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International Conference on Quantum Information

No Advantage to Entanglement in Bit Flip Parameter Estimation

David C. Collins, Michael R. Frey

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We consider optimal estimation of the parameter describing a bit-flip channel. Using the quantum Fisher information as a measure of the accuracy of the parameter estimation, we show that entanglement offers no advantage for multiple uses of the channel. This contrasts with parameter estimation in depolarizing channels, where entanglement offers a modest advantage and unitary channels where entanglement offers a distinct advantage.

DOI
10.1364/icqi.2011.qmi34

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We consider optimal estimation of the parameter describing a bit-flip channel. Using the quantum Fisher information as a measure of the accuracy of the parameter estimation, we show that entanglement offers no advantage for multiple uses of the channel. This contrasts with parameter estimation in depolarizing channels, where entanglement offers a modest advantage and unitary channels where entanglement offers a distinct advantage.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the kkk-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most kkk, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗k^*k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.
We consider protocols for estimating the parameter which characterizes a single qubit depolarizing channel, with the goal of attaining the most accurate estimate per channel use. The accuracy of any quantum estimation protocol will be quantified via the quantum Fisher information (QFI) since the Cramer-Rao bound implies thata larger QFI yields a smaller lower bound on the possible variance in any estimate of the parameter. Within this framework, the choice of input state prior to channel invocation affects estimation accuracy. The known optimal estimation scheme uses pairs of maximally entangled pure input states, giving a gain over any protocol using unentangled input states. Entangling more than two qubits initially in apure state provides no further gain in the QFI per channel use. We ask if,when the available input states are not pure,these gains persist and if correlating more than two qubits is advantageous. We present a protocol using input states correlated over any number of qubits and compare this to an independent channel useprotocol using uncorrelated mixed states. We show that the correlated state protocol yields gains in the QFI per channel use for certain physically reasonable parameter ranges. We show that, unlike the pure state case, using more than two correlated qubits can be advantageous and we show that, as the initial qubit states become highly mixed, adding additional correlated qubits can provide substantial gains in estimation accuracy. We show that for two qubits, such gainsare attained even when the state prior to channel invocationare separable. Thedepolarizing channelmaps a qubit as
We consider estimating the parameter associated with the qubit depolarizing channel when the available initial states that might be employed are mixed. We use quantum Fisher information as a measure of the accuracy of estimation to compare protocols which use collections of qubits in product states to one in which the qubits are in a correlated state. We show that, for certain parameter values and initial states, the correlated-state protocol can yield a greater accuracy per channel invocation than the product-state protocols. We show that, for some parameters and initial states, using more than two qubits and channel invocations is advantageous. These results stand in contrast to the known optimal case that uses pure initial states and a single channel invocation on a pair of entangled qubits.
We ask whether the optimal probe is entangled, and if so, what is its character and amount, for estimating the noise parameter of a large class of local quantum encoding processes that we refer to as vector encoding, examples of which include the local depolarizing and bit-flip channels. We first establish that vector encoding is invariably ``continuously commutative'' for optimal probes. We utilize this result to deal with the queries about entanglement in the optimal probe. We show that for estimating noise extent of the two-party arbitrary-dimensional local depolarizing channel, there is a descending staircase of optimal-probe entanglement for increasing depolarizing strength. For the multi-qubit case, the analysis again leads to a staircase, but which can now be monotonic or not, depending on the multiparty entanglement measure used. We also find that when sufficiently high depolarizing noise is to be estimated, fully product multiparty states are the only choice for being optimal probes. In many cases, for even moderately high depolarizing noise, fully product states are optimal. For two-qubit local bit-flip channels, the continuous commutativity of the channel and optimal probe implies that a product state suffices for obtaining the optimal precision.
We derive a general criterion to detect entangled states in multipartite systems based on the symmetric logarithmic derivative quantum Fisher information. This criterion is a direct consequence of the fact that separable states do not improve the accuracy upon estimating the one-parameter family of quantum channels. Our result is a generalization of the previously known criterion for the one-parameter unitary channel to any one-parameter quantum channel. Several variants of the proposed criterion are also given, and then the general structure is revealed behind this sort of entanglement criteria based on quantum Fisher information. We discuss several examples to illustrate our criterion. In the last part, we briefly show how the proposed criterion can be extended to a more general setting that is applicable for a certain class of open quantum systems, and we discuss how to detect entangled states even in the presence of decoherence.