Randomizing Quantum States: Constructions and Applications
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.
Unable to display preview. Download preview PDF.
- 1.Braunstein, S., Lo, H.-K., Spiller, T.: Forgetting qubits is hot to do. Unpublished manuscript, 1999Google Scholar
- 2.Boykin, P. O., Roychowdhury, V.: Optimal encryption of quantum bits. http://arxiv.org/abs/quant-ph/0003059, 2000
- 3.Ambainis, A., Mosca, M., Tapp, A., de Wolf, R.: Private quantum channels. In IEEE Symposium on Foundations of Computer Science (FOCS), 2000, pp. 547–553Google Scholar
- 9.Bennett, C. H., Hayden, P., Leung, D., Shor, P. W., Winter, A.: Remote preparation of quantum states. http://arxiv.org/abs/quant-ph/0307100, 2003
- 14.Holevo, A. S.: Statistical problems in quantum physics. In: G. Maruyama J. V. Prokhorov, editors, Proceedings of the second Japan-USSR Symposium on Probability Theory, Volume 330 of Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1973, pp. 104–119Google Scholar
- 15.Dembo, A., Zeitouni, O.: Large deviations techniques and applications. New York: Springer-Verlag, 1993Google Scholar
- 18.Hayden, P.: Spin-cycle entanglement. In preparationGoogle Scholar
- 20.Ohya, M., Petz, D.: Quantum entropy and its use. Texts and monographs in physics. Berlin: Springer-Verlag, 1993Google Scholar
- 27.Fannes, M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31, 291–294 (1973)Google Scholar
- 28.Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. Number 1200 in Lecture Notes in Mathematics. Springer-Verlag, 1986Google Scholar
- 30.Young, R. M.: Euler’s constant. Math. Gaz. 75, 187–190 (1991)Google Scholar