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arXiv:quant-ph/0304077 · Physical Review A

von Neumann measurement is optimal for detecting linearly independent mixed quantum states

Yonina C. Eldar

Published Nov 3, 2003• 41 citations
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Abstract

We consider the problem of designing a measurement to minimize the probability of a detection error when distinguishing between a collection of possibly nonorthogonal mixed quantum states. We show that if the quantum state ensemble consists of linearly independent density operators, then the optimal measurement is an orthogonal von Neumann measurement consisting of mutually orthogonal projection operators, and not a more general positive operator-valued measure.

DOI
10.1103/physreva.68.052303

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We consider the problem of designing a measurement to minimize the probability of a detection error when distinguishing between a collection of possibly nonorthogonal mixed quantum states. We show that if the quantum state ensemble consists of linearly independent density operators, then the optimal measurement is an orthogonal von Neumann measurement consisting of mutually orthogonal projection operators, and not a more general positive operator-valued measure.
We consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters (1991) and Hausladen et al. (1996), where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. (see Int. J. Theor. Phys., vol.36, p.1269-88, 1997).
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.
In quantum mechanics, the definition of a Von Neumann measurement can be generalized using positive-operator-valued measures. This modified definition of a quantum measurement allows one to better distinguish between a set of nonorthogo-nal quantum states. In this thesis we examine a quantum detection problem, where we have a physical system whose state is limited to be in only one of a finite number of possibilities. These possible states are not necessarily orthogonal. We want to find the best method of measuring the system in order to distinguish which state the system is in. Mathematically, we want to find a positive-operator-valued measure that minimizes the probability of a detection error. It is shown that all tight-frames with frame constant 1 correspond to positive-operator-valued measures. We reformulate the problem in terms of tight-frames that minimize the detection error. In the finite dimensional case, the problem of finding the tight-frame that minimizes the error can be converted into a Hamiltonian system on the group SO(N). The minimum energy solutions of this Hamiltonian system correspond exactly to the tight-frames that minimize the detection error. In this setting, several numerical methods can be applied to give numerical constructions of the desired tight-frames.
The calculation method proposed here obtains an optimum detection process for an M-ary signal whose quantum states are linearly independent and nonorthogonal. Optimizations are carried out by minimizing the average probability of error or by maximizing the mutual information in the signal detection process. Quantum detection operators, in terms of which the signal detection process is mathematically described, are assumed to be projection operators defined on a Hilbert space H s spanned by the quantum states of the signal; that is, the best projection operators that describe the signal detection process are obtained. The optimization procedure is equivalent to finding an appropriate unitary transformation in the Hilbert space H s. For a binary quantum state signal, the optimum detection processes in which the average probability of error is minimized and in which the mutual information is maximized are obtained. It is found that the optimum detection process minimizing the average probability of error is the same as that maximizing the mutual information.
We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that minimizes the probability of an inconclusive result. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these states share the same symmetries, and can be computed very efficiently by solving a reduced size semidefinite program.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.