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Characterizing Entanglement and Quantum Correlations Constrained by Symmetry pp 73–137Cite as

Nonlocality in Multipartite Quantum States

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Part of the book series: Springer Theses ((Springer Theses))

Abstract

Correlations that go beyond the paradigm of local realism (i.e., those that do not admit a Local Hidden Variable Model (LHVM)) are referred to as nonlocal.

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Notes

  1. 1.

    These correlations are often referred to as classical, local or Einstein–Podolsky–Rosen (EPR) [EPR35]. Throughout this Thesis, we use these terms indistinctively to indicate that correlations can be explained via a LHVM.

  2. 2.

    The term nonlocal is often used in many-body systems to refer to, e.g., the range of interactions. In this Thesis, the term nonlocal will refer to Bell’s nonlocality, unless stated otherwise.

  3. 3.

    Or, at least, they should be independent from the state of the system, which we describe with a hidden variable \(\lambda \).

  4. 4.

    As we shall see in Sect. 6.3.2, free will (what we mathematically quantify as the degree of independence between measurement settings and the state of the system) can be amplified; i.e., one can increase the quality of the randomness used in a Bell experiment, provided that the initial randomness is good enough.

  5. 5.

    In that case, one can never escape the circular argument that, in order to obtain certified randomness by violating a Bell inequality, one would need certified randomness in the first place to run the Bell experiment and choose the measurements. This choice could be done via another Bell experiment, that would need certified randomness to choose its inputs, etc.

  6. 6.

    Take, for instance, lattice spin models that are described by local (here local stands for finite interaction range, or interactions that decay rapidly with the distance) Hamiltonians. In the ground states of such models (the states with the lowest energy), the following properties are true (see [Tur+15a, Tur+15b] and references therein, such as [ECP10]):

    1. 1.

      The reduced density matrix of two spins typically displays entanglement if the spins are close in distance, even at criticality. However, entanglement measures still show signatures of QPTs.

    2. 2.

      One can also try to perform optimized measurements on the rest of the system, in order to concentrate entanglement in two chosen spins. This is the idea behind the concept of localizable entanglement. At standard QPTs, the entanglement length diverges as the correlation length diverges. However, there exist critical systems for which the correlation length remains finite, whereas the entanglement localization length diverges to infinity.

    3. 3.

      For systems that are not at criticality, the low energy states -ground states- exhibit area laws: the entropy (von Neumann or, more generally, Rényi) of the reduced density matrix corresponding to a block scales as the length of the boundary of the block. At criticality, one often needs to apply a logarithmic correction to the growth. If the system is 1-dimensional, these are well studied results; however higher-dimensional cases are full of open questions.

    4. 4.

      Ground states and states that appear as low energy states of physical Hamiltonians can be efficiently described with Matrix Product States (MPS) and, more generally, tensor network techniques.

    5. 5.

      The spectrum of the logarithm of a reduced density matrix of a block is typically referred to as entanglement spectrum. Topological order is typically exhibited in its properties for gapped one- and two-dimensional systems; in 2D, the appearance of the so-called topological entropy gives a negative constant correction to the area laws.

    Most of these results also hold for lattice Bose and Fermi models; even for quantum field theories.

  7. 7.

    One can never violate a Bell inequality with one-body correlators only.

  8. 8.

    The reason is that if \(\vec {P} = \sum _{i}\lambda _i \vec {Q}_i\), where \(\lambda _i\) form a convex combination and \(\vec {Q}_i \in \mathbb {P}\), then \(\pi (\vec {P})=\sum _i{\lambda _i} \pi (\vec {Q}_i)\), where \(\pi (\vec {Q}_i)\in \pi (\mathbb {P})\).

  9. 9.

    Equation (4.5) follows from a counting argument. For a fixed correlator length k, there are \(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) ways to choose the parties involved in the correlator; then, one has to choose k measurements to perform out of m, which can be repeated as they correspond to different parties, and k times \(d-1\) outcomes, because of the normalization condition that makes the last outcome redundant. When \(K=n\), one recovers the general bound \((m(d-1)+1)^n-1\).

