Simplified instantaneous non-local quantum computation with applications to position-based cryptography

Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially with the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given bound and is restricted to classical communication.

[1]  Rafail Ostrovsky,et al.  Position Based Cryptography , 2009, CRYPTO.

[2]  Satoshi Ishizaka,et al.  Quantum teleportation scheme by selecting one of multiple output ports , 2009, 0901.2975.

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

[4]  Schumacher,et al.  Classical information capacity of a quantum channel. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[5]  Lev Vaidman.  Instantaneous measurement of nonlocal variables. , 2003, Physical review letters.

[6]  Y. Aharonov,et al.  Is the usual notion of time evolution adequate for quantum-mechanical systems? I , 1984 .

[7]  Adrian Kent,et al.  Quantum Tagging: Authenticating Location via Quantum Information and Relativistic Signalling Constraints , 2010, ArXiv.

[8]  Satoshi Ishizaka,et al.  Asymptotic teleportation scheme as a universal programmable quantum processor. , 2008, Physical review letters.

[9]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[10]  L. Vaidman,et al.  Nonlocal variables with product-state eigenstates , 2001, quant-ph/0103084.

[11]  H. Lo,et al.  Insecurity of position-based quantum-cryptography protocols against entanglement attacks , 2010, 1009.2256.

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

[13]  Yakir Aharonov,et al.  Can we make sense out of the measurement process in relativistic quantum mechanics , 1981 .

[14]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[15]  Adrian Kent,et al.  Quantum Tagging with Cryptographically Secure Tags , 2010, arXiv.org.

[16]  S. R. Clark,et al.  Entanglement consumption of instantaneous nonlocal quantum measurements , 2010, 1004.0865.

[17]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[18]  Vaidman,et al.  Measurement process in relativistic quantum theory. , 1986, Physical review. D, Particles and fields.

[19]  Adrian Kent.  Quantum tagging for tags containing secret classical data , 2011 .

[20]  Hoi-Kwong Lo,et al.  Is Quantum Bit Commitment Really Possible? , 1996, ArXiv.

[21]  Yakir Aharonov,et al.  States and observables in relativistic quantum field theories , 1980 .

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

[23]  N. Bohr,et al.  Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen , 1933 .

[24]  M. Horodecki,et al.  General teleportation channel, singlet fraction and quasi-distillation , 1998, quant-ph/9807091.

[25]  Vaidman,et al.  Causality constraints on nonlocal quantum measurements. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[26]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[27]  William K. Wootters,et al.  A ‘Pretty Good’ Measurement for Distinguishing Quantum States , 1994 .

[28]  John Watrous,et al.  Semidefinite Programs for Completely Bounded Norms , 2009, Theory Comput..

[29]  Rafail Ostrovsky,et al.  Position-Based Quantum Cryptography: Impossibility and Constructions , 2011, IACR Cryptol. ePrint Arch..

[30]  M. Horodecki,et al.  Locking classical correlations in quantum States. , 2003, Physical review letters.

[31]  Dominic Mayers.  Unconditionally secure quantum bit commitment is impossible , 1997 .

[32]  R. Peierls,et al.  Erweiterung des Unbestimmtheitsprinzips für die relativistische Quantentheorie , 1931 .

[33]  Robert A. Malaney,et al.  Location-dependent communications using quantum entanglement , 2010, 1003.0949.

[34]  Y. Aharonov,et al.  Is the usual notion of time evolution adequate for quantum-mechanical systems? II. Relativistic considerations , 1984 .

[35]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .