We relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".
Quantum data hiding stores classical information in bipartite quantum states that are, in principle, perfectly distinguishable, yet remain almost indistinguishable without access to a quantum communication channel. Here, we investigate whether this limitation can be overcome when the communicating parties are assisted by additional quantum resources. We develop a general framework for state discrimination that unifies catalytic and memory-assisted local discrimination protocols and analyze their power to reveal hidden information. We prove that when the hiding states are separable, neither entangled catalysts nor quantum memory can increase the optimal discrimination probability, establishing the robustness of separable data-hiding schemes. In contrast, for some entangled states, a reusable quantum memory turns locally indistinguishable states into ones that can be discriminated almost perfectly. Our results delineate the fundamental limits of catalytic and memory-assisted state discrimination and identify separable encodings as a robust strategy for quantum data hiding.
One of the many interesting features of quantum nonlocality is that the states of a multipartite quantum system cannot always be distinguished as well by local measurements as they can when all quantum measurements are allowed. In this work, we characterize the distinguishability of sets of multipartite quantum states when restricted to separable measurements, those which contain the class of local measurements but nevertheless are free of entanglement between the component systems. We consider two quantities: the separable fidelity, a truly quantum quantity, which measures how well we can ``clone'' the input state, and the classical probability of success, which simply gives the optimal probability of identifying the state correctly. We obtain lower and upper bounds on the separable fidelity and give several examples in the bipartite and multipartite settings where these bounds are optimal. Moreover the optimal values in these cases can be attained by local measurements. We further show that for distinguishing orthogonal states under separable measurements, a strategy that maximizes the probability of success is also optimal for separable fidelity. We point out that the equality of fidelity and success probability does not depend on an using the optimal strategy, only on the orthogonality of the states. To illustrate this, we present an example where two sets (one consisting of orthogonal states and the other nonorthogonal states) are shown to have the same separable fidelity even though the success probabilities are different.
For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of
N subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to
1 as
N→∞ if the entropy of the probability distribution associated with the ensemble is less than
1/2. In other words, the nonlocality of the
N-fold ensemble asymptotically disappears if the probability distribution associated with each ensemble is concentrated. Furthermore, we show that the discrimination task is a remarkable counterexample of a fundamental open question in theoretical computer science, called a parallel repetition conjecture of interactive games with two classically communicating players. Measurements for the discrimination task include a projective measurement of one party represented by stabilizer states, which enable the other party to perfectly distinguish states that are sampled with high probability.
The minimum-error probability of ambiguous discrimination for two quantum states is the well-known {\it Helstrom limit} presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimum-error probability for ambiguously discriminating arbitrary
m quantum states. In this paper, we obtain a new lower bound on the minimum-error probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two states. Specifically, the main technical contributions are described as follows: (1) We derive a new lower bound on the minimum-error probability for ambiguous discrimination among arbitrary
m mixed quantum states with given prior probabilities, and we present a necessary and sufficient condition to show that this lower bound is attainable. (2) We compare this new lower bound with six other bounds in the literature in detail, and, in some cases, this bound is optimal. (3) It is known that if
m=2, the optimal inconclusive probability of unambiguous discrimination
QU and the minimum-error probability of ambiguous discrimination
QE between arbitrary given
m mixed quantum states have the relationship
QU≥2QE. In this paper, we show that, however, if
m>2, the relationship
QU≥2QE may not hold again in general, and there may be no supremum of
QU/QE for more than two states, which may also reflect an essential difference between discrimination for two-states and multi-states. (4) A number of examples are constructed.
A quantum analogue of the famous Blackwell Theorem in classical statistics has recently been proposed. Given two quantum channels A and B, a set of payoff functions have been proven to have values for B at least as high as they are for A if and only if there exists a quantum garbling channel E such that A=EB. When such a channel E exists, we can globally compare A and B in terms of their `noisiness'. We show that this method of channel noise comparison is equivalent to one obtained by considering the degradation of the distinguishability of states. Here, the channel A is said to be at least as noisy as the channel B if any ensemble of states, fed into each channel and possibly entangled with ancillae, emerges no more distinguishable from A than it does from channel B, where distinguishability is quantified by the minimum error discrimination probability. We also provide a novel application to eavesdropper detection in quantum cryptography.
Two pure orthogonal quantum states can be perfectly distinguished by sequential local action of multiple pairs of parties. However, this process typically leads to the complete dissolution of entanglement in the states being discriminated. We propose a protocol that allows an arbitrary number of pairs of parties to distinguish between any two orthogonal, entangled, two-qubit pure states using local quantum operations and classical communication, with a success probability greater than that of random guessing, while ensuring that at each step, the individual ensemble states retain a finite amount of entanglement. Our protocol employs the minimum-error state discrimination approach. For demonstrating the retention of entanglement in the ensemble states at each step, we use logarithmic negativity as well as the concept of entanglement witnessing. For a large family of sets of the two states, the success probability of discrimination can be as close as required to unity, while sustaining a finite amount of entanglement in each step.