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Some Puzzles and Unresolved Issues About Quantum Entanglement

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Abstract

Schrödinger (Proc Camb Philos Soc 31:555–563, 1935) averred that entanglement is the characteristic trait of quantum mechanics. The first part of this paper is simultaneously an exploration of Schrödinger’s claim and an investigation into the distinction between mere entanglement and genuine quantum entanglement. The typical discussion of these matters in the philosophical literature neglects the structure of the algebra of observables, implicitly assuming a tensor product structure of the simple Type I factor algebras used in ordinary Quantum Mechanics (QM). This limitation is overcome by adopting the algebraic approach to quantum physics, which allows a uniform treatment of ordinary QM, relativistic quantum field theory, and quantum statistical mechanics. The algebraic apparatus helps to distinguish several different criteria of quantum entanglement and to prove results about the relation of quantum entanglement to two additional ways of characterizing the classical versus quantum divide, viz. abelian versus non-abelian algebras of observables, and the ability versus inability to interrogate the system without disturbing it. Schrödinger’s claim is reassessed in the light of this discussion. The second part of the paper deals with the relativity-to-ambiguity threat: the entanglement of a state on a system algebra is entanglement of the state relative to a decomposition of the system algebra into subsystem algebras; a state may be entangled with respect to one decomposition but not another; hence, unless there is some principled way to choose a decomposition, entanglement is a radically ambiguous notion. The problem is illustrated in terms a Realist versus Pragmatist debate, the former claiming that the decomposition must correspond to real as opposed to virtual subsystems, while the latter claims that the real versus virtual distinction is bogus and that practical considerations can steer the choice of decomposition. This debate is applied to the fraught problem of measuring entanglement for indistinguishable particles. The paper ends with some (intentionally inflammatory) remarks about claims in the philosophical literature that entanglement undermines the separability or independence of subsystems while promoting holism.

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Notes

  1. With the rise of quantum computing and quantum information theory there has been a sea change in attitude towards entanglement: it is not something to be feared but rather is a resource to be exploited.

  2. Only separable Hilbert spaces will be considered here.

  3. For a classification of the types of von Neumann algebras, see Sunder (1986). An algebra \({\mathfrak {N}}\) is a factor just in case its center \({\mathcal {Z}}({\mathfrak {N}}):={\mathfrak {N}}\cap {\mathfrak {N}}^{\prime }={\mathbb {C}}I\). Superselection rules involve algebras with non-trivial centers; see Sect. 10.

  4. See Hamhalter (2003) for details.

  5. There is also a more abstract version of the algebraic approach using \(C^{*}\)-algebras instead of von Neumann algebras. Each version has its own advantages. For the present topic the von Neumann algebra version seems the most illuminating.

  6. Silently because texts on ordinary QM only rarely talk explicitly about algebras of observables. The case of indistinguishable particles will be treated in Sect. 11.

  7. Product states are called “factorizable” in the philosophical literature, although this term is used specifically in the case where \({\mathfrak {N}}_{1}\vee {\mathfrak {N}}_{2}\) has a tensor product structure; see below.

  8. See Bratteli and Robinson (1987, Corr. 2.3.21).

  9. Let \((x,y,z,t)\) be an inertial coordinate system for Minkowski spacetime. Then the right Rindler wedge with vertex at the origin consists of those points \(\infty >x>0, x^{2}-t^{2}>0\). Reflecting about the origin gives the left Rindler wedge.

  10. This fact was apparently first noted by Araki; see Buchholz (1974, p. 292).

  11. For an overview of these requirements, see (Summers 1990, 2009).

  12. To make sure that the tensor product algebra is a von Neumann algebra the overbar is added, i.e. \({\mathfrak {N}}_{1}\overline{\otimes }{\mathfrak {N}} _{2}:=({\mathfrak {N}}_{1}\otimes {\mathfrak {N}}_{2})^{\prime \prime }\). For Type I factors \(({\mathfrak {N}}_{1}\otimes {\mathfrak {N}}_{2})^{\prime \prime }=({\mathfrak {N}}_{1}\otimes {\mathfrak {N}}_{2})\).

