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The RT formula and its discontents: spacetime and entanglement

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Abstract

This essay is concerned with a number of related proposals that claim there is a link between spacetime topology and quantum entanglement. I indicate the extent to which these proposals can be understood as stating a duality, and then consider two general approaches to articulating such a duality: a “state-based” approach, under which one attempts to identify relevant topological states as dual to quantum entangled states; and an “observable-based” approach, under which one attempts to identify relevant topological observables as dual to quantum entangement observables. Both approaches are faced with issues, essentially due to the ambiguous nature of quantum entanglement, that remain to be addressed.

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Notes

  1. See, e.g., de Haro (2017, p. 110); de Haro and Butterfield (2019, p. 20). My use of the term in this essay is restricted to the entries (states and/or observables) in the “dictionary” that this mapping sets up.

  2. Swanson (2018) indicates that one route to establishing an equivalence between state-based and observable-based approaches in RQFT requires nothing less than a complete reformulation of algebraic quantum field theory (AQFT), in which the basic object of the latter, a net of local \({\mathcal {C}}^*\)-algebras (the elements of which represent local observables), is replaced with a presheaf of convex oriented sets (the elements of which represent local states). To establish a formal equivalence between these objects requires, in addition, replacing the axioms of AQFT with a set of new axioms that have yet to be fully specified.

  3. A bipartite state \(\rho _{AB}\) is decomposable if and only if it can be written as a convex combination of product states, \(\rho _{AB} = \sum _ip_i\rho _A^i\otimes \rho _B^i\), where \(\sum _ip_i=1\), and \(0 \le p_i \le 1\). (Here I follow Earman 2015, p. 311, in using the term “decomposable” as opposed to “separable”.) For pure states, decomposability reduces to being a product state, and one can show that the above entropic inequality holds if and only if the bipartite state is a product state (Nielson and Chuang 2010, p. 514).

  4. Indecomposability is, arguably, the standard way of defining quantum entanglement in the physics literature (see, e.g., Horodecki et al. 2009, p. 873). On the other hand, Earman (2015) considers four distinct notions of a quantum entangled state, the weakest being a non-product state and the strongest involving a violation of a Bell inequality, with indecomposability ranked in-between.

  5. As Hartman (2015, p. 175) indicates, in a continuum quantum field theory in Minkowski spacetime, there are high-energy modes at small scales across any surface that divides the system into two regions, and this requires a regularization scheme to prevent the entanglement entropy defined with respect to the surface from becoming divergent. Srednicki (1993, p. 669) adopted a high-energy cutoff defined by \(M = a^{-1}\), where a is the spacing between sites on a discrete lattice, and derived the relation \(S_R = \kappa M^2(4\pi r^2)\), where r is the radius of R and \(\kappa \) is a constant. This relation holds specifically for a massless scalar field in flat Minkowski spacetime, and assumedly could be extended (in standard limited cases) to a relation in a curved spacetime using techniques from quantum field theory in curved spacetimes. Equation (1) is formally similar to Bekenstein’s formula for the thermodynamic entropy of a black hole, \(S_{BH} = Area(horizon)/4G\).

  6. These authors were primarily concerned with gapped systems that exhibit “topological order”. The subsequent notion of “topological entanglement entropy” was proposed as a way of characterizing the latter.

  7. More precisely, the gap property entails that the system has a finite correlation length \(\xi \), and this entails that for observables A, B with support on regions X, Y, \(\langle AB \rangle - \langle A \rangle \langle B \rangle \sim e^{-dist(X,Y)/\xi }\), where dist(X, Y) is the distance between X and Y, and expectation values are taken in the ground state. Thus contributions to \(S_R\) should come from a strip on either side of \(\partial R\) of width \(\xi \) (Pachos 2012, p. 179).

  8. Equation (2) is due to Kitaev and Preskill (2006), who use the phrase “topological entropy” in referring to \(\Gamma \).

  9. Kitaev and Preskill (2006) reached the same conclusion using a different configuration of regions and a different linear combination of entanglement entropies.

  10. To show that \((S_1 - S_2) - (S_3 - S_4) = 2\Gamma \), one notes, for instance, that the region \(R_1\) has two disconnected boundaries (\(\partial R_{1a}\) and \(\partial R_{1b}\) in Fig. 1), and assumes each contributes a separate topological correction \(\Gamma \) to \(S_1\). Thus \(S_1\) and \(S_4\) each contain two \(\Gamma \)’s (\(R_4\) also has two disconnected boundaries, \(\partial R_{4a}\) and \(\partial R_{4b}\)), whereas \(S_2\) and \(S_3\) each contain a single \(\Gamma \). The difference \((S_1 - S_2) - (S_3 - S_4)\) thus contains the term \(2\Gamma \), as well as the sum of boundary lengths \(\{( | \partial R_{1a} | + | \partial R_{1b} | ) - | \partial R_2 |\} - \{ | \partial R_3 | - ( | \partial R_{4a} | + | \partial R_{4b} | ) \}\), which is zero.

