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arXiv:quant-ph/0301028 · Journal of Physics A Mathematical and General

Efficient sharing of a continuous-variable quantum secret

Tomáš Tyc, David J Rowe, Barry C. Sanders

Published Jun 24, 2003• 30 citations
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Abstract

We propose an efficient scheme for sharing a continuous-variable quantum secret using passive optical interferometry and squeezers: this efficiency is achieved by showing that a maximum of two squeezers is required to extract the secret state, and we obtain the cheapest configuration in terms of total squeezing cost. Squeezing is a cost for the dealer of the secret as well as for the receivers, and we quantify limitations to the fidelity of the extracted secret state in terms of the squeezing employed by the dealer.

DOI
10.1088/0305-4470/36/27/314

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We propose an efficient scheme for sharing a continuous-variable quantum secret using passive optical interferometry and squeezers: this efficiency is achieved by showing that a maximum of two squeezers is required to extract the secret state, and we obtain the cheapest configuration in terms of total squeezing cost. Squeezing is a cost for the dealer of the secret as well as for the receivers, and we quantify limitations to the fidelity of the extracted secret state in terms of the squeezing employed by the dealer.
We develop the theory of continuous-variable quantum secret sharing and propose its interferometric realization using passive and active optical elements. In the ideal case of infinite squeezing, a fidelity F\mathcal{F}F of unity can be achieved with respect to reconstructing the quantum secret. We quantify the reduction in fidelity for the (2,3) threshold scheme due to finite squeezing and establish the condition for verifying that genuine quantum secret sharing has occurred.
We consider the sharing of quantum secret states using continuous variable systems. Specifically we introduce an encoding procedure where we mix the secret mode with several ancillary squeezed modes through a passive interferometer. We derive simple conditions on the interferometer for this encoding scheme to give a secret sharing protocol and we prove that they are satisfied by almost any interferometer with respect to the Haar measure. This implies that, if the interferometer is chosen uniformly at random, the probability that it may not be used to implement a quantum secret sharing protocol is zero. Furthermore, we show that the decoding operation can be obtained and implemented efficiently in linear optics with an additional squeezer per access party. We analyze the quality of the reconstructed state as a function of the input squeezing.
We consider a protocol for sharing quantum states using continuous variable systems. Specifically we introduce an encoding procedure where bosonic modes in arbitrary secret states are mixed with several ancillary squeezed modes through a passive interferometer. We derive simple conditions on the interferometer for this encoding to define a secret sharing protocol and we prove that they are satisfied by almost any interferometer. This implies that, if the interferometer is chosen uniformly at random, the probability that it may not be used to implement a quantum secret sharing protocol is zero. Furthermore, we show that the decoding operation can be obtained and implemented efficiently with a Gaussian unitary using a number of single-mode squeezers that is at most twice the number of modes of the secret, regardless of the number of players. We benchmark the quality of the reconstructed state by computing the fidelity with the secret state as a function of the input squeezing.
We discuss sharing quantum secrets via optical interferometry and squeezing. A secret quantum state for a single-mode field is encoded into a multimode field as an entangled state and distributed to a set of players so that certain subsets can decode the secret states, and others cannot learn anything about the state. In particular, we discuss the (k,n)-threshold scheme for optics and specifically the (2,3) scheme. An arbitrary (k,n)-threshold scheme can be achieved with no more than two single-mode squeezers.
We present two schemes to perform continuous variable (2, 3) threshold quantum secret sharing (QSS) on the quadrature amplitudes of bright light beams. Both schemes require a pair of entangled light beams. The first scheme utilizes two phase sensitive optical amplifiers, whilst the second uses an electro-optic feedforward loop for the reconstruction of the secret. We examine the efficacy of QSS in terms of fidelity, as well as the signal transfer coefficients and the conditional variances of the reconstructed output state. We show that both schemes in the ideal case yield perfect secret reconstruction.
We present an experimental scheme to perform continuous variable (2,3) threshold quantum secret sharing on the quadratures amplitudes of bright light beams. It requires a pair of entangled light beams and an electro-optic feedforward loop for the reconstruction of the secret. We examine the efficacy of quantum secret sharing in terms of fidelity, as well as the signal transfer coefficients and the conditional variances of the reconstructed output state. We show that, in the ideal limit, perfect secret reconstruction is possible. We discuss two different definitions of quantum secret sharing: the sharing of a quantum secret and the sharing of a classical secret with quantum resources.