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arXiv:quant-ph/0003123 · Physical Review Letters

Fidelity Balance in Quantum Operations

Konrad Banaszek

Published Feb 12, 2001• 165 citations
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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 the 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.

DOI
10.1103/physrevlett.86.1366

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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 the 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.
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.
We derive an analytical formula for the optimal trade-off between the mean estimation fidelity and the mean fidelity of the qubit state after a partial measurement on NNN identically prepared qubits. We also conjecture an analytical expression for the optimal fidelity trade-off in case of a partial measurement on NNN identical copies of a ddd-level system.
In quantum mechanics, there is no measurement process that could gain some information of an unknown quantum state without causing any disturbance. A tradeoff bound between the amount of information gain and the concomitant disturbance in the measurement process of a bipartite entangled state is actually ingrained. Such a bound is fundamental and closely connected with the entangled degree b. In this work, the bound for estimation of a partial entangled state with a local strategy is investigated. It is shown that, with local operation with classical communication, a monotonic change in the picture will be spotted. This is due to the fact that the partial entanglement gradually becomes two individual qubits and, consequently, the optimal operation boils down to local operations. A quantum circuit which achieves the optimal tradeoff is also obtained.