Randomizing Quantum States: Constructions and Applications

The construction of a perfectly secure private quantum channel in dimension d is known to require 2logd shared random key bits between the sender and receiver. We show that if only near-perfect security is required, the size of the key can be reduced by a factor of two. More specifically, we show that there exists a set of roughly dlogd unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d² operators required to randomize perfectly. Aside from the private quantum channel, variations of this construction can be applied to many other tasks in quantum information processing. We show, for instance, that it can be used to construct LOCC data hiding schemes for bits and qubits that are much more efficient than any others known, allowing roughly logd qubits to be hidden in 2logd qubits. The method can also be used to exhibit the existence of quantum states with locked classical correlations, an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key. Our construction also provides the basic building block for a method of remotely preparing arbitrary d-dimensional pure quantum states using approximately logd bits of communication and logd ebits of entanglement.