  13. A double cone region is the interior of the intersection of the forward light cone of a spacetime point \(p\) with the backward light cone of a point \(q\) where \(q\) lies to the chronological future of \(p\).

  14. See the following Section for substantiation of this last point.

  15. In d’Espagnat’s terminology, the mixture would then be “improper.” For a discussion of the distinction between “proper” and “improper” mixtures see d’Espagnat (1971, Sec. 6.3).

  16. Or, more cynically, one simply demands a criterion that will lend itself to formal proofs.

  17. Much of the literature uses “separable” where I use “decomposable.” The former term has become so loaded with (sometimes misleading) associations I prefer to use the latter.

  18. The norm topology is the appropriate topology to use in taking limits since the set of normal states is closed in this topology. If non-normal states are considered then the appropriate topology is the \(w^{*} \)-topology. The set of all states, normal and non-normal, on a von Neumann algebra is the \(w^{*}\)-closure of convex linear combinations of pure states.

  19. Note that, contrary to what formula (1) might suggest, decomposability does not require that in the density operator representation of the state the projectors in the decomposition must be orthogonal. The importance of this point will become apparent in Sect. 7 where the notion of “classicality” of states is introduced; this notion does require orthogonality.

  20. See Bruß (2002) for a survey of various operational and non-operational criteria of entanglement.

  21. Not quite so fast! What one would like is the result that (R1) is equivalent to

    (R2\(^{\prime }\)) no normal state on \({\mathfrak {N}}_{1}\vee {\mathfrak {N}}_{2}\) is (E1) entangled over \({\mathfrak {N}}_{1}\) and \({\mathfrak {N}}_{2}\).

    (R2\(^{\prime }\)) entails (R2) and, thus, (R1). But it is not obvious that (R1) entails (R2\(^{\prime }\)) when \({\mathfrak {N}}_{1}\vee {\mathfrak {N}}_{2}\) is not \(*\)-isomorphic to \({\mathfrak {N}}_{1}\overline{\otimes }{\mathfrak {N}}_{2}\).

  22. So for any resolution of the identity \(I_{{\mathcal {H}}_{1}}\) by a family \(E_{j}\) of mutually orthogonal projectors in \({\mathfrak {N}}_{1}\), it must be the case that \(\sum _{j}{{\mathcal {F}}}_{E_{j}}(\lambda )=1\) for all \(\lambda \in \Lambda \); and similarly for \({{\mathcal {F}}}_{F}\).

  23. The percentage of cases where the entanglement disappears can be upped to 66.6 % but not above; see below.

  24. Alternatively, reflecting on (3) may prompt revisiting the decision to develop a purely formal criterion of quantum entanglement that doesn’t take into account the way in which a state is actually prepared.

  25. What is being (implicitly) used here is the Lüders conditonalization rule for a non-selective projective measurement; see below.

  26. A normal state \(\omega \) on \({\mathfrak {B}}({\mathcal {H}}_{1})\otimes {\mathfrak {B}}({\mathcal {H}}_{2})\) is non-entangled iff its corresponding density operator can be written as \(\rho ^{1}\otimes \rho ^{2}\). Taking spectral decompostions, \(\rho ^{1}=\sum _{i}\lambda _{i}^{1} E_{i}^{1}\) and \(\rho ^{2}=\sum _{j}\lambda _{j}^{2} E_{j}^{2}\), where the \(E_{i}^{1}\) and \(E_{j}^{2}\) are respectively one-dimensional projectors on \(\mathcal {H} _{1}\) and \(\mathcal {H}_{2}\) and \(0\le \lambda _{i}^{1},\lambda _{j}^{2}\le 1\), it follows that \(\rho ^{1}\otimes \rho ^{2}=\sum _{i,j}\lambda _{i} ^{1}\lambda _{j}^{2} E_{i}^{1}\otimes E_{j}^{2}\), which is obviously classical. This argument comes from Li and Luo (2008).

