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arXiv:quant-ph/0104008 · Physical Review A

Fidelity trade-off for finite ensembles of identically prepared qubits

Konrad Banaszek, Igor Devetak

Published Oct 10, 2001• 60 citations
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Abstract

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.

DOI
10.1103/physreva.64.052307

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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.
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.
In order to execute quantum algorithms on bosonic quantum computers, it is necessary to encode their inputs into a desired target quantum state, with a process called state preparation. In real settings, state preparation methods operate in stochastic scenarios, where the prepared state is represented as a mixed state, that is a probabilistic ensemble of pure states. Therefore, existing state preparation methods optimize a variational ansatz by averaging over all possible sequences of photon loss events contained in the prepared ensemble, yielding an output state which approximates the desired target with a certain average fidelity. Such an approach, however, does not take into account directly the distribution of fidelities across states in the ensemble. In this work, we aim to maximize the fidelity of only a subset of pure states sampled from the prepared ensemble, at the cost of reducing the average fidelity overall. This trade-off is relevant when the goal is to obtain high-fidelity states with a certain probability, rather than maximizing the average fidelity of the prepared ensemble. We borrow and adapt objective functions from risk-sensitive optimization, a framework that optimizes specific characteristics of a distribution rather than only its expectation, and that has been successfully applied in Reinforcement Learning. In the considered experimental settings, our results show that the infidelities of some pure states in the prepared ensemble can be reduced by up to four orders of magnitude when using risk-seeking objectives, compared to those achieved using the standard optimization of the average fidelity. Furthermore, we show that it is possible to control the trade-off between the probability of obtaining high-fidelity states and the lowest infidelity threshold reached, providing the flexibility to select the operating point that best suits a certain application.
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.
Present-day quantum devices require precise implementation of desired quantum channels. To characterize the quality of implementation one uses the average operation fidelity FFF, defined as the fidelity between an initial pure state and its image with respect to the analyzed operation, averaged over an ensemble of pure states. We analyze the statistical properties of the operation fidelity for low-dimensional channels and study its extreme values and probability distribution. These results are obtained with the help of the joint numerical range of the set of Kraus operators representing a channel. Analytic expressions for the density P(F)P(F)P(F) are derived in some particular cases including unitary and mixed unitary channels as well as quantum maps represented by commuting Kraus operators. We propose a scheme for fidelity-based channel estimation which uses the distribution of operation fidelity.
Direct fidelity estimation provides a way to estimate the fidelity between an experimentally prepared state and a desired pure target state without performing full tomography. Two influential formulations were introduced in 2011 by Flammia and Liu and by da Silva, Landon-Cardinal, and Poulin. In these protocols, the total estimation error is controlled through two distinct probabilistic steps: first, the fidelity is approximated using randomly sampled Pauli observables; second, each sampled expectation value is estimated from finitely many measurement outcomes. In this work we show that additional structural information about the noise can substantially sharpen the corresponding resource bounds. In particular, for some canonical channels the effective number of sampled Pauli settings can be reduced, leading to lower measurement cost both in the general pure-state setting and in the case of a stabilizer state. These results illustrate a broader point: worst-case confidence bounds in direct fidelity estimation can be significantly conservative when experimentally relevant structure is ignored. As a technical ingredient, we also revisit the allocation of the total accuracy and confidence budgets between the two probabilistic steps. Reformulating the analysis in terms of separate error parameters yields a constrained optimization problem whose solution lowers the average number of measurements in the general pure-state setting. Numerical simulations based on quantum circuits implemented in Qiskit illustrate both the improvement obtained under structured-noise assumptions and the conservativeness of the original worst-case bounds.