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Characterizing Entanglement and Quantum Correlations Constrained by Symmetry pp 13–35Cite as

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

Abstract

In this chapter we present the basics that will be used in the rest of the thesis, as well as the results that represent the state of the art. Expert readers may skip this chapter.

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Notes

  1. 1.

    The authors argued that any complete theory should have an element that describes every ‘element of reality’ (i.e., a physical quantity whose values can be predicted with certainty without disturbing the system).

  2. 2.

    Unless stated otherwise, throughout this Thesis we consider that \(\mathcal{H}\) is a vector space defined over the field of complex numbers, denoted \(\mathbbm {C}\).

  3. 3.

    A problem belongs to the class of complexity NP-hard if any algorithm that solves it can be translated in polynomial time into one solving any problem in NP. Hence, an NP-hard problem is as hard as any problem in NP, although it might be harder.

    NP stands for Nondeterministic Polynomial time and it consists of all problems whose solution can be verified in polynomial time by a deterministic Turing machine.

  4. 4.

    The upper bound \(s \le d_1^2d_2^2\), stems from Carathodory’s theorem [Car11]: Any state expressed as a convex combination like in Eq. (2.2) can be re-expressed as another convex combination of no more than \(\dim _{\mathbbm {R}}\mathcal{D}(\mathcal{H}_{AB})\) terms, as \(\mathcal{D}(\mathcal{H}_{AB})\) can be embedded into the \({\mathbbm {R}}\)-vector space of \(d_1d_2\times d_1d_2\) Hermitian matrices, which has dimension \(d_1^2d_2^2\).

  5. 5.

    A decomposable map \(\Lambda \) can be written as \(\Lambda = \Lambda _{1} + \Lambda _{2}\circ T\), where \(\Lambda _1\) and \(\Lambda _2\) are CP maps. This fact is intimately related to the decomposability of entanglement witnesses via the Choi-Jamiołkowski-Sudarshan isomorphism [Jam72, Cho75].

  6. 6.

    Note that \(\Pi _W\)does not form a subspace; in fact, it can be a finite set.

  7. 7.

    As an example of \(W \in \partial \mathcal{W} \setminus \mathrm {Opt}(\mathcal{W})\), consider the line segment \(W(p)=pW_{+}+(1-p)W_{-} \in M_2 \otimes M_2\) and consider the Bell basis \(| \psi _{\pm } \rangle =(| 00 \rangle \pm | 11 \rangle )/\sqrt{2}\), \(| \phi _{\pm } \rangle =(| 01 \rangle \pm | 10 \rangle )/\sqrt{2}\). Pick \(W_{\pm }=| \psi _{\pm } \rangle \langle \psi _{\pm } |^{T_B} \in \mathrm {Ext}(\mathcal{W})\). For any \(p \in [0,1/2) \cup (1/2,1]\), \(W\in \mathcal{W}\), whereas \(W(1/2)\succeq 0\). Consequently, \(W(p)\notin \mathrm {Opt}(\mathcal{W})\) for any \(0<p<1\). Hence, by moving to one of the extremes of the segment, W(p) can be optimized. \(W(p) \in \partial \mathcal{W}\) because for every \(p\in [0,1]\) and for any \(\varepsilon >0\), \(\mathcal{W}(p) - \varepsilon | \phi _+ \rangle \langle \phi _+ |^{T_B} \notin \mathcal{W}\).

    As an example of \(W\in \mathrm {Opt}(\mathcal{W}) {\setminus }\mathrm {Ext}(\mathcal{W})\), a decomposable witness of the form \(W=Q^{T_A}\) with \(Q\in M_2\otimes M_2\), \(Q \succeq 0\) and \(\mathrm {supp}(Q)\) being a Completely Entangled Subspace (CES) is optimal [Lew+00]; however it is not extremal if \(\mathrm {rank}(Q)>1\). A CES is a subspace containing no product vectors (see e.g. [ATL11]).

  8. 8.

    Exposed EWs form a subset of \(\mathrm {Ext}({\mathcal{W}})\) [HK11]. All extremal decomposable EWs are exposed [CS14].

