Fidelity Balance in Quantum Operations

Abstract

I derive a tight bound between the quality of estimating the state of a single copy of a d-level system, and the degree the initial state has to be altered in course of this procedure. This result provides a complete analytical description of the quantum mechanical trade-off between the information gain and the quantum state disturbance expressed in terms of mean fidelities. I also discuss consequences of this bound for quantum teleportation using nonmaximally entangled states.PACS numbers: 03.67.-a, 03.65.BzAs a general rule, the more information is obtained from an operation on a quantum system, the more its state has to be altered. This heuristic statement was first exemplified by the Heisenberg microscope gedankenexperiment [1], where the spatial resolution of the apparatus was shown to scale inversely with the uncertainty of the momentum transfered during the observation. Presently, the disturbance caused by the information gain has become an issue of practical significance, as it underlies the security of quantum key distribution [2].The balance between the information gain and the state disturbance attracts currently a lot of interest, particularly in the context of quantum cryptography [3]. Information theory provides a selection of concepts to quantify both the information gain and the state disturbance. The choice of measures for these two effects is usually dictated by the relevance to a specific application. In most cases, however, derivation of the actual balance represents a highly nontrivial task, especially if one is tempted to resign from numerical means. The purpose of this Letter is to present a formulation of the information gain versus state disturbance trade-off which is completely solvable using elementary analytical techniques. This formulation is motivated by recent works on quantum state estimation [4], where the information obtained from the operation is converted into an estimate for the initial state of the system.The problem considered in this Letter can be formulated as follows. Suppose we are given a single d-level particle in a completely unknown pure state |ψ . We want to make a guess about the quantum state of this particle, but at the same time we would like to alter the state as little as possible. One can associate two fidelities with such a procedure. The first one, which we will denote by F , describes how much the state after the operation resembles the original one. The second fidelity, denoted by G, characterizes the average quality of our guess. It is natural to expect that these two quantities cannot take simultaneously too large values. What is the actual quantitative bound between them?Two extreme cases are well known: if nothing is done to the particle we have F = 1, but then our guess about the state of the particle has to be random, which yields G = 1/d. On the other hand, the optimal estimation strategy for a single copy [5] yields G = 2/(d + 1), but then the particle after the operation cannot provide any more information on the initial state; thus also F = 2/(...

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