Simple Upper and Lower Bounds on the Ultimate Success Probability for Discriminating Arbitrary Finite-Dimensional Quantum Processes

We consider the problem of discriminating finite-dimensional quantum processes, also called quantum supermaps, that can consist of multiple time steps. Obtaining the ultimate performance for discriminating quantum processes is of fundamental importance, but is challenging mainly due to the necessity of considering all discrimination strategies allowed by quantum mechanics, including entanglement-assisted strategies and adaptive strategies. In the case in which the processes to be discriminated have internal memories, the ultimate performance would generally be more difficult to analyze. In this Letter, we present a simple upper bound on the ultimate success probability for discriminating arbitrary quantum processes. In the special case of multishot channel discrimination, it can be shown that the ultimate success probability increases by at most a constant factor determined by the given channels if the number of channel evaluations increases by one. We also present a lower bound based on Bayesian updating, which has a low computational cost. Our numerical experiments demonstrate that the proposed bounds are reasonably tight. The proposed bounds do not explicitly depend on any quantum phenomena, and can be readily extended to a general operational probabilistic theory.

Figure 1

General protocol of quantum process discrimination. ${\mathcal{E}}_m$ is a process consisting of $T$ channels ${\mathrm{\Lambda }}_m^{(1)},\dots ,{\mathrm{\Lambda }}_m^{(T)}$. Discrimination is characterized by the collection of a state ${\sigma }_1$, channels ${\sigma }_2,\dots ,{\sigma }_T$, and a measurement $\{{\mathrm{\Pi }}_k{\}}_{k=1}^M$.

Figure 2

Proposed protocol based on Bayesian updating. Each channel ${\sigma }_t$ with $2\le t\le T$ is restricted to a measure-and-prepare channel ${\varrho }^{(t)}\circ {\mathrm{\Pi }}^{(t-1)}$. The state preparation ${\varrho }^{(t)}$ and the measurement ${\mathrm{\Pi }}^{(t)}$ depend on the outcome of the previous measurement ${\mathrm{\Pi }}^{(t-1)}$. To provide a tight lower bound on the ultimate success probability, $[{\varrho }^{(1)},{\mathrm{\Pi }}^{(1)}],[{\varrho }^{(2)},{\mathrm{\Pi }}^{(2)}],\dots$ are sequentially optimized.

Figure 3

Success probability in the problem of channel position finding with two AD channels $A_{q_B}$ and $A_{q_T}$, where $T=2$, $M=3$, and $q_B=q_T+0.04$. The ultimate success probability lies in the gray region, between our upper bound $\overline{P_1}$ of Eq. (3) and our lower bound $\underset{\_}{P}$ of Eq. (7). $\overline{P_1^{\prime }}$ is the proposed upper bound described in the SM. ${\overline{P}}_{\mathrm{conv}}$ is the upper bound proposed in Ref. [33]. $P_{\mathrm{PGM}}$ is the success probability achieved by the maximally entangled pure state and the pretty good measurement [70, 71], which is a lower bound on the ultimate success probability.

Figure 4

Success probability in the discrimination between processes consisting of two generalized AD channels with correlated noise, where $T=M=3$. The parameters that are associated with the zero-temperature dissipation rate are set to ${\nu }_0$ and ${\nu }_0+0.04$, respectively. All the other parameters of these channels are the same. The two proposed upper bounds $\overline{P_1}$ of Eq. (3) and $\overline{P_2}$ of Eq. (5) and the proposed lower bound $\underset{\_}{P}$ of Eq. (7) are depicted. In this case, we can numerically compute the ultimate success probability $P^{\star }$. ${\underset{\_}{P}}_{\text{maxent}}$ is the success probability achieved by the maximally entangled pure state and the optimal measurement, which gives a lower bound on $P^{\star }$.