Bell inequalities violated using detectors of low efficiency

We define a family of binary outcome $n$-party $m\leq n$ settings per party Bell inequalities whose members require the least detection efficiency for their violation among all known inequalities of the same type. This gives upper bounds for the minimum value of the critical efficiency --- below which no violation is possible --- achievable for such inequalities. For $m=2$, our family reduces to the one given by Larsson and Semitecolos in 2001. For $m>2$, a gap remains between these bounds and the best lower bounds. The violating state near the threshold efficiency always approaches a product state of $n$ qubits.

[1]  Miguel Navascués,et al.  Robust and versatile black-box certification of quantum devices. , 2014, Physical review letters.

[2]  C. Klempt,et al.  Nonlocality in many-body quantum systems detected with two-body correlators , 2015, 1505.06740.

[3]  S. Wehner,et al.  Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres , 2015, Nature.

[4]  C. Monroe,et al.  Experimental violation of a Bell's inequality with efficient detection , 2001, Nature.

[5]  Nicolas Brunner,et al.  Closing the detection loophole in multipartite Bell tests using Greenberger-Horne-Zeilinger states , 2012, 1208.0622.

[6]  Eberhard,et al.  Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[7]  Jan-AAke Larsson,et al.  Loopholes in Bell inequality tests of local realism , 2014, 1407.0363.

[8]  A. Acín,et al.  A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations , 2008, 0803.4290.

[9]  A. Zeilinger,et al.  Bell violation using entangled photons without the fair-sampling assumption , 2012, Nature.

[10]  C. Branciard.  Detection loophole in Bell experiments: How postselection modifies the requirements to observe nonlocality , 2011 .

[11]  M. Horne,et al.  Experimental Consequences of Objective Local Theories , 1974 .

[12]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

[13]  Stefano Pironio,et al.  Closing the detection loophole in Bell experiments using qudits. , 2009, Physical review letters.

[14]  Adrian Kent,et al.  Private randomness expansion with untrusted devices , 2010, 1011.4474.

[15]  P. Pearle.  Hidden-Variable Example Based upon Data Rejection , 1970 .

[16]  Michael M. Wolf,et al.  Bell inequalities and entanglement , 2001, Quantum Inf. Comput..

[17]  Umesh Vazirani,et al.  Fully device-independent quantum key distribution. , 2012, 1210.1810.

[18]  N. Gisin,et al.  Violation of Bell Inequalities by Photons More Than 10 km Apart , 1998, quant-ph/9806043.

[19]  Gustavo Lima,et al.  Postselection-Loophole-Free Bell Test Over an Installed Optical Fiber Network. , 2015, Physical review letters.

[20]  B. S. Cirel'son.  Quantum generalizations of Bell's inequality , 1980 .

[21]  S. Massar,et al.  Combinatorics and quantum nonlocality. , 2002, Physical review letters.

[22]  Jan-Ake Larsson,et al.  Strict detector-efficiency bounds for n-site Clauser-Horne inequalities , 2001 .

[23]  Tamás Vértesi,et al.  Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems , 2010 .

[24]  G. Vallone,et al.  Bell scenarios in which nonlocality and entanglement are inversely related , 2011, 1106.2240.

[25]  M. Lewenstein,et al.  Detecting nonlocality in many-body quantum states , 2014, Science.

[26]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[27]  A. Aspect.  Bell's inequality test: more ideal than ever , 1999, Nature.

[28]  Aaron J. Miller,et al.  Detection-loophole-free test of quantum nonlocality, and applications. , 2013, Physical review letters.

[29]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[30]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[31]  M. Lewenstein,et al.  Detecting non-locality in multipartite quantum systems with two-body correlation functions , 2013, 1306.6860.

[32]  Stefano Pironio,et al.  Violation of local realism versus detection efficiency , 2003 .

[33]  H. Weinfurter,et al.  Violation of Bell's Inequality under Strict Einstein Locality Conditions , 1998, quant-ph/9810080.

[34]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[35]  T. Vértesi,et al.  Closing the detection loophole in tripartite Bell tests using the W state , 2015, 1504.05934.

[36]  H. Weinfurter,et al.  Heralded Entanglement Between Widely Separated Atoms , 2012, Science.

[37]  M. Lewenstein,et al.  Translationally invariant multipartite Bell inequalities involving only two-body correlators , 2013, 1312.0265.

[38]  Christoph Simon,et al.  Detection loophole in asymmetric bell experiments. , 2007, Physical review letters.

[39]  Andrew Chi-Chih Yao,et al.  Self testing quantum apparatus , 2004, Quantum Inf. Comput..

[40]  A. Acín,et al.  Bounding the set of quantum correlations. , 2006, Physical review letters.

[41]  Adán Cabello,et al.  Minimum detection efficiency for a loophole-free atom-photon bell experiment. , 2007, Physical review letters.

[42]  Erik Lucero,et al.  Violation of Bell's inequality in Josephson phase qubits , 2009, Nature.

[43]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[44]  V. Scarani,et al.  Device-independent security of quantum cryptography against collective attacks. , 2007, Physical review letters.

[45]  G. Lima,et al.  Optimal measurement bases for Bell tests based on the Clauser-Horne inequality , 2011, 1111.0822.