  10. 10.

    This group can be seen as a subgroup of the symmetric group of n elements \({\mathfrak S}_n\), denoted \(G\le {\mathfrak S}_n\), as it applies a subset of all possible permutations to the parties.

  11. 11.

    By simple analogy to the cardinality of the set of functions from X to Y, since \(|Y^X|=|Y|^{|X|}\) in the finite case.

  12. 12.

    This is a well-defined quantity, as every permutation decomposes uniquely into a product of disjoint cycles (modulo a permutation of the cycles).

  13. 13.

    The signed Stirling number of the first kind has an additional factor \((-1)^{n-k}\) in front, and it corresponds to the coefficients of the falling factorial \(x(x-1)\ldots (x-n+1)\) [AS65].

  14. 14.

    In particular, those inequalities that correspond to facets.

  15. 15.

    The convex hull of a set S is the smallest convex set containing S. If the convex hull is a closed set, then it can be defined as the intersection of all closed half-spaces containing S. This happens when S is compact. In particular, in our case, when S is finite.

  16. 16.

    The sets \(\varphi (\partial {\mathbf T}_n)\) and \(\varphi ({\mathbf T}_n)\) are defined through polynomial equalities and inequalities. We say that they are semialgebraic. The characterization of convex hulls of semialgebraic sets is a well studied subject [GT]. Although its exact characterization is an NP-hard problem [BPT], there exist efficient approximations with semi-definite programming techniques in terms of the so-called theta bodies. As we shall discuss in Sect. 4.5, these techniques can be directly applied to any (n, m, d) scenario with K-body correlators when the symmetry group G is \({\mathfrak S}_n\).

    The Navascués-Pironio-Acín (NPA) hierarchy [NPA08] mentioned in Sect. 2.2.2 is also in the spirit of such approximations, although for the case of non-commutative variables. When the NPA hierarchy gives a certificate that a set of correlations is outside \({\mathbf Q}_k\) for some k, then such correlations cannot be realized with quantum resources. In our case, a characterization of \({\mathbf {P}_2^{{\mathfrak S}_n}}\) through theta bodies produces also a hierarchy of sets \({\mathbf {P}_2^{{\mathfrak S}_n}}\subseteq \cdots \subseteq \Theta _2 \subseteq \Theta _1\) that can certify if a set of correlations which is outside of \(\Theta _k\) for some k; in such case, the correlations under study cannot be simulated through shared randomness and they must be necessarily nonlocal. To our knowledge, this technique has never been used in order to decide between local and non-local correlations.

  17. 17.

    It states that, for a square matrix A, a linear system of equations \(A\vec {b}=\vec {c}\) has some solution(s) (is compatible) if, and only if, the rank of A is the same as the rank of the extended matrix \(A|\vec {c}\). This solution is unique, if and only if, \(\det {A}\ne 0\).

  18. 18.

    Take, for example, the Greenberger–Horne–Zeilinger (GHZ) state \(2^{-n/2}(| 0 \rangle ^{\otimes n}+| 1 \rangle ^{\otimes n})\), which is supported on the last block \(J=n/2\) of the decomposition (A.16). The density matrix of the state has only four terms; two of them in the diagonal, and the remaining ones are the coherences \(| D_n^n \rangle \langle D_n^0 |\) and \(| D_n^0 \rangle \langle D_n^n |\), which can be reached only a full-body correlator. Hence, to a not-full-body correlations symmetric Bell inequality, the GHZ state is indistinguishable from the separable mixture \(| 0 \rangle \!\langle 0 |^{\otimes n}/2+| 1 \rangle \!\langle 1 |^{\otimes n}/2\).

  19. 19.

    In other scenarios, such as (n, 3, 2), we have also found other classes of Bell inequalities maximally violated with states supported on the lowest J blocks.

  20. 20.

