Weak Decoupling Duality and Quantum Identification

• Published in: IEEE transactions on information theory Vol. 58; no. 7; pp. 4914 - 4929
• Main Authors: Hayden, Patrick, Winter, Andreas
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Approximation; Classical and quantum physics: mechanics and fields; Decoding; Error correction; Exact sciences and technology; Geometry; Hilbert space; Identification; Identification capacity; Information theory; Information, signal and communications theory; Measurement; Physics; Quantum entanglement; Quantum information; Quantum theory; Receivers; Shannon theory; Telecommunications and information theory; Quantum system; Quantum channel; Information system; Noisy channel; Identification capacity; Quantum information; Error correction; Shannon theory
• ISSN: 0018-9448; 1557-9654; 1557-9654
• DOI: 10.1109/TIT.2012.2191695

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 paper, we develop an analogous duality in the case that the environment approximately forgets the identity of the quantum state, a weaker condition satisfied by -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 nonconstant quantum channel, including classical channels.