Indefinite causal order aids quantum depolarizing channel identification

Abstract

Quantum channel identification is the metrological determination of one or more parameters of a quantum channel. This is accomplished by passing probes in prepared states through the channel and then statistically estimating the parameter(s) from the measured channel outputs. In quantum channel identification, the channel parameters’ quantum Fisher information is a means to assess and compare different probing schemes. We use quantum Fisher information to study a probing scheme in which the channel is put in indefinite causal order (ICO) with copies of itself, focusing our investigation on probing the qudit (d-dimensional) depolarizing channel to estimate its state preservation probability. This ICO arrangement is one in which both the eigenvectors and eigenvalues of the channel output depend on the channel’s state preservation probability. We overcome this complication to obtain the quantum Fisher information in analytical form. This result shows that ICO-assisted probing yields greater information than does the comparable probe re-circulation scheme with definite causal order, that the information gained is greater when the channel ordering is more indefinite, and that the information gained is greatest when the channel ordering is maximally indefinite. This leads us to conclude that ICO is acting here in a strong sense as an aid to channel probing. The effectiveness of ICO for probing the depolarizing channel decreases with probe dimension, being most effective for qubits.

Appendix

This “Appendix” proves inequalities (50) and (51). We assume that $0< \theta < 1$ and that the integers m, d satisfy $m,d\ge 2$. Recall that

$$\begin{aligned} x(\theta )=(d-1)\theta +1, \;\;\;\; y(\theta ) = (d+1)\theta - 1. \end{aligned}$$

Proof of inequality (50): Define the random variable X such that $P\{X=\theta \}=\frac{d-1}{d}$ and $P\{ X=1\} =\frac{1}{d}$. The function $f(x)=x^m$ is strictly convex over the interval $[\theta ,1]$ so $E[X^m] > E[X]^m$ according to Jensen’s inequality. Rewriting these expectations in terms of the distribution of X, we have

$$\begin{aligned} \frac{d-1}{d}\theta ^m+\frac{1}{d} > \left( \frac{d-1}{d} \theta +\frac{1}{d} \right) ^m . \end{aligned}$$

This inequality can be written

$$\begin{aligned} \frac{x(\theta ^m)}{d} > \left( \frac{x(\theta )}{d} \right) ^m , \end{aligned}$$

from which inquality (50) follows.

Proof of inequality (51): We have $x(\theta )/d < \theta$ and $y(\theta )/d < 1$ so

$$\begin{aligned} \sum _{k=0}^{m-1} \theta ^k > \sum _{k=0}^{m-1} \left( \frac{x(\theta )}{d}\right) ^k \left( \frac{y(\theta )}{d}\right) ^{m-1-k} \end{aligned}$$

or, equivalently,

$$\begin{aligned} \frac{1-\theta ^m}{1-\theta } > \frac{1}{d^{m-1}}\frac{x(\theta )^m-y(\theta )^m}{x(\theta )-y(\theta )}. \end{aligned}$$

Then, $x(\theta )-y(\theta ) = 2(1-\theta )$ so

$$\begin{aligned} 2d^{m-1}(1-\theta ^m) > x(\theta )^m-y(\theta )^m , \end{aligned}$$

and inequality (51) follows.