Adaptive quantum channel discrimination using methods of quantum metrology

TL;DR

The paper tackles adaptive quantum channel discrimination by proposing a scalable tensor-network optimization algorithm inspired by quantum metrology, modelled within the quantum comb framework. It formalizes the discrimination task as a semidefinite program over testers and then decomposes the problem into a trainable tensor-network with local optimizations over input states, inter-channel controls, and measurements, enabling analysis beyond small channel uses. A key contribution is casting discrimination bounds in terms of quantum Fisher information and Heisenberg-scaling criteria, linking estimation theory to finite-use discrimination and clarifying when perfect discrimination is possible. The authors demonstrate the method on several noise models, showing ancilla-assisted gains, and validate the tightness of QFI-based bounds, while discussing the practical and theoretical implications for adaptive versus parallel discrimination strategies.

Abstract

We present an efficient tensor-network based algorithm for finding the optimal adaptive quantum channel discrimination strategies inspired by recently developed numerical methods in quantum metrology to find the optimal adaptive channel estimation protocols. We examine the connection between channel discrimination and estimation problems, highlighting in particular an appealing structural similarity between models that admit Heisenberg scaling estimation performance, and models that admit perfect channel discrimination in finite--number of channel uses.

Figures (5)

• Figure 1: Simple diagram representing the task of quantum channel discrimination with single use of the channel. A player (depicted here as the computer) knows the forms of the possible quantum channels from which she has been given one randomly, according to a distribution she also knows. Her task is to choose the state that she inputs into the channel (possibly entangled with some ancillary space) and measurement scheme at the output of the channel that maximises her probability of successfully guessing which channel she has. The lower part represents an adaptive strategy for multiple channels uses, where additional quantum controls are allowed between subsequent channel-uses. The light orange shade represents the quantum comb --- mathematical representation of the strategy.
• Figure 2: Diagrammatic representation of our tensor network algorithm. The boxes represent the unknown channel $C_?$ and the parts of the discrimination strategy. The arrows represent input, output and ancilla spaces, some of them labelled as in Def. \ref{['def:strategy']}. The orange shapes represent combined parts of the comb that remain fixed during one optimisation step. In every step, the whole comb outside the currently optimised tooth is treated as a constant. The repeated optimisation over the teeth (input state $\rho$, inter-channel quantum controls $E_n$ and 'measurement channel' $M$) hopefully leads to an optimal comb.
• Figure 3: Error probability of discrimination between two unitaries (upper) and unitaries with perpendicular dephasing noise (lower) for different number of channel uses and ancilla dimensions 1 and 2. The solid green line represents the theoretical probability of error for unitary discrimination---a lower bound for the numerical results. The effect of the noise can be got rid of with a single qubit ancilla.
• Figure 4: Error probability of discrimination between two unitaries with perpendicular (upper) and parallel (lower) dephasing noise for different number of channel uses and ancilla dimensions 1, 2 and 4. The grey solid lines show the estimation bounds computed with the QMetro++ package, and the green lines are the unitary bound. Logarithmic scale in y-axis.
• Figure 5: Error probability of discrimination between dephasing and amplitude damping channels for different number of channel uses and ancilla dimensions 1, 2 and 4. Single SDP solution for comparison for 1,2 and 3 uses. The grey solid line shows the estimation bound.

Theorems & Definitions (2)

• Definition 1
• Definition 2