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Research in Optical Sciences

Perfect Probabilistic Transformations between Symmetric Sets of Quantum States

Erika Andersson, Vedran Dunjko

Published Jan 1, 2012• 0 citations
Publisher

Abstract

We study probabilistic transforms between sets of quantum states. An example is a multiprobabilistic transform from symmetric coherent states to symmetric qubit states. We suggest an asymptotically optimal linear optical realization based on quantum scissors.
DOI
10.1364/hilas.2012.jt2a.21

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We investigate probabilistic transformations of quantum states from a `source' set to a `target' set of states. Such transforms have many applications. They can be used for tasks which include state-dependent cloning or quantum state discrimination, and as interfaces between systems whose information encodings are not related by a unitary transform, such as continuous-variable systems and finite-dimensional systems. In a probabilistic transform, information may be lost or leaked, and we explain the concepts of leak and redundancy. Following this, we show how the analysis of probabilistic transforms significantly simplifies for symmetric source and target sets of states. In particular, we give a simple linear program which solves the task of finding optimal transforms, and a method of characterizing the introduced leak and redundancy in information-theoretic terms. Using the developed techniques, we analyse a class of transforms which convert coherent states with information encoded in their relative phase to symmetric qubit states. Each of these sets of states on their own appears in many well studied quantum information protocols. Finally, we suggest an asymptotic realization based on quantum scissors.
The perfect NOT transformation, probabilistic perfect NOT transformation and conjugate transformation are studied. Perfect NOT transformation criteria on a quantum state set SSS of a qubit are obtained. Two necessary and sufficient conditions for realizing a perfect NOT transformation on SSS are derived. When these conditions are not satisfied we discuss a probabilistic perfect NOT transformation (gate). We construct a probabilistic perfect NOT machine (gate) by a general unitary-reduction operation. With a postselection of the measurement outcomes, the probabilistic NOT gate yields perfectly complements of the input states. We prove that one can realize probabilistically the NOT gate of the input states secretly chosen from a certain set S={∣Ψ1>,∣Ψ2>,...,∣Ψn>}S=\{|Ψ_1>, |Ψ_2>,..., |Ψ_n>\}S={∣Ψ1​>,∣Ψ2​>,...,∣Ψn​>} if and only if ∣Ψ1>,∣Ψ2>,...,|Ψ_1>, |Ψ_2>,...,∣Ψ1​>,∣Ψ2​>,..., and ∣Ψn>|Ψ_n>∣Ψn​> are linearly independent. We also generalize the probabilistic NOT transformation to the conjugate transformation in the multi-level quantum system. The lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.
Abstract We obtain the perfect NOT transformation criteria, two necessary and sufficient conditions for realizing a perfect NOT transformation on a quantum state set S of a qubit. Furthermore, we discuss a probabilistic perfect NOT transformation when there is no perfect NOT transformation on a state set S. We also generalize the perfect NOT transformation to the conjugate transformation in the multi-level quantum system and a lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.
We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explicitly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our examples reveal that the probabilistic cloner may yield higher fidelity than the best deterministic cloner even when the states that should be cloned are linearly dependent and are drawn from a continuous set.
Quantum state separation is a probabilistic map that transforms a given set of pure states into another set of more distinguishable ones. Here we investigate such a map acting onto uniparametric families of symmetric linearly dependent or independent quantum states. We obtained analytical solutions for the success probability of the maps---which is shown to be optimal---as well as explicit constructions in terms of positive operator valued measures. Our results can be used for state discrimination strategies interpolating continuously between minimum-error and unambiguous (or maximum-confidence) discrimination, which, in turn, have many applications in quantum information protocols. As an example, we show that quantum teleportation through a nonmaximally entangled quantum channel can be accomplished with higher probability than the one provided by unambiguous (or maximum-confidence) discrimination and with higher fidelity than the one achievable by minimum-error discrimination. Finally, an optical network is proposed for implementing parametric state separation.