A new method for quantifying entanglement of multipartite entangled states

We propose a new way for quantifying entanglement of multipartite entangled states which have a symmetrical structure and can be expressed as valence-bond-solid states. We put forward a new concept ‘unit.’ The entangled state can be decomposed into a series of units or be reconstructed by multiplying the units successively, which simplifies the analyses of multipartite entanglement greatly. We compute and add up the generalized concurrence of each unit to quantify the entanglement of the whole state. We verify that the new method coincides with concurrence for two-partite pure states. We prove that the new method is a good entanglement measure obeying the three necessary conditions for all good entanglement quantification methods. Based on the method, we compute the entanglement of multipartite GHZ, cluster and AKLT states.

[1]  Robert Raussendorf,et al.  Fault-tolerant quantum computation with high threshold in two dimensions. , 2007, Physical review letters.

[2]  V. Vedral,et al.  Entanglement in Many-Body Systems , 2007, quant-ph/0703044.

[3]  H. Bechmann-Pasquinucci,et al.  Quantum cryptography , 2001, quant-ph/0101098.

[4]  J. Eisert,et al.  Schmidt measure as a tool for quantifying multiparticle entanglement , 2000, quant-ph/0007081.

[5]  M. Horodecki,et al.  Quantum entanglement , 2007, quant-ph/0702225.

[6]  A. Zeilinger,et al.  Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons. , 2015, Physical review letters.

[7]  S. Wehner,et al.  Bell Nonlocality , 2013, 1303.2849.

[8]  E. Knill,et al.  A strong loophole-free test of local realism , 2015, 2016 Conference on Lasers and Electro-Optics (CLEO).

[9]  R. Mcweeny.  On the Einstein-Podolsky-Rosen Paradox , 2000 .

[10]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

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

[13]  V. Vedral.  The role of relative entropy in quantum information theory , 2001, quant-ph/0102094.

[14]  J I Cirac,et al.  Diverging entanglement length in gapped quantum spin systems. , 2004, Physical review letters.

[15]  M. Lewenstein,et al.  Volume of the set of separable states , 1998, quant-ph/9804024.

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

[17]  Yi Huang,et al.  Concurrence vectors of multipartite states based on coefficient matrices , 2012, Quantum Inf. Process..

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

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

[20]  S. Fei,et al.  A note on invariants and entanglements , 2001, quant-ph/0109073.

[21]  E. Schrödinger.  Discussion of Probability Relations between Separated Systems , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[23]  J. Cirac,et al.  Entanglement percolation in quantum networks , 2006, quant-ph/0612167.

[24]  H. Barnum,et al.  Monotones and invariants for multi-particle quantum states , 2001, quant-ph/0103155.

[25]  E. Lieb,et al.  Valence bond ground states in isotropic quantum antiferromagnets , 1988 .

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

[27]  M. Genovese.  Research on hidden variable theories: A review of recent progresses , 2005, quant-ph/0701071.