A lower bound on the probability of error in quantum state discrimination

We give a lower bound on the probability of error in quantum state discrimination. The bound is a weighted sum of the pairwise fidelities of the states to be distinguished.Comment: 4 pages; v2 fixes typos and adds remarks; v3 adds a new referenc

Quantum target detection using entangled photons

We investigate performances of pure continuous variable states in discriminating thermal and identity channels by comparing their M-copy error probability bounds. This offers us a simplified mathematical analysis for quantum target detection with slightly modified features: the object -- if it is present -- perfectly reflects the signal beam irradiating it, while thermal noise photons are returned to the receiver in its absence. This model facilitates us to obtain analytic results on error-probability bounds i.e., the quantum Chernoff bound and the lower bound constructed from the Bhattacharya bound on M-copy discrimination error-probabilities of some important quantum states, like photon number states, N00N states, coherent states and the entangled photons obtained from spontaneous parametric down conversion (SPDC). Comparing the $M$-copy error-bounds, we identify that N00N states indeed offer enhanced sensitivity than the photon number state system, when average signal photon number is small compared to the thermal noise level. However, in the high signal-to-noise scenario, N00N states fail to be advantageous than the photon number states. Entangled SPDC photon pairs too outperform conventional coherent state system in the low signal-to-noise case. On the other hand, conventional coherent state system surpasses the performance sensitivity offered by entangled photon pair, when the signal intensity is much above that of thermal noise. We find an analogous performance regime in the lossy target detection (where the target is modeled as a weakly reflecting object) in a high signal-to-noise scenario.Comment: 8 pages, RevTex, 4 figure

Discrimination of quantum states under locality constraints in the many-copy setting

We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability and the optimal trade-off between the type-I and type-II errors, thus establishing the associated Chernoff, Stein, Hoeffding, and the strong converse exponents. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB.Comment: Comments are welcom

On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states

Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d x d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations.Comment: 11 pages; v2: Journal version; Minor errors fixed in Section I

Advances in the theory of channel simulation: from quantum communication to quantum sensing

In this thesis we investigate the fundamental limitations that the laws of the quantum nature impose on the performance of quantum communications, quantum metrology and quantum channel discrimination. In a quantum communication scenario, the typical tasks are represented by the simple transmission of quantum bits, the distribution of entangle- ment and the sharing of quantum secret keys. The ultimate rates for each of these protocols are given by the two-way quantum capacities of the quantum channel which are in turn defined by considering the most general adaptive strategies that can be implemented over the channel. To assess these quantum capacities, we combine the simulation of quantum channels, suitably generalized to systems of arbitrary dimension, with quantum telepor- tation and the relative entropy of entanglement. This procedure is called teleportation stretching. Relying on this, we are able to reduce any adaptive protocols into simpler block ones and to determine the tightest upper bound on the two-way quantum capacities. Re- markably, we also prove the existence of a particular class of quantum channel for which the lower and the upper bounds coincide. By employing a slight modification of the tele- portation scheme, allowing the two parties to share a multi-copy resource state, we apply our technique to simplify adaptive protocols for quantum metrology and quantum channel discrimination. In the first case we show that the modified teleportation stretching implies a quantum Cram ́er-Rao bound that follows asymptotically the Heisenberg scaling. In the second scenario we are able to derive the only known so far fundamental lower bound on the probability of error affecting the discrimination of two arbitrary finite-dimensional quantum channels

Discrimination of quantum states under locality constraints in the many-copy setting

We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability, thus establishing the associated Chernoff exponent. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite asymptotic separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB

Quantum statistical inference and communication

This thesis studies the limits on the performances of inference tasks with quantum data and quantum operations. Our results can be divided in two main parts. In the first part, we study how to infer relative properties of sets of quantum states, given a certain amount of copies of the states. We investigate the performance of optimal inference strategies according to several figures of merit which quantifies the precision of the inference. Since we are not interested in obtaining a complete reconstruction of the states, optimal strategies do not require to perform quantum tomography. In particular, we address the following problems: - We evaluate the asymptotic error probabilities of optimal learning machines for quantum state discrimination. Here, a machine receives a number of copies of a pair of unknown states, which can be seen as training data, together with a test system which is initialized in one of the states of the pair with equal probability. The goal is to implement a measurement to discriminate in which state the test system is, minimizing the error probability. We analyze the optimal strategies for a number of different settings, differing on the prior incomplete information on the states available to the agent. - We evaluate the limits on the precision of the estimation of the overlap between two unknown pure states, given N and M copies of each state. We find an asymptotic expansion of a Fisher information associated with the estimation problem, which gives a lower bound on the mean square error of any estimator. We compute the minimum average mean square error for random pure states, and we evaluate the effect of depolarizing noise on qubit states. We compare the performance of the optimal estimation strategy with the performances of other intuitive strategies, such as the swap test and measurements based on estimating the states. - We evaluate how many samples from a collection of N d-dimensional states are necessary to understand with high probability if the collection is made of identical states or they differ more than a threshold according to a motivated closeness measure. The access to copies of the states in the collection is given as follows: each time the agent ask for a copy of the states, the agent receives one of the states with some fixed probability, together with a different label for each state in the collection. We prove that the problem can be solved with O(pNd=2) copies, and that this scaling is optimal up to a constant independent on d;N; . In the second part, we study optimal classical and quantum communication rates for several physically motivated noise models. - The quantum and private capacities of most realistic channels cannot be evaluated from their regularized expressions. We design several degradable extensions for notable channels, obtaining upper bounds on the quantum and private capacities of the original channels. We obtain sufficient conditions for the degradability of flagged extensions of channels which are convex combination of other channels. These sufficient conditions are easy to verify and simplify the construction of degradable extensions. - We consider the problem of transmitting classical information with continuous variable systems and an energy constraint, when it is impossible to maintain a shared reference frame and in presence of losses. At variance with phase-insensitive noise models, we show that, in some regimes, squeezing improves the communication rates with respect to coherent state sources and with respect to sources producing up to two-photon Fock states. We give upper and lower bounds on the optimal coherent state rate and show that using part of the energy to repeatedly restore a phase reference is strictly suboptimal for high energies

Minimum-error discrimination between mixed quantum states

We derive a general lower bound on the minimum-error probability for {\it ambiguous discrimination} between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out a upper bound on the minimum-error probability for ambiguous discrimination of mixed quantum states. Therefore, some sufficient conditions are obtained for the minimum-error probability attaining this bound. Finally, under the condition of the minimum-error probability attaining this bound, we compare the minimum-error probability for {\it ambiguously} discriminating arbitrary $m$ mixed quantum states with the optimal failure probability for {\it unambiguously} discriminating the same states.Comment: A further revised version, and some results have been adde