  27. Perhaps then it is proper to denote “non-classical” by (E\(\dfrac{1}{2}\)) to indicate that it lies between (E1) and (E0).

  28. Needless to say, a selective measurement will generally result in a change of state, the post measurement state being either \(\omega (E\cdot E)/\omega (\cdot )\) or \(\omega ((I-E)\cdot (I-E))/\omega (I-E)\) depending upon whether the outcome is respectively Yes or No. Since it is not clear what, if any, information is gained in a non-selective measurement I have changed “allows acquisition of information” to “allows interrogation.”

  29. That \(\omega \) is faithful to \({\mathfrak {N}}\) means that \(\omega (A)=0\) \(\Rightarrow \) \(A=0\) for all \(A\in {\mathfrak {N}}\). If, as assumed, \({\mathfrak {N}}\) acts on a separable Hilbert space then it admits a faithful normal state. This is not necessarily so if \({\mathfrak {N}}\) acts on a non-separable \({\mathcal {H}}\).

  30. And “vector state” can be substituted for “ pure state” since in the case at hand the normal pure states coincide with the vector states.

  31. Since the function \(\beta (\omega )\) that determines a violation of the Bell inequalities is linear, for a mixed state \(\omega =\sum _{j}\lambda _{j}\varphi _{j}\), \(\beta (\omega )=\sum _{j}\lambda _{j}\beta (\varphi _{j} )\). So if each of the \(\varphi _{j}\) gives a maximal violation of the Bell inequalities then so does the mixed state \(\omega \).

  32. The converse is not true. What is true is that if the restriction of a vector state on \({\mathfrak {B}}(\mathcal {H}_{1})\otimes {\mathfrak {B}} ({\mathcal {H}}_{2})\otimes \cdots \otimes {\mathfrak {B}}({\mathcal {H}}_{N})\) to any bi-partite subsystem algebra \({\mathfrak {B}}({\mathcal {H}}_{m})\otimes {\mathfrak {B}}({\mathcal {H}}_{n})\), \(m\ne n\), is not only a classical state but also takes the special form \( {\sum\nolimits _{i}} p_{i}E_{i}^{m}\otimes E_{j}^{n}\), where the \(E_{i}^{m}\) and \(E_{j}^{n}\) are minimal resolutions of the identities of \(\mathcal {H}_{m}\) and \(\mathcal {H} _{m}\) respectively, then the state vector of the \(N\)-partite state admits a generalized Schmidt decomposition.

  33. Of course, how strong this result is will depend on what technical explication is given for “generic.” The decoherence approach to resolving the measurement problem would presumably welcome (or even require?) a strong no-trickle down result. For an overview of decoherence and its consequences, see Schlosshauer (2007).

  34. Does this present a problem for the decoherence program?

  35. For the composite system Hilbert space Schrödinger used the space of \(L_{{\mathbb {C}}}^{2}\) functions \(\psi (x,y)\) where \(x\) and \(y\) are the coordinates of the two subsystems. Then the condition for a product state is that \(\psi (x,y)\) is a product of a function of \(x\) and a function of \(y\).

  36. In order to identify the component algebras in the tensor product algebra as subalgebras of the system algebra \({\mathfrak {B}}(L_{{\mathbb {C}}} ^{2}({\mathbb {R}}^{2d}))\) it is necessary to choose a particular isomorphism between the system algebra and the tensor product algebra. A natural choice is dictated if the position and momentum operators \(Q\) and \(P\) of the two-particle system are identified respectively with \(q_{1}\otimes q_{2}\) and \(p_{1}\otimes p_{2}\) where the \(q\)’s and \(p\)’s are the one-particle position and momentum operators. See also Ex. 4 below, where there is no natural choice of isomorphism.

