Square-root measurement is a very useful suboptimal measurement in many applications. It was shown that the square-root measurement minimizes the squared error for pure states. In this paper, the least squared error problem is reformulated and a new proof is provided. It is found that the least squared error depends only on the average density operator of the input states. The properties of the least squared error are then discussed, and it is shown that if the input pure states are uniformly distributed, the average probability of error has an upper bound depending on the least squared error, the rank of the average density operator, and the number of the input states. The aforementioned properties help explain why the square-root measurement can be effective in decoding processes.
Square-root measurement is a very useful suboptimal measurement in many applications. It was shown that the square-root measurement minimizes the squared error for pure states. In this paper, the least squared error problem is reformulated and a new proof is provided. It is found that the least squared error depends only on the average density operator of the input states. The properties of the least squared error are then discussed, and it is shown that if the input pure states are uniformly distributed, the average probability of error has an upper bound depending on the least squared error, the rank of the average density operator, and the number of the input states. The aforementioned properties help explain why the square-root measurement can be effective in decoding processes.
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).
Distinguishing assigned quantum states with assigned probabilities via quantum measurements is a crucial problem for the transmission of classical information through quantum channels. Measurement operators maximizing the probability of correct discrimination have been characterized by Helstrom, Holevo, and Yuen since the 1970s. On the other hand, closed-form solutions are available only for particular situations enjoying high degrees of symmetry. As a suboptimal solution to the problem, measurement operators, directly determined from states and probabilities and known as square-root measurements (SRMs), were introduced by Hausladen and Wootters. These operators were also recognized to be optimal for pure states equipped with geometrical uniform symmetry (GUS). In this paper we discuss the optimality of the SRM and find necessary and sufficient conditions in order that SRM maximize the correct decision probabilities for the set of states formed by several constellations of GUS states. The results are applied to some specific examples concerning double constellations of quantum phase-shift keying and pulse-position modulation states, with possible applications to practical systems of quantum communications.
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.
Given a quantum pure state chosen from a set with some a priori probabilities, what is the optimal measurement needed to correctly guess the given state? We show that a good choice is the family of square-root or ``pretty good'' measurements, as each measurement in the family is optimal for at least one discrimination problem with the same quantum states but possibly different a priori probabilities. Furthermore, the map from measurement to discrimination problems can be explicitly described. In fact, for linearly independent states, every pair of discrimination problem and optimal measurement can be explicitly generated this way.
We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.
The Uhlmann fidelity F(ρ0,ρ1)=tr∣ρ0ρ1∣ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error ε requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of ρ0 and ρ1. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on 1/ε. However, the known optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure.
In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of trA(∣ψ0⟩⟨ψ1∣) (or its inverse) to the reference register R of the purification ∣ψ1⟩ (or ∣ψ0⟩) on the registers A and R, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.