Probabilistic quantum multimeters

We propose quantum devices that can realize probabilistically different projective measurements on a qubit. The desired measurement basis is selected by the quantum state of a program register. First we analyze the phase-covariant multimeters for a large class of program states, then the universal multimeters for a special choice of program. In both cases we start with deterministic but erroneous devices and then proceed to devices that never make a mistake but from time to time they give an inconclusive result. These multimeters are optimized (for a given type of a program) with respect to the minimum probability of inconclusive result. This concept is further generalized to the multimeters that minimize the error rate for a given probability of an inconclusive result (or vice versa). Finally, we propose a generalization for qudits.Comment: 12 pages, 3 figure

CP^n, or, entanglement illustrated

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP^n as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP^3 and give a few new results concerning them.Comment: 28 pages, 9 figure

Experimental Demonstration of Optimal Unambiguous State Discrimination

We present the first full demonstration of unambiguous state discrimination between non-orthogonal quantum states. Using a novel free space interferometer we have realised the optimum quantum measurement scheme for two non-orthogonal states of light, known as the Ivanovic-Dieks-Peres (IDP) measurement. We have for the first time gained access to all three possible outcomes of this measurement. All aspects of this generalised measurement scheme, including its superiority over a standard von Neumann measurement, have been demonstrated within 1.5% of the IDP predictions

Distinguishing two-qubit states using local measurements and restricted classical communication

The problem of unambiguous state discrimination consists of determining which of a set of known quantum states a particular system is in. 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 has been shown to be the same as if the 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 these procedures, in particular to secret sharing, are discussed.Comment: 18 pages, two figure

Strategies and Networks for State-Dependent Quantum Cloning

State-dependent cloning machines that have so far been considered either deterministically copy a set of states approximately, or probablistically copy them exactly. In considering the case of two equiprobable pure states, we derive the maximum global fidelity of $N$ approximate clones given $M$ initial exact copies, where $N>M$. We also consider strategies which interpolate between approximate and exact cloning. A tight inequality is obtained which expresses a trade-off between the global fidelity and success probability. This inequality is found to tend, in the limit as $N{\to}{\infty}$, to a known inequality which expresses the trade-off between error and inconclusive result probabilities for state-discrimination measurements. Quantum-computational networks are also constructed for the kinds of cloning machine we describe. For this purpose, we introduce two gates: the distinguishability transfer and state separation gates. Their key properties are describedComment: 12 pages, 6 eps figures, submitted to Phys. Rev.

Nonlocality without inequalities has not been proved for maximally entangled states

Two approaches to extend Hardy's proof of nonlocality without inequalities to maximally entangled states of bipartite two-level systems are shown to fail. On one hand, it is shown that Wu and co-workers' proof [Phys. Rev. A 53, R1927 (1996)] uses an effective state which is not maximally entangled. On the other hand, it is demonstrated that Hardy's proof cannot be generalized by the replacement of one of the four von Neumann measurements involved in the original proof by a generalized measurement to unambiguously discriminate between non-orthogonal states.Comment: 7 pages, 2 figures. To appear in Phys. Rev.

Maximal Entanglement, Collective Coordinates and Tracking the King

Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime $\ne 2$. The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics, notions used are reviewed in an attempt to have the paper self contained.Comment: 8. arXiv admin note: substantial text overlap with arXiv:1206.3884, arXiv:1206.035

An expectation value expansion of Hermitian operators in a discrete Hilbert space

We discuss a real-valued expansion of any Hermitian operator defined in a Hilbert space of finite dimension N, where N is a prime number, or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement of the discrete Wigner function. We expect the expansion to be of use in quantum information applications since qubits typically are represented by a discrete, and finite-dimensional physical system of dimension N=2^p, where p is the number of qubits involved. As a particular example we use the expansion to prove that an intermediate measurement basis (a Breidbart basis) cannot be found if the Hilbert space dimension is 3 or 4.Comment: A mild update. In particular, I. D. Ivanovic's earlier derivation of the expansion is properly acknowledged. 16 pages, one PS figure, 1 table, written in RevTe