Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.

[1]  Debbie W. Leung,et al.  Quantum data hiding , 2002, IEEE Trans. Inf. Theory.

[2]  Laura Mančinska,et al.  A Framework for Bounding Nonlocality of State Discrimination , 2012, Communications in Mathematical Physics.

[3]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[4]  Masahito Hayashi,et al.  Two-way classical communication remarkably improves local distinguishability , 2007, 0708.3154.

[5]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[6]  E. Rains.  Entanglement purification via separable superoperators , 1997, quant-ph/9707002.

[7]  M. Horodecki,et al.  Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature? , 1998, quant-ph/9801069.

[8]  P. Horodecki.  Separability criterion and inseparable mixed states with positive partial transposition , 1997, quant-ph/9703004.

[9]  C. H. Bennett,et al.  Unextendible product bases and bound entanglement , 1998, quant-ph/9808030.

[10]  Eric M. Rains.  A semidefinite program for distillable entanglement , 2001, IEEE Trans. Inf. Theory.

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

[12]  H. Lo,et al.  Random bipartite entanglement from W and W-like states. , 2006, Physical review letters.

[13]  M. Nielsen.  Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[14]  Runyao Duan,et al.  Tripartite entanglement transformations and tensor rank. , 2008, Physical review letters.

[15]  John Watrous,et al.  Notes on super-operator norms induced by schatten norms , 2004, Quantum Inf. Comput..

[16]  M. Horodecki,et al.  The Uniqueness Theorem for Entanglement Measures , 2001, quant-ph/0105017.

[17]  Howard Barnum,et al.  Separable balls around the maximally mixed multipartite quantum states , 2003 .

[18]  Hermann Kampermann,et al.  Asymptotically perfect discrimination in the local-operation-and-classical-communication paradigm , 2011 .

[19]  D. Leung,et al.  Hiding bits in bell states. , 2000, Physical Review Letters.

[20]  Eric Chitambar,et al.  Local quantum transformations requiring infinite rounds of classical communication. , 2011, Physical review letters.

[21]  W. Wootters,et al.  Optimal detection of quantum information. , 1991, Physical review letters.

[22]  W. Wootters.  Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

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

[24]  Runyao Duan,et al.  Local distinguishability of orthogonal 2 ⊗ 3 pure states , 2008 .

[25]  H. Lo,et al.  Concentrating entanglement by local actions: Beyond mean values , 1997, quant-ph/9707038.

[26]  S. D. Rinaldis.  Distinguishability of complete and unextendible product bases , 2003, quant-ph/0304027.

[27]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[28]  Frank Verstraete,et al.  Local vs. joint measurements for the entanglement of assistance , 2003, Quantum Inf. Comput..

[29]  Masato Koashi,et al.  ‘Quantum Nonlocality without Entanglement’ in a Pair of Qubits , 2007, OSA Workshop on Entanglement and Quantum Decoherence.

[30]  Martin B. Plenio,et al.  An introduction to entanglement measures , 2005, Quantum Inf. Comput..

[31]  S. Turgut,et al.  Transformations of W-type entangled states , 2010, 1003.2118.

[32]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[33]  Ognyan Oreshkov,et al.  Weak measurements are universal. , 2005, Physical review letters.

[34]  Yuan Feng,et al.  Distinguishability of Quantum States by Separable Operations , 2007, IEEE Transactions on Information Theory.

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

[36]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[37]  Erika Andersson,et al.  Binary search trees for generalized measurements , 2007, 0712.2665.

[38]  Hoi-Kwong Lo,et al.  Proof of security of quantum key distribution with two-way classical communications , 2001, IEEE Trans. Inf. Theory.

[39]  Massar,et al.  Optimal extraction of information from finite quantum ensembles. , 1995, Physical review letters.

[40]  R F Werner,et al.  Hiding classical data in multipartite quantum states. , 2002, Physical review letters.

[41]  J. Cirac,et al.  Reflections upon separability and distillability , 2001, quant-ph/0110081.

[42]  C. H. Bennett,et al.  Quantum nonlocality without entanglement , 1998, quant-ph/9804053.

[43]  Scott M. Cohen.  Local distinguishability with preservation of entanglement , 2007 .

[44]  Charles H. Bennett,et al.  Exact and asymptotic measures of multipartite pure-state entanglement , 1999, Physical Review A.

[45]  J. Kowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[46]  Hoi-Kwong Lo,et al.  Randomly distilling W-class states into general configurations of two-party entanglement , 2011 .

[47]  M. Horodecki,et al.  Limits for entanglement measures. , 1999, Physical review letters.

[48]  E. Stachow.  An Operational Approach to Quantum Probability , 1978 .

[49]  Hoi-Kwong Lo,et al.  Increasing entanglement monotones by separable operations. , 2012, Physical review letters.

[50]  Anthony Chefles.  Condition for unambiguous state discrimination using local operations and classical communication , 2004 .

[51]  J. Cirac,et al.  Entangling operations and their implementation using a small amount of entanglement. , 2000, Physical review letters.

[52]  M. Horodecki,et al.  Separability of mixed states: necessary and sufficient conditions , 1996, quant-ph/9605038.

[53]  M. Lewenstein,et al.  Classification of mixed three-qubit states. , 2001, Physical review letters.

[54]  Mikhail N. Vyalyi,et al.  Classical and quantum codes , 2002 .

[55]  Michal Horodecki,et al.  Entanglement measures , 2001, Quantum Inf. Comput..