  37. Each of \(\sigma _{1}^{x}, \sigma _{1}^{y}, \sigma _{1}^{z}\) commutes with each of \(\sigma _{2}^{x}, \sigma _{2}^{y}, \sigma _{2}^{z}\). The other defining properties of the Pauli operators are that \((\sigma _{j}^{x} )^{2}=(\sigma _{j}^{y})^{2}=(\sigma _{j}^{z})^{2}=I\), and \(\sigma _{j}^{x} \sigma _{j}^{y}=-\sigma _{j}^{y}\sigma _{j}^{x}=i\sigma _{j}^{z}\), \(\sigma _{j} ^{y}\sigma _{j}^{z}=-\sigma _{j}^{z}\sigma _{j}^{y}=i\sigma _{j}^{x}\), \(\sigma _{j}^{z}\sigma _{j}^{x}=-\sigma _{j}^{x}\sigma _{j}^{z}=i\sigma _{j}^{y}\) for \(j=1,2\).

  38. Zanardi et al. (2004) use this example to motivate the slogan that “[E]ntanglement is always relative to a particular set of experimental capabilities” (p. 1; italics in the original); see also Zanardi (2001, 2002). It is unclear, however, whether they mean to fully endorse what I am calling Pragmatism.

  39. If \({\mathcal {I}}:{\mathfrak {N}}_{1}\overline{\otimes }{\mathfrak {N}} _{2}\rightarrow {\mathfrak {N}}\) is a \(*\)-isomorphism, then \(A\in \mathfrak {N}_{1}\) (respectively, \(B\in {\mathfrak {R}}_{2}\)) is identified with \({\mathcal {I}}(A\otimes I)\) (respectively, \({\mathcal {I}}(I\otimes B)\)). Without such an identification the entanglement of a composite system state over the components of the tensor product algebra becomes inscrutable.

  40. The reader interested in getting a sense of the recent physics literature on how to define and quantify entanglement for indistinguishable particles can start with the following sample: Amico et al. (2008), Bañuls et al. (2007), Dowling et al. (2006), Eckert et al. (2002), Ghirardi et al. (2002), Ghirardi and Marinatto (2003), Ghirardi and Marinatto (2004a, b, 2005), Giddings and Fisher (2002), Horodecki et al. (2009), Kaplan (2005), Li et al. (2001), Paškauskas and You (2001), Plastino et al. (2005), Schliemann et al. (2001), Shi (2003), Shi (2004), Wang et al. (2006), Wiseman and Vaccaro (2003), Zanardi (2001, 2002), Zanardi and Wang (2002), and Zanardi et al. (2004).

  41. It is sometimes said that superselection rules place limitations on the superposition principle. Strictly speaking this is false: a vector space is a vector space, is a vector space, \(\ldots \) , i.e. the linear combination of any two vectors belongs to the space. What is true that some linear combinations do not produce coherent superpositions, i.e. the linear combination is a vector that generates a mixed algebraic state. See Earman (2008).

  42. The postulate of the commutativity of superselection rules asserts the commutativity of all unitary operators that commute with the set of \(\Theta \) of all self-adjoint operators corresponding to genuine observables. This postulate fails for a system of three or more indistinguishable particles since the unitaries representing pair interchanges do not commute. For a discussion of these matters, see Earman (2008).

  43. Without citing Schmidt (1906), Schrodinger (1935) states a version of the bi-orthogonal decomposition theorem, and he takes the equality of all of the \(\lambda _{i}\) of the bi-orthogonal decomposition to be a necessary condition for the “Einstein-Podolsky-Rosen case” (p. 558).

  44. Note that this procedure does not generalize beyond the two-fermion case, which should already arouse suspicion about whether it captures something fundamental.

  45. Such entanglement is sometimes dubbed “accessible entanglement” (Wiseman and Vaccaro 2003) and sometimes “quantum entanglement” (Eckert et al. 2002)—which should not be confused with the sense of quantum entanglement discussed above.

