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arXiv:quant-ph/0501174 · Physical Review A

Unambiguous discrimination of special sets of multipartite states using local measurements and classical communication

Jihane Mimih, Mark S. Hillery

Published Jan 20, 2005• 13 citations
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Abstract

We initially consider a quantum system consisting of two qubits, which can be in one of two nonorthogonal states $\ensuremath{\mid}{\ensuremath{\Psi}}_{0}⟩$ and $\ensuremath{\mid}{\ensuremath{\Psi}}_{1}⟩$. We distribute the qubits to two parties, Alice and Bob. They each measure their qubits and then compare their measurement results to determine which state they were sent. This procedure is error-free, which implies that it must sometimes fail. In addition, no quantum memory is required; it is not necessary for one of the qubits to be stored until the result of the measurement on the other is known. We consider the cases in which, should failure occur, both parties receive a failure signal or only one does. In the latter case, if the two states share the same Schmidt basis, the states can be discriminated with the same failure probability that would be obtained if the qubits were measured together. This scheme is sufficiently simple that it can be generalized to multipartite qubit, and qudit, states. Applications to quantum secret sharing are discussed. Finally, we present an optical scheme to experimentally realize the protocol in the case of two qubits.

DOI
10.1103/physreva.71.012329

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We initially consider a quantum system consisting of two qubits, which can be in one of two nonorthogonal states \ensuremath∣\ensuremathΨ0⟩\ensuremath{\mid}{\ensuremath{\Psi}}_{0}⟩\ensuremath∣\ensuremathΨ0​⟩ and \ensuremath∣\ensuremathΨ1⟩\ensuremath{\mid}{\ensuremath{\Psi}}_{1}⟩\ensuremath∣\ensuremathΨ1​⟩. We distribute the qubits to two parties, Alice and Bob. They each measure their qubits and then compare their measurement results to determine which state they were sent. This procedure is error-free, which implies that it must sometimes fail. In addition, no quantum memory is required; it is not necessary for one of the qubits to be stored until the result of the measurement on the other is known. We consider the cases in which, should failure occur, both parties receive a failure signal or only one does. In the latter case, if the two states share the same Schmidt basis, the states can be discriminated with the same failure probability that would be obtained if the qubits were measured together. This scheme is sufficiently simple that it can be generalized to multipartite qubit, and qudit, states. Applications to quantum secret sharing are discussed. Finally, we present an optical scheme to experimentally realize the protocol in the case of two qubits.
We consider a quantum system of two nonorthogonal bipartite quantum states. We distribute the qubits between two parties, Alice and Bob. They each measure their qubits and then compare their measurement results to determine which state they were sent. This procedure is error-free, which implies that it must sometimes fail. In addition, no quantum memory is required; it is not necessary for one of the qubits to be stored until the result of the measurement on the other is known. We consider the cases in which, should a failure occur, both parties receive a failure signal or only one does. In the latter case, if the two states share the same Schmidt basis, the states can be discriminated with the same failure probability as would be obtained if the qubits were measured together. This scheme is sufficiently simple that it can be generalized to multipartite qubit and qudit states. Applications to quantum secret sharing are discussed.
The problem of unambiguous state discrimination consists of determining in which of a set of known quantum states a particular system is. One is allowed to fail but not to make a mistake. The optimal procedure is the one with the lowest failure probability. This procedure has been extended to bipartite states where the two parties Alice and Bob are allowed to manipulate their particles locally and communicate classically in order to determine which of two possible two-particle states they have been given. The failure probability of this local procedure is the same as if the two particles were together in the same location. Here we examine the effect of restricting the classical communication between the parties, either allowing none or eliminating the possibility that one party's measurement depends on the result of the other party's. These issues are studied for two-qubit states, and optimal procedures are found. In some cases the restrictions cause increases in the failure probability, but in other cases they do not. Applications of this procedure, in particular to secret sharing, are discussed.
Abstract We discuss a scheme in which sequential state-discrimination measurements are performed on qudits to determine the quantum state in which they were initially prepared. The qudits belong to a set of nonorthogonal quantum states and hence cannot be distinguished with certainty. Unambiguous state discrimination allows error-free measurements at the expense of occasionally failing to give a conclusive answer about the state of the qudit. Qudits have the potential to carry more information per transmission than qubits. We considered the situation in which Alice sends one of N qudits, where the dimension of the qudits is also N . We looked at two cases, one in which the states all have the same overlap and one in which the qudits are divided into two sets, with qudits in different sets having different overlaps. We also studied the robustness of our scheme against a simple eavesdropping attack and found that by using qudits rather than qubits, there is a greater probability that an eavesdropper will introduce errors and be detected.
We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination measurement, we never make a mistake, i.e., misidentify the state, but the measurement may fail, in which case we gain no information about which state was sent. We find that there is a nonzero probability for both Bob and Charlie to identify the state, and we maximize this probability. The probability that Charlie's measurement succeeds depends on how much information about the state Alice sent is left in the qubit after Bob's measurement, and this information can be quantified by the overlap between the two possible states in which Bob's measurement leaves the qubit. This Letter is a first step toward developing a theory of nondestructive sequential quantum measurements, which could be useful in quantum communication schemes.
One notion of nonlocality in quantum theory is the fact that information may be encoded in a composite system in such a way that it is not accessible through local measurements, even with the assistance of classical communication. Thus, contrary to the classical case, there exists information in quantum many-body systems which cannot be accessed locally. We show, however, that, remarkably, two-dimensional subspaces do not have this property: Any physically allowed measurement on information encoded in any two-dimensional subspace, regardless of entanglement or multipartite structure, may be performed locally. Further, this requires only local measurement and feed-forward of classical information, readily achievable in many experimental platforms. As an application to quantum secret sharing we suggest a twist on a well-known quantum information splitting protocol, which ensures that no receiving party ever has access to the full state sent, but parties must work together to perform measurements on the state. These results may have practical applications to the measurement of encoded qubits in, e.g., quantum secret sharing, quantum error correction, and reveal a fundamental property unique to two-dimensional subspaces.
We consider one copy of a quantum system prepared in one of two nonorthogonal pure product states of multipartite distributed among separated parties. We show that there exist protocols that achieve optimal conclusive discrimination by means of local operations and classical communications as good as by global measurements. Also, we show a protocol that minimizes the average number of local operations. Our result implies that two pure product multipartite states might not have the nonlocal property although more than two can have.