    Since numerically we could easily solve the polytope \(\mathbb {P}_2^{\mathfrak {S}_n}\) for \(n\le 33\), the best inequalities (in terms of the maximal quantum violation over the classical bound) were found to be those from Theorem 4.4.

  21. 21.

    One simply renames the outcomes of all the observables.

  22. 22.

    We have numerically checked that the inequalities presented are optimal (with respect to the quantum violation relative to the classical bound) for the Dicke states. But the Dicke states need not be the optimal states for such inequalities; in general, they are not.

  23. 23.

    Take, for instance, \(K=4\). There are two partitions of 4 with 2 elements: (2, 2) and (3, 1). For the (2, 2) partition, there are 3 permutations of 4 elements of the form \((\cdot \cdot )(\cdot \cdot )\) (recall that disjoint cycles commute) and, for the (3, 1) partition, one finds 8 permutations of the form \((\cdot \cdot \cdot )(\cdot )\). Hence, \(\left[ \begin{array}{c}4 \\ 2 \end{array}\right] =3+8=11\).

  24. 24.

    According to Eq. (4.122), for the (4, 2, 2) scenario we should have \(|Y^X/\mathbb {Z}_4|=(1\times 4^4+1\times 4^2+2\times 4)/4=70\) candidates to vertex. However, explicitly solving the polytope with, for example, the CDD algorithm [Fuk14], gives that \(|\mathrm {Ext}(\mathbb {P}_2^{{\mathbb {Z}}_4})|=68\). This is because the DLSs \(f: A\mapsto (+,+),\ B\mapsto (-,+),\ C \mapsto (+,-),\ D \mapsto (-,-)\) and \(\tilde{f}: A \mapsto (+,+),\ B\mapsto (-,-),\ C \mapsto (+,-),\ D \mapsto (-,+)\) give exactly the same values on the translationally invariant correlators (4.117, 4.118). The same happens for \(g: A\mapsto (+,+),\ B\mapsto (+,-),\ C \mapsto (-,+),\ D \mapsto (-,-)\) and \(\tilde{g}: A \mapsto (+,+),\ B\mapsto (-,-),\ C \mapsto (-,+),\ D \mapsto (+,-)\). Thus, f and \(\tilde{f}\) (as well as g and \(\tilde{g}\)) are projected to the same element in \(\mathrm {Ext}(\mathbb {P}_2^{{\mathbb {Z}}_4})\).

  25. 25.

    Let us recall that for n-partite states \(| \varphi \rangle \) in \(\mathcal{H}=({\mathbb {C}^d})^{\otimes n}\), the geometric measure of entanglement is defined as \(E_G(| \varphi \rangle )=1-\max _{| \phi _{\mathrm {prod}} \rangle }|\langle \phi _{\mathrm {prod}} | \varphi \rangle |^2\), where \(| \varphi _{\mathrm {prod}} \rangle =| e_1 \rangle \cdots | e_n \rangle \) [WG03].

  26. 26.

    Noticeably, such inequalities have some similarity with the Guess-Your-Neighbor-Input (GYNI) Bell inequalities [Alm+10]. Operationally, they can be regarded as distributed tasks for which the aid of quantum resources does not provide any advantage over classical ones. However, there exist no-signalling correlations which are beyond the quantum set of correlations \(\mathbf {Q}\) that perform better at such task. From the geometrical point of view, GYNI inequalities constitute facets of \(\mathbf {P}_L\); however only the 21st inequality has this property for \(\mathbb {P}_2^{{\mathbb {Z}}_4}\).

  27. 27.

    A translationally invariant n-qubit state \(| \psi \rangle \) is such that, for any k, \(\tau ^k(| \psi \rangle )=| \psi \rangle \), where \(\tau \in {\mathfrak S}_n\) is the permutation that shifts to the right.

  28. 28.

    We have used that \(\mathcal{S}_{xz}=2\{J_x,J_z\}\).

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    Jordi Tura i Brugués

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Tura i Brugués, J. (2017). Nonlocality in Multipartite Quantum States. In: Characterizing Entanglement and Quantum Correlations Constrained by Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-49571-2_4

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