and diagonal representation , matrix trace and density operator or matrix . The concept of density matrix makes it possible to provide a very first and brief hint of the analog of Shannon's entropy in the quantum world, referred to as von Neumann's entropy , to be further developed in Chapter 21. Once we have all the required tools, we can focus on quantum measurement and analyze three different types referred to as basis-state measurements , projection or von Neumann measurements , and POVM measurements. In particular, POVM measurements are shown to possess a remarkable property of unambiguous quantum state discrimination (UQSD), after which it is possible to derive “absolutely certain” information from unknown system states. The more complex case of quantum measurements in composite systems described by joint or tensor states is then considered.
We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.
We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing among a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.