Quantum Privacy and Quantum Coherence
Benjamin W. Schumacher, Michael D. Westmoreland · Physical Review Letters · 1998
We derive a simple relation between a quantum channel’s capacity to convey coherent (quantum) information and its usefulness for quantum cryptography. 1 Typeset using REVTEX A quantum communication channel can be used to perform a variety of tasks, including: • Conveying classical information from a sender to a receiver. • Conveying quantum information (including quantum entanglement) from a sender to a receiver. • Creating shared information between a sender and receiver, information that is reliably secret from any third party and can thus be used as a cryptographic key for later private communication. (The use of quantum channels to aid in cryptographic tasks such as key distribution is called quantum cryptography.) Each of these tasks can be performed in the presence of noise. Indeed, in quantum cryptography the noise is of central importance in revealing the activity of an eavesdropper. Deutsch et al. [1] examined the security of quantum cryptographic schemes over quantum channels that contain noise. They pointed out that any protocol which allowed “entanglement purification ” between two parties automatically provided a means of communicating secret information that no third party could share. Here we will continue this line of thought by showing that the privacy of the channel, measured by the amount of information available to the receiver that is not available to any eavesdropper, can be made at least as great as the channel’s coherent information [2]. Suppose Alice prepares a quantum system Q in an initial state ρQ. Alice conveys the system Q through a noisy quantum channel to Bob. The noisy channel may be described by a superoperator E Q, so that the final state ρQ ′ = E Q (ρQ). The evolution of the channel given by the superoperator E Q is in fact unitary evolution on a larger quantum system that includes the environment E of the system. This environment may be considered to be initially in a pure state � � E �0 �. In this case, the superoperator is given by