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Optical Fiber Communication Conference

Fidelity trade-off in quantum operations

Konrad Banaszek

Published Jan 1, 2001• 0 citations
Publisher

Abstract

It is a general quantum mechanical rule that the more information is obtained from an operation on a quantum system, the more its state has to be altered in course of such a procedure. In this talk, we analyze how this rule results in a trade-off between the quality of estimating the state of a quantum system and the disturbance of the original state. We show that the trade-off can be described by a tight analytical inequality quantifying these two effects in terms of mean fidelities. We apply this result to discuss the flow of information in teleportation protocols based on nonmaximally entangled states.

DOI
10.1364/icqi.2001.fqipa4

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It is a general quantum mechanical rule that the more information is obtained from an operation on a quantum system, the more its state has to be altered in course of such a procedure. In this talk, we analyze how this rule results in a trade-off between the quality of estimating the state of a quantum system and the disturbance of the original state. We show that the trade-off can be described by a tight analytical inequality quantifying these two effects in terms of mean fidelities. We apply this result to discuss the flow of information in teleportation protocols based on nonmaximally entangled states.
We calculate the trade-off between the quality of estimating the quantum state of an ensemble of identically prepared qubits and the minimum level of disturbance that has to be introduced by this procedure in quantum mechanics. The trade-off is quantified using two mean fidelities: the operation fidelity, which characterizes the average resemblance of the final qubit state to the initial one, and the estimation fidelity, describing the quality of the obtained estimate. We analyze properties of quantum operations saturating the achievability bound for the operation fidelity versus the estimation fidelity, which allows us to reduce substantially the complexity of the problem of finding the trade-off curve. The reduced optimization problem has the form of an eigenvalue problem for a set of tridiagonal matrices, and it may be easily solved using standard numerical tools.
The trade-off between the information gain and the state disturbance is derived for quantum operations on a single qubit prepared in a uniformly distributed pure state. The derivation is valid for a class of measures quantifying the state disturbance and the information gain which satisfy certain invariance conditions. This class includes in particular the Shannon entropy versus the operation fidelity. The central role in the derivation is played by efficient quantum operations, which leave the system in a pure output state for any measurement outcome. It is pointed out that the optimality of efficient quantum operations among those inducing a given operator-valued measure is related to Davies' characterization of convex invariant functions on hermitian operators.
It is of general interest how a quantum measurement may disturb a quantum system while it gives information on the state of the system. We study a trade-off relation between the information gain and the output fidelity for a quantum nondemolition measurement scheme for photon numbers. Toward this aim, we obtain general expressions for the information gain and the output fidelity for an arbitrary initial state. We particularly investigate how these two quantities vary with measurement strength for some specific classes of states, through a single measurement or successive measurements.
It is of general interest how a quantum measurement may disturb a quantum system while it gives information on the state of the system. We study a trade-off relation between the information gain and the output fidelity for a quantum nondemolition measurement scheme for photon numbers. Toward this aim, we obtain general expressions for the information gain and the output fidelity for an arbitrary initial state. We particularly investigate how these two quantities vary with measurement strength for some specific classes of states, through a single measurement or successive measurements.