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arXiv:1208.6049 · Physical Review A

Mixed-state Pauli-channel parameter estimation

David C. Collins

Published Mar 4, 2013• 18 citations
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Abstract

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 $m$ channel invocations, uses initial states for the systems which are pure and unentangled and provides an uncertainty of $O(1/\sqrt{m})$. 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.

DOI
10.1103/physreva.87.032301

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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 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.
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.
The optimal measurement configuration, i.e. the optimal input quantum state and measurement in the form of a positive operator-valued measure with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/N, where N is the number of used measurements.
The optimal measurement configuration, i.e. the optimal input quantum state and measurement in the form of a positive operator-valued measure with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/ N , where N is the number of used measurements.
We built a new set of suitable measures of correlations for bipartite quantum states based upon a recently introduced theoretical framework [Bussandri et al. in Quantum Inf. Proc. 18:57, 2019]. We applied these measures to examine the behavior of correlations in two-qubit states with maximally mixed marginals independently interacting with non-dissipative decohering environments in different dynamical scenarios of physical relevance. In order to get further insight about the physical meaning of the behavior of these correlation measures we compared our results with those obtained by means of well-known correlation measures such as quantum mutual information and quantum discord. On one hand, we found that the behaviors of total and classical correlations, as assessed by means of the measures introduced in this work, are qualitatively in agreement with the behavior displayed by quantum mutual information and the measure of classical correlations typically used to calculate quantum discord. We also found that the optimization of all the measures of classical correlations depends upon a single parameter and the optimal value of this parameter turns out to be the same in all cases. On the other hand, regarding the measures of quantum correlations used in our studies, we found that in general their behavior does not follow the standard quantum discord D . As the quantification by means of standard quantum discord and the measures of quantum correlations introduced in this work depends upon the assumption that total correlations are additive, our results indicate that this property needs a deeper and systematic study in order to gain a further understanding regarding the possibility to obtain reliable quantifiers of quantum correlations within this additive scheme.