Maximum $N$-body correlations do not in general imply genuine multipartite entanglement

Christopher Eltschka1 and Jens Siewert2,3

1Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
2Departamento de Química Física, Universidad del País Vasco UPV/EHU, E-48080 Bilbao, Spain
3IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain

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Abstract

The existence of correlations between the parts of a quantum system on the one hand, and entanglement between them on the other, are different properties. Yet, one intuitively would identify strong $N$-party correlations with $N$-party entanglement in an $N$-partite quantum state. If the local systems are qubits, this intuition is confirmed: The state with the strongest $N$-party correlations is the Greenberger-Horne-Zeilinger (GHZ) state, which does have genuine multipartite entanglement. However, for high-dimensional local systems the state with strongest $N$-party correlations may be a tensor product of Bell states, that is, partially separable. We show this by introducing several novel tools for handling the Bloch representation.

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[1] U. Fano, A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics, Phys. Rev. 93, 121 (1954).
https:/​/​doi.org/​10.1103/​PhysRev.93.121%20%20%20%20%20

[2] U. Fano, Description of States in Quantum Mechanics by Density Matrix and Operator Techniques, Rev. Mod. Phys. 29, 74 (1957).
https:/​/​doi.org/​10.1103/​RevModPhys.29.74%20%20%20%20%20

[3] G. Mahler and V.A. Weberruß, Quantum Networks, 2nd Edition (Springer, Berlin, 2004).
https:/​/​doi.org/​10.1007/​978-3-662-03669-3%20%20%20%20%20

[4] C. Klöckl and M. Huber, Characterizing multipartite entanglement without shared reference frames, Phys. Rev. A 91, 042339 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.042339%20%20%20%20%20

[5] C. Eltschka and J. Siewert, Monogamy equalities for qubit entanglement from Lorentz invariance, Phys. Rev. Lett. 114, 140402 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.140402

[6] M.-C. Tran, B. Dakic, F. Arnault, W. Laskowski, and T. Paterek, Quantum entanglement from random measurements, Phys. Rev. A 94, 042302 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.92.050301%20%20%20%20%20

[7] P. Appel, M. Huber, and C. Klöckl, Monogamy of correlations and entropy inequalities in the Bloch picture, J. Phys. Commun. (2020), doi:10.1088/​2399-6528/​ab6fb.
https:/​/​doi.org/​10.1088/​2399-6528/​ab6fb4%20%20%20%20%20

[8] F. Huber, O. Gühne, and J. Siewert, Absolutely Maximally Entangled States of Seven Qubits Do Not Exist, Phys. Rev. Lett. 118, 200502 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.200502%20%20%20%20%20

[9] C. Eltschka and J. Siewert, Distribution of entanglement and correlations in all finite dimensions, Quantum 2, 64 (2018).
https:/​/​doi.org/​10.22331/​q-2018-05-22-64%20%20%20%20%20

[10] N. Wyderka, F. Huber, and O. Gühne, Constraints on correlations in multiqubit systems, Phys. Rev. A 97, 060101 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.060101%20%20%20%20%20

[11] F. Huber, C. Eltschka, J. Siewert, and O. Gühne, Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity, J. Phys. A: Math. Theor. 51, 175301 (2018).
https:/​/​doi.org/​10.1088/​1751-8121/​aaade5%20%20%20%20%20

[12] C. Eltschka, F. Huber, O. Gühne, and J. Siewert, Exponentially many entanglement and correlation constraints for multipartite quantum states Phys. Rev. A 98, 052317 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.052317%20%20%20%20%20

[13] T. Cox and P.C.E. Stamp, Partitioned density matrices and entanglement correlators, Phys. Rev. A 98, 062110 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.062110%20%20%20%20%20

[14] N. Wyderka and O. Gühne, Characterizing quantum states via sector lengths, (2019).
arXiv:1905.06928
https:/​/​arxiv.org/​abs/​1905.06928

[15] C. Eltschka and J. Siewert, Joint Schmidt-type decomposition for two bipartite pure states, Phys. Rev. A 101, 022302 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.022302%20%20%20%20%20

[16] J. Schlienz and G. Mahler, Description of entanglement, Phys. Rev. A 52, 4396 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.52.4396%20%20%20%20%20

[17] J. Schlienz and G. Mahler, The maximal entangled three-particle state is unique, Phys. Lett. A 224, 39 (1996).
https:/​/​doi.org/​10.1016/​S0375-9601(96)00803-1%20%20%20%20%20

[18] M. Żukowski and C. Brukner, Bell's theorem for general $N$-qubit states, Phys. Rev. Lett. 88, 210401 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.210401%20%20%20%20%20

[19] M. Teodorescu-Frumosu and G. Jaeger, Quantum Lorentz-group invariants of $n$-qubit systems, Phys. Rev. A 67, 052305 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.052305%20%20%20%20%20