  11. See, e.g., Teh (2013), De Haro (2017), Vistarini (2017) for reviews of aspects of the AdS/CFT correspondence. See Jaksland (2018) for a discussion of aspects of the RT formula and van Raamsdonk’s interpretation of it.

  12. As Headricks (2019, p. 48) reports, the RT formula (4) without the second term on the right is limited in three ways: The bulk theory must be a (i) classical (ii) Einsteinian theory of gravity, and (iii) the bulk spacetime (in addition to being asymptotically AdS) must have a time-reflection symmetry under which the boundary subregion R is invariant (in order to pick out a bulk timeslice with the relevant properties). These restrictions can be relaxed in various ways: A covariant version of the RT formula can be derived for bulk spacetimes without assuming any symmetries (Hubeny et al. 2007). Higher derivative corrections to the bulk gravitational action can be included for non-Einsteinian theories of gravity, which subsequently requires including terms beyond the area term. Finally, corrections to the Newtonian gravitational constant can be included to move away from classical theories. One result of the latter is the second term on the right in Eq.  (4) due to Faulkner et al. (2013).

  13. As Hartman (2015, p. 175) notes, “Quantum field theory is strictly speaking not bipartite”. To address this, Hartman recommends inserting a UV cutoff to render the theory finite. On the other hand, according to Earman (2015, p. 309), the restriction of any discussion of quantum entanglement to finite tensor product Hilbert spaces “...is to be deplored because it neglects possibilities that need to be explored”. Moreover, even granted this restriction, there still remain various ambiguities associated with quantum entanglement. While I won’t attempt to address Earman’s general concern in this essay, Sect. 5.3 attempts to address some of the latter concerns.

  14. van Raamsdonk (2010, p. 2326) describes the limit as a process in which “the two regions of space are pinching off from each other”, but immediately qualifies this by cautioning “Here and below, we should keep in mind that the spacetime will likely cease to have a completely geometrical description before the entanglement is strictly zero”. Elsewhere, he suggests that “without entanglement, we have a product state in two non-interacting systems, and the only possible interpretation would be two disconnected spacetimes” (van Raasmdonk 2016, p. 22). Further motivation for this interpretation of the limit comes from an argument that relates the mutual information I(C, D) of subsystems localized in bulk subregions \(C \subset H_R\) and \(D \subset \overline{H_R}\) near the boundary, on the one hand, to the spatiotemporal distance d(p, q) between bulk points \(p \in C\) and \(q \in D\) on the other hand: the argument shows that as I(C, D) decreases to zero, d(p, q) increases to infinity (van Raamsdonk 2010, p. 2327). Thus, insofar as I(C, D) is a measure of the degree of entanglement of subsystems localized in C and D, as the entanglement near the boundary goes to zero, the distance between bulk points near the boundary increases. This suggests a process in which “the two regions of [bulk] spacetime pull apart and pinch off from each other” (van Raamsdonk 2010, p. 2327).

  15. One way of carrying this out is suggested by Bao et al. (2015a) who attempt to identify “no-go” theorems for Einstein-Rosen wormholes that are dual to no-go theorems associated with quantum entangled states.

  16. See, e.g., Alagic et al. (2016, p. 2) for a discussion of this theorem due to Alexander.

  17. This definition of a quantum entangled state vector (i.e., a pure state) as a non-product state is sufficient for the purpose of an initial comparison with Def. 1. However, it does not fully capture the sense of non-locality associated with the correlations that subsystems in a quantum entangled state may exhibit, as Sect. 5 below indicates.

  18. See, e.g., Alagic et al. (2016). Conditions (11) and (12) guarantee that R satisfies the braiding relations in the definition of \(B_n\) in Sect. 3 above.

  19. Recall from Sect. 3 that an n-braid is topologically entangled just when it contains at least one pairwise set of terms \(\sigma _i \sigma _j\) such that \(\sigma _i \sigma _j \ne 1\). Recall, too, that we are (still!) bracketing off the concern over defining an entangled state as a non-product state.

