Randomizing Quantum States: Constructions and Applications

Abstract

The construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d 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 d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d 2 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 log d qubits to be hidden in 2 log d 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 log d bits of communication and log d ebits of entanglement.

Entanglement and the foundations of statistical mechanics

“…Levy's Lemma has already been used in quantum information theory to study entanglement and other correlation properties of random states in large bipartite systems [5]. It provides a very powerful tool with which to evaluate functions of randomly chosen quantum states.…”

Exact and approximate unitary 2-designs and their application to fidelity estimation

“…The importance of generating random states and random unitary operators in quantum information processors has become increasingly clear from the growing number of algorithms and protocols that presume such a resource [1][2][3][4][5][6][7][8]. For many algorithms and protocols the invariant (Haar) measure on the unitary group U (D) is the natural randomization measure [3,5,7,8].…”

Robust quantum data locking from phase modulation

“…Until now, the codewords used in QDL have been restricted to either Haar-distributed random bases [2,5,6] or approximate mutually unbiased bases [5] (the role of mutually unbiased bases being not yet completely understood [1,33]). This paper showed that codewords modulated by random phase shifts in a given basis suffice to guarantee strong and robust QDL properties.…”

“…In QDL the knowledge of a relatively short key of constant size allows one to (un)lock an arbitrarily long message encoded in a quantum system. Otherwise, without knowledge of the key, only a negligible amount of information about the message can be accessed [2][3][4][5][6]. Such an exponential disproportion between the length of the key and that of the message is impossible in classical information theory, according to which the bits of secret key should be at least as many as the bits of encrypted information [7].…”