The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of U(n) group, we construct the Jordan basis to derive the analytical results for both the optimal unambiguous discrimination and minimum-error discrimination. The POVM operators for unambiguous discrimination and orthogonal measurement operators for minimum-error discrimination are obtained. We find that the optimal failure probability and minimum-error probability for the discrimination between the mean input mixd states are dependent on the dimension of the unknown qudit states. We applied the approach to generalize the results of He and Bergou (Phys. Rev. A {\bf 75}, 032316 (2007)) from qubit to qudit case, and we further solve the problem of programmable dicriminators with arbitrary copies of unknown states in both program and data systems.
The universal programmable discriminator is a device for discrimination between unknown quantum states. It has two kinds of registers: the program register and the data register. The states that are stored in registers are all unknown. It is assured that the data state is identical with one of the program states with the certain probability. The aim is to optimally identify the state stored in the data register with one of the program states that it is done by discriminating between unknown states of the combined system contains program and data registers. We design an optimal universal unambiguous programmable discriminator that has
N qubit systems in the program register, to unambiguously identify the qubit state of data with one of
N program qubit states. All unknown qubit states are pure and
ND copies of the data state are available. We determine the optimal positive-operator valued measure (POVM) elements in two different ways. First, we find optimal POVMs that maximize the average of the overall success probability of discrimination between
N unknown pure states of the combined system. In another way, the problem of discrimination between
N unknown pure states of the combined system is reduced to discrimination between
N known average mixed states. It is shown that theses two different ways lead to the same results. We show that at least $N\ensuremath{-}1$ copies of the data state are necessary for discriminating between
N unknown states of the combined system.
Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: One in which the purity of the possible states is a priori known, and the fully universal one where the machine operates over generic mixed states of unknown purity. We find analytical results for both the unambiguous and minimum error discrimination strategies. This allows us to calculate the asymptotic performance of programmable discrimination machines when a large number of copies are provided and to recover the standard state discrimination and state comparison values as different limiting cases.
This work investigates which sets of quantum states give rise to the highest achievable success probability in minimum-error state discrimination if multiple copies of the unknown state are given. Specifically, we consider uniformly distributed ensembles of the form
{N1,ρi⊗k}i=1N, where
N states in dimension
d are provided in
k identical copies, and derive universal limits in this scenario. For pure state ensembles, we prove that whenever
N is large enough to support a state
k-design, these designs will exactly give rise to the maximally discriminable sets. We further show that when
N exceeds the size required for a
k-design, mixed states can outperform all pure state ensembles. We then recognise that the problem of most discriminable classical states in the multi-copy regime is in one-to-one correspondence to the concept of the multiplicative Bayes capacity of independent uses of classical channels, a concept that emerges naturally in the context of classical information leakage. This connection allows us to completely solve the classical analogue of our problem when
N≥(kd+k−1), and to prove that quantum systems offer a quadratic advantage (in number of copies
k) over classical ones. Then, we prove that this quantum over classical advantage is strongly reduced when one is restricted to real quantum states, more precisely, when
N≥k+1, pure real qubits only offer a constant advantage over classical bits. Finally, we introduce computational techniques to find sets of most discriminable ensembles and to obtain rigorous universal upper bounds on the maximal success probability for multi-copy state discrimination in cases that are analytically intractable.
A universal programmable discriminator can perform discrimination between two unknown states, and the optimal solution can be approached via discrimination between the two averages over the uniformly distributed unknown input pure states, as has been widely discussed in previous works. In this paper, we consider the success probabilities of the optimal universal programmable unambiguous discriminators when applied to the pure input states. More precisely, the analytic results of the success probabilities are derived with the expressions of the optimal measurement operators for the universal discriminators, and we find that the success probabilities are independent of the dimension
d while the amount of copies in the two program registers is equal. The success probability of the programmable unambiguous discriminator can asymptotically approach that of the usual unambiguous discrimination (state comparison) as the number of copies in the program registers (data register) goes to infinity.