  46. The full story can be more complicated. For example, if the fermions are charged then charge superselection limits the genuine observables to a subalgebra \({\mathfrak {B}}({\mathcal {H}}_{\mathcal {A}}(L))\), namely the von Neumann algebra generated by all self-adjoint operators on \({\mathcal {H}}_{\mathcal {A}}(L)\) that commute with the number operator.

  47. I realize that in saying this I am waving a red flag in front of the thundering herd of philosophers who pronounce on these matters. So be it. These bulls can only succeed in goring themselves more than they already have.

  48. The EPR paradox is first presented in Einstein et al. (1935). A number of years later in “Quantum Mechanics and Reality” (1948) Einstein made it clear his real worry was what he perceived as the challenge that quantum physics poses to the “independent existence (the ‘being-thus’)” of (sub)systems. For attempts to clarify Einstein’s worry, see Howard (1989) and Fine (1996).

  49. Those who want to blur the line between separability/independence and entanglement end up not doing justice to either. An example of what I have in mind comes from Howard’s (1989), which is concerned with a separability principle that is “a fundamental ontological principle governing the individuation of physical systems and their associated states” (p. 225). This principle requires that

    (H) The joint state of the two systems is wholly determined by the separate states of the two component systems.

    In a footnote Howard explains:

    How the joint state is determined by the separate states depends on the details of a theory’s mathematical formulation. At a minimum, the idea is that no information is contained in the joint state that is not already contained in the separate states \(\ldots \) (fn 2, p. 226)

    None of the extant mathematical formulations of quantum physics gives a prescription for going from states on subsystems to a state on the composite system. But in the present context (H) can be taken to require that for any pair of subsystem states there must be a unique extension to a joint state—otherwise there is a loss of information in passing from the composite system state to the component system states. This condition is indeed violated for entangled states. But this information loss criterion does not capture the difference between simple entanglement (E0) and genuine quantum entanglement ((E1), (E2), or (E3)) since information loss can occur for both types of states on \({\mathfrak {N}}_{1}\overline{\otimes }{\mathfrak {N}}_{2}\) when the subsystem algebras are non-abelian. When both subsystem algebras \({\mathfrak {N}}_{1}\) and \({\mathfrak {N}}_{2}\) are classical (= abelian) we know that there are no states (normal or non-normal) on \({\mathfrak {N}}_{1} \overline{\otimes }{\mathfrak {N}}_{2}\) that are (E1) entangled over \({\mathfrak {N}}_{1}\) and \({\mathfrak {N}}_{2}\). Nevertheless, even for abelian subsystem algebras there are normal entangled states on \({\mathfrak {N}} _{1}\overline{\otimes }{\mathfrak {N}}_{2}\) for which there is information loss in passing to the reduced subsystem states. But no one thinks that separability/independence of classical systems is threatened by such information loss.

    Howard (1989) has been criticized by Fogel (2007) and Winsberg and Fine (2003) but on rather different grounds than the ones offered here.

  50. One might also worry that in relativistic QFT the entanglement between relatively spacelike regions cannot be destroyed by local operations (see Clifton and Halvorson 2001). Here I have three comments. First, Valente (2013)) argues that the Clifton-Halvorson no-go result rests on a overly restrictive notion of local operation. Second, the point at issue seems to me to relate less to the question of separability/independence of subsystems than to the question of how entrenched entanglement is in QFT. Third, if (contrary to what I am suggesting) there is a defensible no-go result here and if that result can be shown to compromise the “being thus” of subsystems, so be it. That is what would create a problem for separability/independence, not the existence or even ubiquity of entanglement per se.

  51. For an authoritative overview, see Healey (2008).

References

  • Amico, L., Fazio, R., Osterloh, A., & Derdal, V. (2008). Entanglement in many-body systems. Reviews of Modern Physics, 80, 517–576.

