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arXiv:quant-ph/0509211 · Journal of Modern Optics

Entangled eavesdropping in quantum key distribution

Howard E. Brandt

Published Oct 25, 2006• 6 citations
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Abstract

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.

DOI
10.1080/09500340600895383

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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.
Abstract The mathematical physical bases are given for quantum cryptographic probes which exploit entanglement to eavesdrop on quantum key distribution. The quantum circuits and designs are presented for two different optimized entangling probes for attacking the BB84 Protocol of quantum key distribution (QKD) and yielding maximum information to the probes. Probe photon polarization states become optimally entangled with the signal states on their way between the legitimate transmitter and receiver. The designs are based on different optimum unitary transformations, each yielding the same maximum information on the pre-privacy amplified key. 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 states of the probe. 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. Symmetric projective measurements of the probe yield maximum information on the pre-privacy amplified key once basis information becomes available during basis reconciliation.
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.
For a general entangling probe attacking the BB84 protocol in quantum key distrobution, I show that the simplest quantum circuit representing the optimal entangling probe, and yielding the maximum information gain on the pre-privacy-amplified key, consisting of a single CNOT gate in which the control qubit consists of two polarization-basis states of the signal, the target qubit consists of two probe-photon polarization basis states, and the initial state of the probe is set by the error rate induced by the probe. A method is determined for measuring the appropriate correlated state of the probe. Finally, a possible implementation of the entangling probe is described. the device is a simple special-purpose quantum cryptography.
An alternative design is given for an optimized quantum cryptographic entangling probe for attacking the BB84 protocol of quantum key distribution. The initial state of the probe has a simpler analytical dependence on the set error rate to be induced by the probe than in the earlier design. The new device yields the same maximum information to the probe for a full range of induced error rates. As in the earlier design, the probe contains a single CNOT gate which produces the optimum entanglement between the BB84 signal states and the correlated probe states.
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.
It is argued that the BB84 (Bennett–Brassard 1984) protocol of quantum key distribution has a vulnerability similar to the well-known vulnerability of the B92 (Bennett 1992) protocol in the presence of losses. The quantum circuit and design are first reviewed 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 optimally entangled with the signal states on their way between the legitimate transmitter and receiver. Although standard von Neumann projective measurements of the probe yield maximum information on the pre-privacy amplified key, if instead the probe measurements are performed with a certain positive operator valued measure, then the measurement results are unambiguous, at least some of the time, resulting in complete information gain by the probe for sufficient loss in the key distribution channel.