  20. Kauffman and Lomonaco (2002, p. 8). Let \(\{|0 \rangle , |1 \rangle \}\) be a basis for \({\mathbf {C}}^2\), and let \(|\Phi \rangle = \{|0 \rangle + |1 \rangle \}\{|0 \rangle + |1 \rangle \}\). Then \(R|\Phi \rangle = \alpha |00 \rangle + \gamma |10 \rangle + \delta |01 \rangle + \beta |11 \rangle \). This is a product state just when \(\alpha |00 \rangle + \gamma |10 \rangle + \delta |01 \rangle + \beta |11 \rangle = \{X|0 \rangle + Y|1 \rangle \}\{X'|0 \rangle + Y'|1 \rangle \}\), and this requires \(\alpha = XX'\), \(\gamma = X'Y\), \(\delta = XY'\), \(\beta = YY'\). This holds just when \(\alpha \beta = \delta \gamma \).

  21. The appearance of \(\mu \) in the trace ensures that the latter is invariant under the Markov moves, which are a set of transformations that leave the closure of a braid invariant.

  22. Alagic et al. (2016, p. 10) explicitly show this by counterexample.

  23. See, e.g., Kauffman and Lomonaco (2002, p. 10). The trivial 2-link can be obtained from the closure of the 2-braid \(\sigma _1 \sigma _1^{-1}\), which is not topologically entangled. The Whitehead 2-link can be obtained from the closure of the 3-braid \(\sigma _1 \sigma _2^{-1}\sigma _1 \sigma _2^{-2}\), which is topologically entangled.

  24. For a pure bipartite state, one can show that indecomposability is both necessary and sufficient for a violation of a Bell inequality (Brunner et al. 2014, p. 437). On the other hand, a pure bipartite state is indecomposable if and only if it is a non-product state, and “non-productness” is both necessary and sufficient for a violation of an entropic inequality (Nielson and Chuang 2010, p. 514, as noted above in footnote 2).

  25. Statements to this effect can be found in, e.g., Bovino et al. (2005, p. 1), Walborn et al. (2006, p. 1022), Mintet and Buchleitner (2007, p. 1).

  26. Recall that a linear operator O on a Hilbert space \({\mathcal {H}}\) is a map \(O : {\mathcal {H}} \rightarrow {\mathcal {H}}\) such that \(O(\alpha |\phi \rangle + \beta |\psi \rangle ) = \alpha O | \phi \rangle + \beta O | \psi \rangle \), for any \(\alpha , \beta \in {\mathbf {C}}\) and any \(|\phi \rangle , |\psi \rangle \in {\mathcal {H}}\). Thus if O is linear and responds “yes” to two entangled states separately, then it should respond “yes” to their sum; but the latter may not be an entangled state.

  27. The notion of linearity here is that described in the previous footnote 26. Note that a Wilson loop operator is distinct from a loop state. The latter is a state with respect to which a loop operator has a non-zero expectation value. Issues concerning how to characterize an appropriate space of loop states (arising in “loop representations” of gauge theories, and in loop quantum gravity) are distinct from the linear nature of a loop operator.

  28. For instance, let \(A_1 = {\hat{e}}_1 \cdot \mathbf {\sigma }\), \(A_2 = {\hat{e}}_2 \cdot \mathbf {\sigma }\), \(B_1 = -\sqrt{1/2}({\hat{e}}_1 + {\hat{e}}_2) \cdot \mathbf {\sigma }\), and \(B_2 = -\sqrt{1/2}(-{\hat{e}}_1 + {\hat{e}}_2) \cdot \mathbf {\sigma }\), where \({\hat{e}}_1, {\hat{e}}_2\) are any two choices of spin measurement axes, and \(\mathbf {\sigma } = (\sigma _x, \sigma _y, \sigma _z)\) encodes the Pauli operators.

  29. Here I follow Bain’s (2019, p. 25) distinction between two types of non-locality claimed to be present in intrinsic topologically ordered condensed matter systems.

  30. Indeed, the conditional statistical independence condition (17) is sometimes referred to as “Bell locality”, and the requirement that causal signal propagation not exceed an appropriate bound is sometimes referred to as “Einstein locality”. Quantum entanglement non-locality, Def. 4, can then be thought of as the denial of both Bell locality and Einstein locality.

  31. There is a limited sense in which topological non-locality entails quantum entanglement non-locality in condensed matter systems that exhibit topological order (see, e.g., Bain 2019).

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Acknowledgements

Thanks to Gabriela Avila, Sam Granade, and two anonymous reviewers for very helpful discussion and comments.

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  1. Department of Technology, Tandon School of Engineering , Culture and Society, New York University, 6 Metrotech Center, Brooklyn, NY, 11201, USA

    Jonathan Bain

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Bain, J. The RT formula and its discontents: spacetime and entanglement. Synthese 198, 11833–11860 (2021). https://doi.org/10.1007/s11229-020-02836-4

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