    Article  Google Scholar 

  • Ancin, A., Andrianov, A., Costa, L., Jané, E., Latorre, J. I., & Tarrach, R. (2000). Generalized Schmidt decompositon and classification of three-quantum-bit states. Physical Review Letters, 85, 1560–1563.

    Article  Google Scholar 

  • Bañuls, M.-C., Cirac, J. I., & Wolf, M. M. (2007). Entanglement in fermionic systems. Physical Review A, 76, 022311-1–13.

    Article  Google Scholar 

  • Bratteli, O., & Robinson, D. W. (1987). Operator algebras and quantum statistical mechanics 1 (2nd ed.). New York: Springer.

    Book  Google Scholar 

  • Brauenstein, S. L., Mann, A., & Revzen, M. (1992). Maximal violations of Bell inequalities for mixed states. Physical Review Letters, 68, 3259–3261.

    Article  Google Scholar 

  • Bruß, D. (2002). Characterizing entanglement. Journal of Mathematical Physics, 43, 4237–4251.

    Article  Google Scholar 

  • Buchholz, D. (1974). Product states for local algebras. Communications in Mathematical Physics, 36, 287–304.

    Article  Google Scholar 

  • Clifton, R., & Halvorson, H. (2001). Entanglement and open systems in algebraic quantum field theory. Studies in History and Philosophy of Modern Physics, 32, 1–31.

    Article  Google Scholar 

  • d’Espagnat, B. (1971). Conceptual foundations of quantum mechanics. Menlo Park, CA: W. A. Benjamin.

    Google Scholar 

  • de la Torre, A. C., Goyeneche, D., & Leitao, L. (2010). Entanglement for all quantum states. European Journal of Physics, 31, 325–332.

    Article  Google Scholar 

  • Dowling, M. R., Doherty, A. C., & Wiseman, H. M. (2006). Entanglement of indistinguishable particles in condensed matter physics. Physical Review A, 73, 052323-1–12.

    Article  Google Scholar 

  • Earman, J. (2008). Superselection rules for philosophers. Erkenntnis, 69, 377–414.

    Article  Google Scholar 

  • Eckert, K., & Schliemann, J. (2002). Quantum correlations in systems of indistinguishable particles. Annals of Physics, 299, 88–127.

    Article  Google Scholar 

  • Einstein, A. (1948). Quantum mechanics and reality. Dialectica, 2, 323–324.

    Article  Google Scholar 

  • Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47, 777–780.

    Article  Google Scholar 

  • Fine, A. (1996). The shaky game: Einstein, realism, and the quantum theory (2nd ed.). Chicago: University of Chicago Press.

    Book  Google Scholar 

  • Fogel, B. (2007). Formalizing the separability condition on Bell’s theorem. Studies in History and Philosophy of Modern Physics, 38, 920–937.

    Article  Google Scholar 

  • Garg, A., & Mermin, N. D. (1982). Correlation inequalities and hidden variables. Physical Review Letters, 49, 1220–1223.

    Article  Google Scholar 

  • Ghirardi, G., Marinatto, L., & Weber, T. (2002). Entanglement and properties of composite quantum systems: A conceptual and mathematical analysis. Journal of Statistical Physics, 108, 49–122.

    Article  Google Scholar 

  • Ghirardi, G., & Marinatto, L. (2003). Entanglement and properties. Fortschritte der Physik, 51, 379–387.

    Article  Google Scholar 

  • Ghirardi, G., & Marinatto, L. (2004a). Criteria for the entanglement of composite systems with identical particles. Fortschritte der Physik, 52, 1045–1051.

    Article  Google Scholar 

  • Ghirardi, G., & Marinatto, L. (2004b). General criteria for the entanglement of two indistinguishable particles. Physical Review A, 70, 102109-1–102109-10.

    Article  Google Scholar 

  • Ghirardi, G., & Marinatto, L. (2005). Identical particles and entanglement. Optics and Spectroscopy, 99, 386–390.

