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arXiv:1312.5205 · Journal of Physics A Mathematical and Theoretical

Minimum-cost quantum measurements for quantum information

Petros Wallden, Vedran Dunjko, Erika Andersson

Published Mar 10, 2014• 18 citations
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Abstract

Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.

DOI
10.1088/1751-8113/47/12/125303

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Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.
The optimum quantum measurements that minimize the error probability for mixed‐state signals are considered. The square‐root measurement (SRM) is known to be the optimal measurement for many pure‐state signals, e.g., symmetric states, linear codes, etc. However, when the states are mixed, the SRM is no longer guaranteed to be optimal even if the states are symmetric. Here, we define the ‘pure’ square‐root measurement (pure‐SRM) which is the SRM for pure states and its optimality for mixed states. It is shown that the pure‐SRM is satisfied the necessary and sufficient conditions of the quantum Bayes strategy when the signals are mixed by transmission errors, and therefore the pure‐SRM is the optimal measurement.
A key concept of quantum information theory is that accessing information encoded in a quantum system requires us to discriminate between several possible states the system could be in. A natural generalization of this problem, namely, quantum sequence discrimination, appears in various quantum information processing tasks, the objective being to determine the state of a finite sequence of quantum states. Since such a sequence is a composite quantum system, the fundamental question is whether an optimal measurement is local, i.e., comprising measurements on the individual members, or collective, i.e., requiring joint measurement(s). In some known instances of this problem, the optimal measurement is local, whereas in others, it is collective. But, so far, a definite prescription based solely on the problem description has been lacking. In this paper, we prove that if the members of a given sequence are secretly and independently drawn from an ensemble or even from different ensembles, the optimum success probability is achievable by fixed local measurements on the individual members of the sequence, and no collective measurement is necessary. This holds for both minimum-error and unambiguous state discrimination paradigms.
Minimum probability of error (MPE) measurements are quantum mechanical measurements that discriminate between a set of candidate states and achieve the minimum error allowed by quantum mechanics. Conditions for a measurement to be an MPE measurement have been derived by Yuen, Kennedy and Lax. But finding explicit measurements that satisfy these conditions is a hard problem in general and even in cases where an explicit measurement is known, calculating the MPE of the measurement is not easy. In some cases, MPE measurements have been found such as when the states form a single orbit under a group action i.e., there is a transitive group action on the states. For such state sets, termed Geometrically Uniform (GU) in [1], it was shown that the `pretty good measurement' (PGM) is optimal. However, calculating the MPE and other performance metrics for the PGM involves inverting large matrices, and it is therefore not easy in general to evaluate. Our first contribution is a formula for the MPE and conditional probabilities of GU sets using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). CGU sets appear in many practical problems in quantum communication and imaging. For example, they include all linear codes formed using pure-state modulation constellations, which are known to achieve the ultimate (Holevo) capacity of optical communication. Optimal (MPE) measurement for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the optimal measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs-which is what is needed to solve for MPE measurements for an arbitrary set of pure states using the Yuen-Kennedy-Lax conditions [2]. We show examples of the evaluation of optimal measurements for CGU states.
We find the optimal measurement for distinguishing between symmetric multi-mode phase-randomized coherent states. A motivation for this is that phase-randomized coherent states can be used for quantum communication, including quantum cryptography. The so-called square-root measurement is optimal for pure symmetric states, but is not always optimal for mixed symmetric states. When phase-randomizing a multi-mode coherent state, the state becomes a mixture of pure multi-mode states with different total photon numbers. We find that the optimal measurement for distinguishing between any set of phase-randomised coherent states can be realised by first counting the total number of photons, and then distinguishing between the resulting pure states in the corresponding photon-number subspace. If the multi-mode coherent states we started from are symmetric, then the optimal measurement in each subspace is a square-root measurement. The overall optimal measurement in the cases we consider is also a square-root measurement. In some cases, we are able to present a simple linear optical circuit that realizes the overall optimal measurement.
Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.