Symmetries in Quantum Key Distribution and the Connection between Optimal Attacks and Optimal Cloning

We investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack where the eavesdropper employs an optimal cloner to attack the quantum key distribution (QKD) protocol. The analysis is done in the context of the security proof in Refs. [1, 2] for discrete variable protocols in d-dimensional Hilbert spaces. We consider a scenario in which the protocols and cloners are equipped with symmetries. These symmetries are used to dene a quantum cloning scenario. We nd that, in general, it does not hold that the optimal attack is an optimal cloner. However, there are classes of protocols, where we can identify an optimal attack by an optimal cloner. We analyze protocols with 2, d and d + 1 mutually unbiased bases where d is a prime, and show that for the protocols with 2 and d + 1 MUBs the optimal attack is an optimal cloner, but for the protocols with d MUBs, it is not 1 . Finally, we give criteria to identify protocols which have dierent signal states, but the same optimal attack. Using these criteria, we present qubit protocols which have the same optimal attack as the BB84 protocol or the 6-state protocol.

[1]  H. Bechmann-Pasquinucci,et al.  Incoherent and coherent eavesdropping in the six-state protocol of quantum cryptography , 1998, quant-ph/9807041.

[2]  Nicolas J. Cerf,et al.  Asymmetric phase covariant d-dimensional cloning , 2005, Quantum Inf. Comput..

[3]  G. D’Ariano,et al.  Optimal nonuniversally covariant cloning , 2001, quant-ph/0101100.

[4]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[5]  G. M. D'Ariano,et al.  Extremal quantum cloning machines , 2005 .

[6]  N. Gisin,et al.  Optimal Eavesdropping in Quantum Cryptography. I , 1997, quant-ph/9701039.

[7]  D. Bruß,et al.  Optimal eavesdropping in cryptography with three-dimensional quantum states. , 2001, Physical review letters.

[8]  Anders Karlsson,et al.  Security of quantum key distribution using d-level systems. , 2001, Physical review letters.

[9]  Nicolas J. Cerf,et al.  Asymmetric quantum cloning in any dimension , 1998, quant-ph/9805024.

[10]  G. Koster,et al.  The Properties of the Thirty-Two Point Groups , 1963 .

[11]  P. Butler.  Point Group Symmetry Applications , 1981 .

[12]  Cerf,et al.  Pauli cloning of a quantum Bit , 2000, Physical review letters.

[13]  P. Butler.  Point Group Symmetry Applications: Methods and Tables , 1981 .

[14]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .

[15]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[16]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[17]  S. Massar,et al.  Optimal Quantum Cloning Machines , 1997, quant-ph/9705046.

[18]  I. D. Ivonovic.  Geometrical description of quantal state determination , 1981 .

[19]  Matthias Christandl,et al.  One-and-a-Half Quantum de Finetti Theorems , 2007 .

[20]  H Bechmann-Pasquinucci,et al.  Quantum cryptography with 3-state systems. , 2000, Physical review letters.

[21]  B Kraus,et al.  Lower and upper bounds on the secret-key rate for quantum key distribution protocols using one-way classical communication. , 2004, Physical review letters.

[22]  R. Renner,et al.  An information-theoretic security proof for QKD protocols , 2005, quant-ph/0502064.

[23]  R. Renner.  Symmetry of large physical systems implies independence of subsystems , 2007 .

[24]  M. Żukowski,et al.  Security of Quantum Key Distribution with entangled Qutrits. , 2002, quant-ph/0207057.

[25]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[26]  Buzek,et al.  Quantum copying: Beyond the no-cloning theorem. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[27]  I. D. Ivanović,et al.  Unbiased projector basis over C3 , 1997 .

[28]  A. Winter,et al.  Distillation of secret key and entanglement from quantum states , 2003, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  R. Renner,et al.  Information-theoretic security proof for quantum-key-distribution protocols , 2005, quant-ph/0502064.

[30]  N. Gisin,et al.  OPTIMAL EAVESDROPPING IN QUANTUM CRYPTOGRAPHY. I. INFORMATION BOUND AND OPTIMAL STRATEGY , 1997 .

[31]  Valerio Scarani,et al.  Security proof for quantum key distribution using qudit systems , 2010, 1003.5464.

[32]  D. Bruß.  Optimal Eavesdropping in Quantum Cryptography with Six States , 1998, quant-ph/9805019.

[33]  P. Oscar Boykin,et al.  A New Proof for the Existence of Mutually Unbiased Bases , 2002, Algorithmica.