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arXiv:quant-ph/0302033

Optimum Probe Parameters for Entangling Probe in Quantum Key Distribution

Howard E. Brandt

Published Feb 4, 2003• 4 citations
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Abstract

For the four-state protocol of quantum key distribution, optimum sets of probe parameters are calculated for the most general unitary probe in which each individual transmitted photon is made to interact with the probe so that the signal and the probe are left in an entangled state, and projective measurement by the probe, made subsequent to projective measurement by the legitimate receiver, yields information about the signal state. The probe optimization is based on maximizing the Renyi information gain by the probe on corrected data for a given error rate induced by the probe in the legitimate receiver. An arbitrary angle is included between the nonorthogonal linear polarization states of the signal photons. Two sets of optimum probe parameters are determined which both correspond to the same optimization. Also, a larger set of optimum probe parameters is found than was known previously for the standard BB84 protocol. A detailed comparison is made between the complete and incomplete optimizations, and the latter simpler optimization is also made complete. Also, the process of key distillation from the quantum transmission in quantum key distribution is reviewed, with the objective of calculating the secrecy capacity of the four-state protocol in the presence of the eavesdropping probe. Emphasis is placed on information leakage to the probe.

DOI
10.48550/arxiv.quant-ph/0302033

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For the four-state protocol of quantum key distribution, optimum sets of probe parameters are calculated for the most general unitary probe in which each individual transmitted photon is made to interact with the probe so that the signal and the probe are left in an entangled state, and projective measurement by the probe, made subsequent to projective measurement by the legitimate receiver, yields information about the signal state. The probe optimization is based on maximizing the Renyi information gain by the probe on corrected data for a given error rate induced by the probe in the legitimate receiver. An arbitrary angle is included between the nonorthogonal linear polarization states of the signal photons. Two sets of optimum probe parameters are determined which both correspond to the same optimization. Also, a larger set of optimum probe parameters is found than was known previously for the standard BB84 protocol. A detailed comparison is made between the complete and incomplete optimizations, and the latter simpler optimization is also made complete. Also, the process of key distillation from the quantum transmission in quantum key distribution is reviewed, with the objective of calculating the secrecy capacity of the four-state protocol in the presence of the eavesdropping probe. Emphasis is placed on information leakage to the probe.
The standard four-state (BB84) protocol of key distribution in quantum cryptography, Slutsky, Rao, Sun, and Fainman performed an eavesdropping probe optimization, which an average yields the most information to the eavesdropper for a given error rate caused by the probe. The most general possible probe consistent with unitary was considered, in which each individual transmitted bit is made to interact with the probe so that the carrier and the probe are left in an endangered state, and measurement by the probe, made subsequent to measurement by the legitimate receiver, yield information about the carrier state. The probe optimization is based on maximizing the Renyi information gain by the probe on corrected data for a given error rate induced by the probe in a legitimate receiver. The results were obtained for the standard protocol with an angle of 45 degrees between the signal bases. In more recent work, a larger set of optimum probe parameters was found than was known previously, and although they all yield the same maximum Renyi information gain by the probe, alternative options are made available for optimum probe design. In the present work, the corresponding optimized unitary transformation, representing the action of the probe on the signal, are calculated. I have determined three classes of unitary transformations yielding the same maximum information to the probe. The simplest one corresponds to a probe having a two-dimensional Hilbert space of states, and is uniquely determined by the error rate. The second class corresponds to a probe having a four-dimensional Hilbert space of states, and is determined by the error rate and two continuous angle parameters, which are mutually constrained by the error rate. The third class corresponds to a probe having a four-dimensional Hilbert space, and is determined by the error rate and two continuous angle parameters, one of which is constrained by the error rate. This work will be useful in the design of an optimum-entangling probe.
The process of key distillation from the quantum transmission in quantum key distribution is reviewed, with the objective of calculating the secrecy capacity of the four-state protocol in the presence of an individual attack in which the eavesdropping probe is entangled with the signal states, and states of the probe become correlated with the states measured by the legitimate receiver. Emphasis is placed on information leakage to the eavesdropping probe. The calculation generalizes earlier work to include an arbitrary angle between the signal bases.
Following a review of the measure of Renyi information gain and its optimization, the quantum circuits and designs are reviewed of optimized entangling probes for attacking the BB84 protocol of quantum key distribution (QKD), extracting maximum Renyi information on the pre-privacy amplified key. Probe photon polarization states become optimally entangled with the signal states on their way between the legitimate transmitter and receiver. Standard symmetric von Neumann projective measurements of the probe yield maximum information on the pre-privacy amplified key.