[20] H. Aschauer, J. Calsamiglia, M. Hein, and H.J. Briegel, Local invariants for multi-partite entangled states allowing for a simple entanglement criterion, Quantum Inf. Comput. 4, 383 (2004); journal link; arXiv.org link.
https:/​/​doi.org/​10.5555/​2011586.2011590%20%20%20%20%20
arXiv:quant-ph/0306048

[21] A. J. Scott, Multipartite entanglement, quantum error correcting codes, and entangling power of quantum evolutions, Phys. Rev. A 69, 052330 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.052330%20%20%20%20%20

[22] J.I. de Vicente, Separability criteria based on the Bloch representation of density matrices, Quantum Inf. Comput. 7, 624 (2007); journal link; arXiv.org link.
https:/​/​doi.org/​10.5555/​2011734.2011739%20%20%20%20%20
arXiv:quant-ph/0607195

[23] J.I. de Vicente, Further results on entanglement detection and quantification from the correlation matrix criterion, J. Phys. A: Math. Theor. 41, 065309 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​6/​065309%20%20%20%20%20

[24] P. Badziag, C. Brukner, W. Laskowski, T. Paterek, and M. Żukowski, Experimentally Friendly Geometrical Criteria for Entanglement, Phys. Rev. Lett. 100, 140403 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.100.140403%20%20%20%20%20

[25] W. Laskowski, M. Markiewicz, T. Paterek, and M. Żukowski, Correlation-tensor criteria for genuine multiqubit entanglement, Phys. Rev. A 84, 062305 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.062305%20%20%20%20%20

[26] J.I. de Vicente and M. Huber, Multipartite entanglement detection from correlation tensors, Phys. Rev. A 84, 062306 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.062306%20%20%20%20%20

[27] We will use the term ``$k$-sector length'' instead of ``squared $k$-sector length'' following Ref. Tran2016. In the present context this does not lead to confusion.

[28] One may imagine very different correlation quantifiers, e.g., D. Girolami, T. Tufarelli, and C.E. Susa, Quantifying Genuine Multipartite Correlations and their Pattern Complexity, Phys. Rev. Lett. 119, 140505 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.140505%20%20%20%20%20

[29] J. Kaszlikowski, A. Sen De, U. Sen, V. Vedral, A. Winter, Quantum Correlation Without Classical Correlations, Phys. Rev. Lett. 101, 070502 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.070502%20%20%20%20%20

[30] C. Schwemmer, L. Knips, M.C. Tran, A. de Rosier, W. Laskowski, T. Paterek, and H. Weinfurter, Genuine Multipartite Entanglement without Multipartite Correlations, Phys. Rev. Lett. 114, 180501 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.180501%20%20%20%20%20

[31] M.C. Tran, M. Zuppardo, A. de Rosier, L. Knips, W. Laskowski, T. Paterek, and H. Weinfurter, Genuine $N$-partite entanglement without $N$-partite correlation functions, Phys. Rev. A 95, 062331 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.062331%20%20%20%20%20

[32] W. Klobus, W. Laskowski, T. Paterek, M. Wiesniak, and H. Weinfurter, Higher dimensional entanglement without correlations, Eur. Phys. J. D 73, 29 (2019).
https:/​/​doi.org/​10.1140/​epjd/​e2018-90446-6%20%20%20%20%20

[33] This relation corresponds to a special case of the quantum MacWilliams identity, cf. Ref. Huber2018.

[34] V. Coffman, J. Kundu, and W.K. Wootters, Distributed entanglement, Phys. Rev. A 61, 052306 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.61.052306%20%20%20%20%20

[35] P. Rungta, V. Buzek, C.M. Caves, M. Hillery, and G.J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A 64, 042315 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.042315

[36] W. Hall, Multipartite reduction criteria for separability, Phys. Rev. A 72, 022311 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.022311

[37] M. Lewenstein, R. Augusiak, D. Chruściński, S. Rana, and J. Samsonowicz, Sufficient separability criteria and linear maps, Phys. Rev. A 93, 042335 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.042335

[38] An in-depth analysis of this projection operator will be carried out in forthcoming work.

[39] D. Goyeneche and K. Życzkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A 90, 022316 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.022316%20%20%20%20%20

[40] D. Goyeneche, D. Alsina, J.I. Latorre, A. Riera, and K. Życzkowski, Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices, Phys. Rev. A 92, 032316 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.032316%20%20%20%20%20

Cited by

[1] Andreas Ketterer, Nikolai Wyderka, and Otfried Gühne, "Entanglement characterization using quantum designs", Quantum 4, 325 (2020).

[2] Cornelia Spee, "Certifying the purity of quantum states with temporal correlations", Physical Review A 102 1, 012420 (2020).

[3] Daniel Miller, "Small quantum networks in the qudit stabilizer formalism", arXiv:1910.09551.

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