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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Optimized unitary transformation for BB84 entangling probe

Howard E. Brandt

Published Aug 24, 2004• 1 citation
Publisher

Abstract

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.

DOI
10.1117/12.540875

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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.
For a general entangling probe attacking the Bennett-Brassard 1984 protocol in quantum key distribution, I calculate three classes of optimized unitary transformations, all 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 induced by the probe in the legitimate receiver. 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. Furthermore, I show that the simplest quantum circuit representing the optimal entangling probe consists of a single controlled-NOT gate in which the control qubit consists of two polarization-basis states of the signal, the target qubit consists of two probe-basis states, and the initial state of the probe is set by the error rate. A method is determined for measuring the appropriate correlated state of the probe. Finally, a possible implementation of the entangling probe is described.
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.
I present the design for an optimized entangling probe attacking the BB84 protocol of quantum key distribution and yielding maximum information to the probe. Probe photon polarization states become entangled with the signal states on their way between the legitimate transmitter and receiver. Based on an earlier optimization of the unitary transformation representing the probe for a set error rate induced by the probe, I derive the simplest quantum circuit which faithfully yields the correct entangled state produced by the probe. The quantum circuit consists of a single CNOT gate in which the control qubit is a photon polarization-basis state of the signal, the target is a probe photon polarization basis state, and the initial state of the probe is set by the error rate. A method is determined for measuring the appropriate correlated state of the probe, yielding maximum information to the probe. Finally, a possible implementation of the entangling probe is given.
Recent designs are summarized for some optimized quantum cryptographic entangling probes to be used in attacking the BB84 protocol of quantum key distribution. They are all based on alternative optimum unitary transformations, each yielding the same maximum information on the pre-privacy amplified key. One of the optimum unitary transformations, not previously addressed in terms of implementation, is shown, for a certain choice of free parameters, to again yield one of the recent designs. In each of the designs, the main quantum circuit consists of a single CNOT gate which produces the optimum entanglement between the BB84 signal states and the correlated probe states. For each design a different well-defined single-photon polarization state serves as the initial state of the probe, and in each case the probe is projectively measured.