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arXiv:1506.06147 · Physical Review A

Depolarizing-channel parameter estimation using noisy initial states

David C. Collins, Jaimie Stephens

Published Sep 25, 2015• 21 citations
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Abstract

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.

DOI
10.1103/physreva.92.032324

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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 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 an arbitrary qubit channel depending on a single parameter, which is to be estimated by a physical process. Using the quantum Fisher information per channel invocation to quantify the estimation accuracy, we consider various estimation protocols when the available initial states are mixed with very low purity rrr. We compare a protocol using a single channel invocation on one out of nnn qubits prepared in a particular correlated input state to the optimal protocol using uncorrelated input states, with the same initial-state purity. We show that, to lowest order in initial-state purity, for a unital channel this correlated-state protocol enhances the estimation accuracy by a factor between $n\ensuremath{-}1$ and nnn, provided that $n{r}^{2}\ensuremath{\ll}1$. We also show that to lowest order in initial-state purity, a broad class of nonunital channels yields no gain regardless of the input state.
The accuracy of any physical scheme used to estimate the parameter describing the strength of a single qubit Pauli channel can be quantified using standard techniques from quantum estimation theory. It is known that the optimal estimation scheme, with mmm channel invocations, uses initial states for the systems which are pure and unentangled and provides an uncertainty of O(1/m)O(1/\sqrt{m})O(1/m​). This protocol is analogous to a classical repetition and averaging scheme. We consider estimation schemes where the initial states available are not pure and compare a protocol involving quantum correlated states to independent state protocols analogous to classical repetition schemes. We show, that unlike the pure state case, the quantum correlated state protocol can yield greater estimation accuracy than any independent state protocol. We show that these gains persist even when the system states are separable and, in some cases, when quantum discord is absent after channel invocation. We describe the relevance of these protocols to nuclear magnetic resonance measurements.
We consider protocols for estimating the parameter in a single-parameter unital qubit channel, assuming that the available initial states are highly mixed with very low purity. We compare two protocols, each querying the channel once, via their quantum Fisher informations. One uses nnn qubits prepared in a particular correlated input state and subsequently queries the channel on one qubit. The other uses a single qubit. We extend the results of Collins [Phys. Rev. A 99, 012123 (2019)] by allowing for additional noisy evolution on the spectator qubits in the nnn-qubit protocol. We provide simple algebraic expressions that will determine when the nnn-qubit protocol is superior. We provide a technique that can alleviate certain types of noise. We show that for certain types of noisy evolution the nnn-qubit protocol will be inferior but for others it will be superior.
We consider the problem of identifying the degree of correlation θ in the depolarization rates of an array of qubits, such as might be found in a holding register in a quantum information processing architecture. We treat this as a problem in quantum channel identification and use the quantum Fisher information about θ in the channel output to compare different candidate probe states. We consider channel probes in general pure N-qubit cat states and find that partial entanglement is always suboptimal; the best cat probe state is either separable or maximally entangled, and we lay out for any given depolarizing rate p and correlation θ, the (p, θ)-regions for which separable and maximally entangled probes each yield the most information. In the case of very large qubit arrays, we find that the quantum Fisher information obtained about θ using any pure cat probe state is asymptotically independent of the state and identically equal to the information available about a classical coin's bias from a single coin toss. Finally, we turn our attention to arrays of N qudits (with common dimension D) that are depolarizing with some correlation θ. We consider both separable and maximally entangled cat probe states and find and compare their quantum Fisher informations in both the large N and large D regimes.