Quantum dimension test using the uncertainty principle

We propose a test for certifying the dimension of a quantum system: store in it a random $n$-bit string, in either the computational or the Hadamard basis, and later check that the string can be mostly recovered. The protocol tolerates noise, and the verifier only needs to prepare one-qubit states. The analysis is based on uncertainty relations in the presence of quantum memory, due to Berta et al. (2010).

[1]  Mario Berta,et al.  Quantum coding with finite resources , 2015, Nature Communications.

[2]  Matej Pivoluska,et al.  Measurements in two bases are sufficient for certifying high-dimensional entanglement , 2017, Nature Physics.

[3]  Matthias Christandl,et al.  Uncertainty, monogamy, and locking of quantum correlations , 2005, IEEE Transactions on Information Theory.

[4]  Gustavo Lima,et al.  Device-independent certification of high-dimensional quantum systems. , 2014, Physical review letters.

[5]  William Slofstra.  A group with at least subexponential hyperlinear profile , 2018 .

[6]  Henry Yuen,et al.  Noise-Tolerant Testing of High Entanglement of Formation , 2017, ICALP.

[7]  Anand Natarajan,et al.  A quantum linearity test for robustly verifying entanglement , 2017, STOC.

[8]  Wen-Hao Zhang,et al.  Experimental self-testing of entangled states , 2018, 1803.10961.

[9]  Nilanjana Datta,et al.  The Quantum Capacity of Channels With Arbitrarily Correlated Noise , 2009, IEEE Transactions on Information Theory.

[10]  R. Prevedel,et al.  Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement , 2010, 1012.0332.

[11]  E. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[12]  Anand Natarajan,et al.  Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[13]  Paul Skrzypczyk,et al.  Dimension of physical systems, information processing, and thermodynamics , 2014, 1401.4488.

[14]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[15]  Thomas Vidick,et al.  A three-player coherent state embezzlement game , 2018, Quantum.

[16]  Jean-Daniel Bancal,et al.  Device-independent certification of one-shot distillable entanglement , 2017, New Journal of Physics.

[17]  Laura Mančinska,et al.  Multidimensional quantum entanglement with large-scale integrated optics , 2018, Science.

[18]  V. Scarani,et al.  Testing the dimension of Hilbert spaces. , 2008, Physical review letters.

[19]  Guang-Can Guo,et al.  Experimental realization of dimension witnesses based on quantum state discrimination , 2016 .

[20]  Mario Berta,et al.  Quantum to Classical Randomness Extractors , 2011, IEEE Transactions on Information Theory.

[21]  Rodrigo Gallego,et al.  Device-independent tests of classical and quantum dimensions. , 2010, Physical review letters.

[22]  A. Lichnerowicz.  Proof of the Strong Subadditivity of Quantum-Mechanical Entropy , 2018 .

[23]  Miguel Navascués,et al.  Dimension witnesses and quantum state discrimination. , 2012, Physical review letters.

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

[25]  Patrick J. Coles,et al.  Entropic uncertainty relations and their applications , 2015, 1511.04857.

[26]  E. Lieb,et al.  Proof of the strong subadditivity of quantum‐mechanical entropy , 1973 .

[27]  Zhaohui Wei,et al.  On the minimum dimension of a Hilbert space needed to generate a quantum correlation , 2015, Physical review letters.

[28]  Marcin Pawłowski,et al.  Experimental tests of classical and quantum dimensionality. , 2013, Physical review letters.

[29]  G. Guo,et al.  Experimental investigation of the entanglement-assisted entropic uncertainty principle , 2010, 1012.0361.

[30]  Mohamed Bourennane,et al.  Experimental device-independent tests of classical and quantum dimensions , 2011, Nature Physics.

[31]  J. Boileau,et al.  Conjectured strong complementary information tradeoff. , 2008, Physical review letters.

[32]  Christof Wunderlich,et al.  High-Fidelity Preservation of Quantum Information During Trapped-Ion Transport. , 2017, Physical review letters.

[33]  F. Schmidt-Kaler,et al.  Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation , 2017, 1705.02771.

[34]  Andrea Coladangelo,et al.  Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH and the magic square game , 2016, Quantum Inf. Comput..

[35]  Matthew Coudron,et al.  The Parallel-Repeated Magic Square Game is Rigid , 2016, 1609.06306.

[36]  Andrea Coladangelo,et al.  A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations , 2019, Quantum.

[37]  Thomas Vidick,et al.  Test for a large amount of entanglement, using few measurements , 2018, Quantum.

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

[39]  Matthias Christandl,et al.  Lower bound on the dimension of a quantum system given measured data , 2008, 0808.3960.

[40]  N. Brunner,et al.  Experimental estimation of the dimension of classical and quantum systems , 2011, Nature Physics.

[41]  Thomas Vidick,et al.  Low-degree testing for quantum states , 2018 .

[42]  Michele Dall'Arno,et al.  Robustness of device-independent dimension witnesses , 2012, 1207.2574.

[43]  Chris Sutherland,et al.  Overlapping Qubits , 2017, ITCS.

[44]  V. Scarani,et al.  A new device-independent dimension witness and its experimental implementation , 2016, 1606.01602.

[45]  Marco Tomamichel,et al.  Capacity estimation and verification of quantum channels with arbitrarily correlated errors , 2016, Nature Communications.

[46]  Jonathan Barrett.  Information processing in generalized probabilistic theories , 2005 .

[47]  Miguel Navascués,et al.  Bounding the Set of Finite Dimensional Quantum Correlations. , 2014, Physical review letters.

[48]  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.

[49]  Matthew McKague,et al.  Self-testing in parallel , 2015, 1511.04194.