Deterministic Quantum State Transformations

We derive a necessary condition for the existence of a completely-positive, linear, trace-preserving map which deterministically transforms one finite set of pure quantum states into another. This condition is also sufficient for linearly-independent initial states. We also examine the issue of quantum coherence, that is, when such operations maintain the purity of superpositions. If, in any deterministic transformation from one linearly-independent set to another, even a single, complete superposition of the initial states maintains its purity, the initial and final states are related by a unitary transformation.Comment: Minor cosmetic change

Distributed implementation of standard oracle operators

The standard oracle operator corresponding to a function f is a unitary operator that computes this function coherently, i.e. it maintains superpositions. This operator acts on a bipartite system, where the subsystems are the input and output registers. In distributed quantum computation, these subsystems may be spatially separated, in which case we will be interested in its classical and entangling capacities. For an arbitrary function f, we show that the unidirectional classical and entangling capacities of this operator are log_{2}(n_{f}) bits/ebits, where n_{f} is the number of different values this function can take. An optimal procedure for bidirectional classical communication with a standard oracle operator corresponding to a permutation on Z_{M} is given. The bidirectional classical capacity of such an operator is found to be 2log_{2}(M) bits. The proofs of these capacities are facilitated by an optimal distributed protocol for the implementation of an arbitrary standard oracle operator.Comment: 4.4 pages, Revtex 4. Submitted to Physical Review Letter

Quantum State Discrimination

There are fundamental limits to the accuracy with which one can determine the state of a quantum system. I give an overview of the main approaches to quantum state discrimination. Several strategies exist. In quantum hypothesis testing, a quantum system is prepared in a member of a known, finite set of states, and the aim is to guess which one with the minimum probability of error. Error free discrimination is also sometimes possible, if we allow for the possibility of obtaining inconclusive results. If no prior information about the state is provided, then it is impractical to try to determine it exactly, and it must be estimated instead. In addition to reviewing these various strategies, I describe connections between state discrimination, the manipulation of quantum entanglement, and quantum cloning. Recent experimental work is also discussed.Comment: Contemporary Physics 2000, in pres

Optimum Unambiguous Discrimination Between Linearly Independent Symmetric States

The quantum formalism permits one to discriminate sometimes between any set of linearly-independent pure states with certainty. We obtain the maximum probability with which a set of equally-likely, symmetric, linearly-independent states can be discriminated. The form of this bound is examined for symmetric coherent states of a harmonic oscillator or field mode.Comment: 9 pages, 2 eps figures, submitted to Physics Letters

Unambiguous Discrimination Between Linearly-Independent Quantum States

The theory of generalised measurements is used to examine the problem of discriminating unambiguously between non-orthogonal pure quantum states. Measurements of this type never give erroneous results, although, in general, there will be a non-zero probability of a result being inconclusive. It is shown that only linearly-independent states can be unambiguously discriminated. In addition to examining the general properties of such measurements, we discuss their application to entanglement concentration

Unambiguous Discrimination Between Linearly Dependent States with Multiple Copies

A set of quantum states can be unambiguously discriminated if and only if they are linearly independent. However, for a linearly dependent set, if C copies of the state are available, then the resulting C particle states may form a linearly independent set, and be amenable to unambiguous discrimination. We obtain necessary and sufficient conditions for the possibility of unambiguous discrimination between N states given that C copies are available and that the single copies span a D dimensional space. These conditions are found to be identical for qubits. We then examine in detail the linearly dependent trine ensemble. The set of C>1 copies of each state is a set of linearly independent lifted trine states. The maximum unambiguous discrimination probability is evaluated for all C>1 with equal a priori probabilities.Comment: 12 Pages RevTeX 4, 1 EPS figur