Experimentally efficient methods for estimating the performance of quantum measurements

Efficient methods for characterizing the performance of quantum measurements are important in the experimental quantum sciences. Ideally, one requires both a physically relevant distinguishability measure between measurement operations and a well-defined experimental procedure for estimating the distinguishability measure. Here, we propose the average measurement fidelity and error between quantum measurements as distinguishability measures. We present protocols for obtaining bounds on these quantities that are both estimable using experimentally accessible quantities and scalable in the size of the quantum system. We explain why the bounds should be valid in large generality and illustrate the method via numerical examples.Comment: 20 pages, 1 figure. Expanded details and typos corrected. Accepted versio

A Theoretical Analysis of NDCG Type Ranking Measures

A central problem in ranking is to design a ranking measure for evaluation of ranking functions. In this paper we study, from a theoretical perspective, the widely used Normalized Discounted Cumulative Gain (NDCG)-type ranking measures. Although there are extensive empirical studies of NDCG, little is known about its theoretical properties. We first show that, whatever the ranking function is, the standard NDCG which adopts a logarithmic discount, converges to 1 as the number of items to rank goes to infinity. On the first sight, this result is very surprising. It seems to imply that NDCG cannot differentiate good and bad ranking functions, contradicting to the empirical success of NDCG in many applications. In order to have a deeper understanding of ranking measures in general, we propose a notion referred to as consistent distinguishability. This notion captures the intuition that a ranking measure should have such a property: For every pair of substantially different ranking functions, the ranking measure can decide which one is better in a consistent manner on almost all datasets. We show that NDCG with logarithmic discount has consistent distinguishability although it converges to the same limit for all ranking functions. We next characterize the set of all feasible discount functions for NDCG according to the concept of consistent distinguishability. Specifically we show that whether NDCG has consistent distinguishability depends on how fast the discount decays, and 1/r is a critical point. We then turn to the cut-off version of NDCG, i.e., NDCG@k. We analyze the distinguishability of NDCG@k for various choices of k and the discount functions. Experimental results on real Web search datasets agree well with the theory.Comment: COLT 201

Macroscopicity of quantum superpositions on a one-parameter unitary path in Hilbert space

We analyze quantum states formed as superpositions of an initial pure product state and its image under local unitary evolution, using two measurement-based measures of superposition size: one based on the optimal quantum binary distinguishability of the branches of the superposition and another based on the ratio of the maximal quantum Fisher information of the superposition to that of its branches, i.e., the relative metrological usefulness of the superposition. A general formula for the effective sizes of these states according to the branch distinguishability measure is obtained and applied to superposition states of $N$ quantum harmonic oscillators composed of Gaussian branches. Considering optimal distinguishability of pure states on a time-evolution path leads naturally to a notion of distinguishability time that generalizes the well known orthogonalization times of Mandelstam and Tamm and Margolus and Levitin. We further show that the distinguishability time provides a compact operational expression for the superposition size measure based on the relative quantum Fisher information. By restricting the maximization procedure in the definition of this measure to an appropriate algebra of observables, we show that the superposition size of, e.g., N00N states and hierarchical cat states, can scale linearly with the number of elementary particles comprising the superposition state, implying precision scaling inversely with the total number of photons when these states are employed as probes in quantum parameter estimation of a 1-local Hamiltonian in this algebra

On measures of non-Markovianity: divisibility vs. backflow of information

We analyze two recently proposed measures of non-Markovianity: one based on the concept of divisibility of the dynamical map and the other one based on distinguishability of quantum states. We provide a toy model to show that these two measures need not agree. In addition, we discuss possible generalizations and intricate relations between these measures.Comment: 7 pages; new section is adde

Hilbert's projective metric in quantum information theory

We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or PPT operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under LOCC protocols or other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.Comment: 32 pages including 3 appendices and 3 figures; v2: minor changes, published versio

Estimating distinguishability measures on quantum computers

The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper, we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as ``quantum combs''). The fidelity-based algorithms offer novel physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate many of these algorithms by using a variational approach with parameterized quantum circuits. We find that the simulations converge well in both the noiseless and noisy scenarios, for all examples considered. Furthermore, the noisy simulations exhibit a parameter noise resilience. Finally, we establish a strong relationship between various quantum computational complexity classes and distance estimation problems.Comment: v3: 45 pages, 17 figures, includes new complexity-theoretic results, showing that several fidelity and distance estimation promise problems are complete for BQP, QMA, and QMA(2