The standard definition of quantum state randomization, which is the quantum analog of the classical one-time pad, consists in applying some transformation to the quantum message conditioned on a classical secret key k. We investigate encryption schemes in which this transformation is conditioned on a quantum encryption key state ρ k instead of a classical string, and extend this symmetric-key scheme to an asymmetric-key model in which copies of the same encryption key ρ k may be held by several different people, but maintaining information-theoretical security. We find bounds on the message size and the number of copies of the encryption key which can be safely created in these two models in terms of the entropy of the decryption key, and show that the optimal bound can be asymptotically reached by a scheme using classical encryption keys. This means that the use of quantum states as encryption keys does not allow more of these to be created and shared, nor encrypt larger messages, than if these keys are purely classical.
Randomization of quantum states is the quantum analogue of the classical one‐time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d e2) Pauli operators to map any d‐dimensional state to a state that is within trace distance e of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter, and Ambainis and Smith.Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d‐dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small e. 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 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 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. A quantum encryption scheme (also called private quantum channel, or state randomization protocol) is a one-time pad for quantum messages. If two parties share a classical random string, one of them can transmit a quantum state to the other so that an eavesdropper gets little or no information about the state being transmitted. Perfect encryption schemes leak no information at all about the message. Approximate encryption schemes leak a non-zero (though small) amount of information but require a shorter shared random key. Approximate schemes with short keys have been shown to have a number of applications in quantum cryptography and information theory [8]. We investigate how a classical private key can be used by two players, connected by an insecure one-way quantum channel, to perform private communication of quantum information. In particular we show that in order to transmit n qubits privately, 2n bits of shared private key are necessary and sufficient. This result may be viewed as the quantum analogue of the classical one-time pad encryption scheme. From the point of view of the eavesdropper, this encryption process can be seen as a randomization of the original state. We thus also obtain strict bounds on the amount of entropy necessary for randomizing n qubits. We show that 2n random classical bits are both necessary and sufficient for encrypting any unknown state of n quantum bits in an informationally secure manner. We also characterize the complete set of optimal protocols in terms of a set of unitary operations that comprise an orthonormal basis in a canonical inner product space. Moreover, a connection is made between quantum encryption and quantum teleportation that allows for a different proof of optimality of teleportation.