  9. 9.

    The typical example is the transposition map, which detects all \(2\otimes 2\) and \(2\otimes 3\) states, whereas its corresponding entanglement witness detects just a subset of them [HHH96].

  10. 10.

    There exist other frameworks in which can study nonlocality, such as the ones considered in Sect. 5.5.1. In this Thesis, we consider the typical framework in which parties perform a single measurement on a single copy of their resource and repeat the experiment in the same conditions.

  11. 11.

    For instance, if Alice has to choose between measuring the spin of a electron in the direction x and measuring the spin in the direction z, her choice has to be independent on the state of the electron; in other words, the electron cannot know what Alice is going to measure. This situation is relevant in the framework of quantum cryptography tasks, where the manufacturer of the devices and/or the provider of entangled particles is untrusted and can use this information to fake the statistics \(P(\vec {a}|\vec {x})\), compromising security (see Sect. 6.3.2).

  12. 12.

    A convex polyhedra admits this dual description as well, if we allow for vertices to be at infinity. Some programs avoid this by working in the Projective space instead of the Affine space, by treating points as rays, for example [Fuk14].

  13. 13.

    This follows from a simple combinatorical argument: The number of independent components of \(\vec {P}\in \mathbf {P}_{NS}\) is given by the normalization conditions of probabilities and the number of different marginals because of the NS principle: For every party, one can choose whether to measure it or not; if it is indeed measured, there are m possible measurements to perform, and for each measurement there are \(d-1\) outcomes to specify (because the last outcome can always be recasted as a function of the rest by means of the normalization conditions). If nobody measures, there is no value needed to specify, so we rule out this possibility.

  14. 14.

    A spectrahedron is the feasible set of a Semi-Definite Program (SDP).

  15. 15.

    Note, however, that in polytope theory, a tight inequality is one which just touches the polytope.

  16. 16.

    See also [Fri12] for an interesting duality relation between the vertices and facets of \(\mathbf {P}_{NS}\) and \(\mathbf {P}_L\) in the (n, 2, 2) scenario. A repository of the currently known Bell inequalities can be found in [RBG14].

  17. 17.

    The same argument applies to PR-boxes [Bar+05].

  18. 18.

    There is another formulation of measure of nonlocality formulated by Elitzur–Popescu–Rohrlich (EPR2) [EPR92], which measures the nonlocal content of \(\vec {P}\) by decomposing it as a convex combination of a no-signalling distribution \(\vec {P}_{NS}\in \mathbf {P}_{NS}\) and a local distribution \(\vec {P}_L \in \mathbf {P}_{L}\) with maximal p: \(\vec {P}=p\vec {P}_{NS}+(1-p)\vec {P}_{L}\).

  19. 19.

    A nice way to see this is via the so-called Majorana representation [Maj32], which assigns a product state to every pure Dicke state; when taking a superposition of all permutations of this product state, one recovers the original Dicke state. For \(d=2\) this assignment is unique and it can be easily visualized in the Bloch sphere. Then, the action of \(U^{\otimes n}\) is just a rotation of the Bloch sphere [Mar11].

  20. 20.

    It was shown in [HHH98] that bipartite PPT states cannot be distilled; i.e., no matter how many copies of a non-distillable state are available, there is no protocol that would produce a pure maximally entangled state \(| \psi ^+ \rangle \). This is the reason why bound entanglement (i.e., entanglement of undistillable states) is considered the weakest form of entanglement.

    Interestingly, a conjecture by Peres related the concepts of nonlocality and bound entanglement, claiming that all bound entangled states admit a local model. The intuition that bound entanglement is too weak to violate a Bell inequality was proven to be false both in the multipartite [VB12] and the bipartite [VB14] scenarios very recently.

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  1. ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860, Castelldefels (Barcelona), Spain

    Jordi Tura i Brugués

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  1. Jordi Tura i Brugués
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Tura i Brugués, J. (2017). Background. In: Characterizing Entanglement and Quantum Correlations Constrained by Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-49571-2_2

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