Weak Decoupling Duality and Quantum Identification

TL;DR

This paper introduces weak decoupling duality, linking the environment's forgetfulness of quantum states to the preservation of pairwise fidelities, and explores implications for quantum identification and channel capacities.

Contribution

It develops a new duality principle for approximate quantum state forgetfulness and applies it to quantum identification and channel capacity analysis.

Findings

Quantum identification cannot be cloned.

Optimal identification rate equals entanglement-assisted classical capacity.

Rate is positive for all non-constant channels.

Abstract

If a quantum system is subject to noise, it is possible to perform quantum error correction reversing the action of the noise if and only if no information about the system's quantum state leaks to the environment. In this article, we develop an analogous duality in the case that the environment approximately forgets the identity of the quantum state, a weaker condition satisfied by epsilon-randomizing maps and approximate unitary designs. Specifically, we show that the environment approximately forgets quantum states if and only if the original channel approximately preserves pairwise fidelities of pure inputs, an observation we call weak decoupling duality. Using this tool, we then go on to study the task of using the output of a channel to simulate restricted classes of measurements on a space of input states. The case of simulating measurements that test whether the input state is an arbitrary pure state is known as equality testing or quantum identification. An immediate consequence of weak decoupling duality is that the ability to perform quantum identification cannot be cloned. We furthermore establish that the optimal amortized rate at which quantum states can be identified through a noisy quantum channel is equal to the entanglement-assisted classical capacity of the channel, despite the fact that the task is quantum, not classical, and entanglement-assistance is not allowed. In particular, this rate is strictly positive for every non-constant quantum channel, including classical channels.