    Article  Google Scholar 

  • Giddings, J. R., & Fisher, A. J. (2002). Describing mixed spin-space entanglement of pure states of indistinguishable particles using occupation-number basis. Physical Review A, 66, 032305-1–11.

    Google Scholar 

  • Gisin, N. (1991). Bell’s inequality holds for all non-product states. Physics Letters A, 154, 201–202.

    Article  Google Scholar 

  • Gisin, N., & Peres, A. (1992). Maximal violation of Bell’s inequality for arbitrarily large spin. Physics Letters A, 162, 15–17.

    Article  Google Scholar 

  • Haag, R. (1992). Local quantum physics. Berlin: Springer.

    Book  Google Scholar 

  • Halvorson, H., & Clifton, R. (2000). Generic Bell correlation between arbitrary local algebras in quantum field theory. Journal of Mathematical Physics, 41, 1711–1718.

    Article  Google Scholar 

  • Hamhalter, J. (2003). Quantum measure theory. Dordrecht: Kluwer Academic.

    Book  Google Scholar 

  • Harshman, N. L., & Wickramasekara, S. (2007). Tensor product structures, entanglement, and particle scattering. Open Systems and Information Dynamics, 14, 341–351.

    Article  Google Scholar 

  • Healey, R. (2008). Holism and nonseparability in physics. In Stanford encyclopedia of philosophy. URL: http://plato.stanford.edu/entries/physics-holism/.

  • Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009). Quantum entanglement. Reviews of Modern Physics, 81, 865–942.

    Article  Google Scholar 

  • Horuzhy, S. S. (1990). Introduction to algebraic quantum field theory. Dordrecht: Kluwer Academic.

    Google Scholar 

  • Howard, D. (1989). Holism, separability, and the metaphysical implications of the Bell Experiments. In J. Cushing & E. McMullin (Eds.), Philosophical consequences of quantum theory: Reflections on Bell’s theorem (pp. 224–253). Notre Dame, IN: University of Nature Dame Press.

    Google Scholar 

  • Kaplan, T. A. (2005). On fermion entanglement. Fluctuation and Noise Letters, 5, C15–C22.

    Article  Google Scholar 

  • Li, N., & Luo, S. (2008). Classical states versus separable states. Physical Review A, 78, 024303-1–4.

    Google Scholar 

  • Li, Y. S., Zeng, B., Liu, X. S., & Long, G. L. (2001). Entanglement in a two-identical-particle system. Physical Review A, 64, 054302-1–4.

    Google Scholar 

  • Luo, S. (2008). Using measurement-induced disturbance to characterize correlations as classical or quantum. Physical Review A, 77, 022301-1–5.

    Google Scholar 

  • Messiah, A. M., & Greenberg, O. W. (1964). Symmetrization postulate and its experimental foundation. Physical Review, 136, 248–267.

    Article  Google Scholar 

  • Modi, K., Paterek, T., Son, W., Vederal, V., & Williamson, M. (2010). Unified view of quantum and classical correlations. Physical Review Letters, 104, 080501-1–4.

    Article  Google Scholar 

  • Ollivier, H., & Zurek, H. (2002). Quantum discord: A measure of quantumness of correlations. Physical Review Letters, 88, 017901-1–4.

    Google Scholar 

  • Paškauskas, R., & You, L. (2001). Quantum correlation in two-boson wave functions. Physical Review A, 64, 042310-1–4.

    Google Scholar 

  • Peres, A. (1995). Higher order Schmidt decompositions. Physics Letters A, 202, 16–17.

    Article  Google Scholar 

  • Plastino, A. R., Manzano, D., & Dehesa, J. S. (2005). Separability criteria and entanglement measures for pure states of N identical particles. European, Physics Letters, 86, 2005-p1–2005-p5.

    Google Scholar 

  • Popescu, S. (1994). Bell’s inequalities versus teleportation: What is nonlocality? Physical Review Letters, 72, 797–799.

    Article  Google Scholar 

  • Raggio, G. A. (1988). A remark on Bell’s inequality and decomposable normal states. Letters in Mathematical Physics, 15, 27–29.

    Article  Google Scholar 

  • Schliemann, J., Cirac, I. I., Kuś, M., Lewenstein, M., & Loss, D. (2001). Quantum correlations in two-fermion systems. Physical Review A, 64, 022303-1–9.

    Article  Google Scholar 

  • Schlosshauer, M. (2007). Decoherence and the quantum-to-classical transition. Berlin: Springer.

    Google Scholar 

  • Schmidt, E. (1906). Zur theorie linearen und nichtlinearen Integralgleichungen. Mathmatische Annalen, 63, 433–476.

    Article  Google Scholar 

  • Schrödinger, E. (1935). Discussion of probability relations between separated systems. Proceedings of the Cambridge Philosophical Society, 31, 555–563; 32 (1936), 446–451.

  • Shi, Y. (2003). Quantum entanglement of identical particles. Physical Review A, 67, 02430-1–02430-4.

    Article  Google Scholar 

  • Shi, Y. (2004). Quantum entanglement in second-quantized condensed matter physics. Journal of Physics A, 37, 6807–6822.

    Article  Google Scholar 

  • Summers, S. J. (1990). On the independence of local algebras in quantum field theory. Reviews in Mathematical Physics, 2, 201–247.

    Article  Google Scholar 

  • Summers, S. J. (2009). Subsystems and independence in relativistic microscopic physics. Studies in History and Philosophy of Modern Physics, 40, 133–141.

    Article  Google Scholar 

  • Sunder, V. (1986). An invitation to von Neumann algebras. Berlin: Springer.

    Google Scholar 

  • Tommasini, P., & Timmermans, E. (1998). The hydrogen atom as an entangled electron-positron pair. American Journal of Physics, 66, 881–886.

    Article  Google Scholar 

  • Valente, G. (2013). Local disentanglement in relativistic quantum field theory. Pre-print.

  • Wang, X.-H., Fei, S.-M., & Wu, K. (2006). Separability and entanglement of identical bosonic systems. Journal of Physics A, 39, L555–557.

    Article  Google Scholar 

  • Werner, W. F. (1989). Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden variable model. Physical Review A, 40, 4277–4281.

    Article  Google Scholar 

  • Winsberg, E., & Fine, A. (2003). Quantum life: Interaction, entanglement, and separation. Journal of Philosophy, C, 80–97.

    Google Scholar 

  • Wiseman, H. M., & Vaccaro, J. A. (2003). Entanglement of indistinguishable particles shared between two parties. Physical Review Letters, 91, 097902-1–097902-4.

    Article  Google Scholar 

  • Zanardi, P. (2001). Virtual quantum systems. Physical Review Letters, 87, 077901-1–077901-4.

    Article  Google Scholar 

  • Zanardi, P. (2002). Quantum entanglement for fermionic states. Physical Review A, 65, 042101-1–042101-5.

    Article  Google Scholar 

  • Zanardi, P., Lidar, D. A., & Lloyd, S. (2004). quantum tensor product structures are observable induced. Physical Review Letters, 92, 06042-1–06042-4.

    Article  Google Scholar 

  • Zanardi, P., & Wang, X. (2002). Fermionic entanglement in itererant systems. Journal of Physics A, 35, 7947–7959.

    Article  Google Scholar 

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Acknowledgments

I am grateful to an anonymous referee for suggestions that led to improvements of a earlier version of this article.

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  1. Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    John Earman

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  1. John Earman
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Correspondence to John Earman.

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Earman, J. Some Puzzles and Unresolved Issues About Quantum Entanglement. Erkenn 80, 303–337 (2015). https://doi.org/10.1007/s10670-014-9627-8

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  • DOI: https://doi.org/10.1007/s10670-014-9627-8

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