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Entanglement, Complexity, and Causal Asymmetry in Quantum Theories

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Abstract

It is often claimed that one cannot locate a notion of causation in fundamental physical theories. The reason most commonly given is that the dynamics of those theories do not support any distinction between the past and the future, and this vitiates any attempt to locate a notion of causal asymmetry—and thus of causation—in fundamental physical theories. I argue that this is incorrect: the ubiquitous generation of entanglement between quantum systems grounds a relevant asymmetry in the dynamical evolution of quantum systems. I show that by exploiting a connection between the amount of entanglement in a quantum state and the algorithmic complexity of that state, one can use recently developed tools for causal inference to identify a causal asymmetry—and a notion of causation—in the dynamical evolution of quantum systems.

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Notes

  1. One might wonder why I describe this as a problem of inferring causal direction rather than temporal direction. I will return to this question in Sect. 6.

  2. This generalizes straightforwardly to probabilistic settings: X causes Y if and only if there is an intervention that can be performed on \(p_X(x)\), the probability distribution over the values of X, while holding all other variables fixed, that produces a change in \(p_Y(y)\). See [102] for a much richer philosophical development of the interventionist account of causation than I will need here.

  3. Well, four things. The fourth thing is that throughout the paper \(\log = \log _2\) and I adopt units where \(\hbar =c=1\).

  4. See [28] for an overview. For a clarifying sketch of several types of projects one might pursue when thinking about causation, see [103].

  5. This doesn’t mean that all causal inference puzzles are solved once one allows departures from unitary evolution. Far from it: it is only after considering such departures that one encounters the most vexing causal inference problem posed by quantum theories: the explanation of EPR-type correlations.

  6. This presumes time-translation invariance. Time-translation invariance will always be assumed in this paper, a reflection of the fact that I am considering closed quantum systems.

  7. See [26, section 4] or [33] for a discussion of the importance of distinguishing between the two when considering how the time-symmetry of a theory bears on our ability to distinguish cause and effect in microphysics.

  8. The distinct but related question of whether the two unitaries U(t)and U(t)\(^{-1}\) are equally implementable in practice is complicated [46, 48, 49].

  9. The appropriate understanding of time-reversal in quantum mechanics has received a fair amount of philosophical attention in recent years [1, 3, 12, 13, 24,25,26, 32, 78, 92]. The initial stimulation for much of this work were the arguments for a non-standard definition of time reversal by [1] and [12]. For reasons compactly summarized in [79], I remain partial to the traditional account and will adopt it throughout this paper.

  10. This is a special case of the fact that the state of any degree of freedom that is odd under time-reversal—i.e. \({\mathcal{T}}^2|\psi \rangle = -|\psi \rangle\)—is orthogonal to its time-reversed state [83, section 3.2].

  11. For two arguments for the same conclusion in classical and quantum statistical mechanics that have points of contact with the argument I offer here, but which are ultimately distinct, see [59, chapter 4] or [67].

  12. For a related point about the relationship between time-reversal invariance and determinism to the past and future and additional discussion, see [26, section 4].

  13. See [68, section 2.5] for a pedagogical presentation.

  14. It also reflects the mathematical fact that the amount of entanglement between two systems is independent of a unitary change of basis.

  15. The squares of the Schmidt coefficients \(\lambda _i^2\) in the Schmidt decomposition of a bipartite pure state \(|\Psi \rangle\) are the eigenvalues of the density operator \(\sigma = |{\Psi }\rangle \langle {\Psi }|\). A convenient fact about the Schmidt decomposition is that the \(\lambda _i^2\) are also the eigenvalues of each of the reduced density operators \(\rho _A\) and \(\rho _B\) representing each entangled subsystem. Since these eigenvalues fully determine the entanglement entropy, the Schmidt decomposition reveals that one can calculate the entanglement entropy of the bipartite system in two equivalent ways: by computing the Shannon entropy of the probability distribution generated by the square of the coefficients of \(\sigma\) for the full bipartite system or, as is more common, computing the von Neumann entropy of the reduced density operators \(\rho _A\) or \(\rho _B\).

  16. The Schmidt rank of a bipartite pure state is sometimes referred to interchangeably as its Schmidt number. This is unfortunate since I will occasionally mention a quantity introduced by [93] that they call the Schmidt number, which is an extension of the Schmidt rank to mixed states. I’ve altered Preskill’s terminology to cohere with mine in this paper.

  17. For example, see the remarks in [85] or, more substantively, the proof in [96] mentioned below. This is not to say that there is never any relationship between the entanglement entropy and the Schmidt rank; for example, see [90, section III]. For philosophical discussion of some of the multiple notions of entanglement see [27].

  18. See [95, chapters 2.2 & 3.6] for a clear presentation and philosophical discussion of the interpretation of the Shannon and von Neumann entropies as measures of information.

  19. See [41] for a helpful comparison of Shannon information and algorithmic information.

  20. See [9] for an elementary proof. The fact that all norms on a finite-dimensional vector space are equivalent justifies the statement that this is true for “any norm” on \({\mathcal{H}}_{AB}\).

  21. In the finite-dimensional case, the restriction to pure states is important: for mixed states of a bipartite system, the separable states are no longer as sparse, and the set of separable mixed states always contains an open ball around the maximally mixed state [108]. For an infinite-dimensional Hilbert space, this is no longer true: as [19] showed, the set of mixed states is nowhere dense in the set of all states, in the topology induced by the trace-norm.

  22. I will drop the “non-trivial” qualifier for the remainder of the paper. Considering time evolutions that are non-trivial in this sense is equivalent to requiring that the time evolution operator U(t) does not factorize into a product \(\text{U(t)}_A \otimes \text{U(t)}_B\) of operators evolving Alice and Bob’s subsystems independently. Any unitary operator that factorizes in this way cannot change the Schmidt coefficients of a quantum state, as I mentioned previously, and so cannot create entanglement between subsystems.

  23. A general argument is sketched in [7, section 6.1]. There are an immense number of concrete examples of dynamically generated entanglement; see, for instance, the discussions in many-body physics [4, 11, 30], quantum field theory [15, 53, 71], and non-relativistic quantum mechanics [61, 87].

  24. This is clearest in the algebraic approach to quantum theories, where a quantum system is defined by a C\(^*\) algebra. The GNS reconstruction theorem enables one to move from the abstract C\(^*\) algebra to a Hilbert space representation for the system, but this requires data about the full C\(^*\) algebra, including the Hamiltonian.

  25. See [99] for proofs that in any finite-dimensional quantum system, and any infinite-dimensional quantum system that satisfies modest constraints on the Hamiltonian, there exists a single time \(\tau _R\) such that after \(\tau _R\) every state in \({\mathcal{H}}_{AB}\) will have returned to a state arbitrarily close to itself.

  26. Note that this excludes the case where both \(|\alpha \rangle\) and \(|\beta \rangle\) are entangled but neither has full Schmidt rank. I’m not aware of any demonstration that within the set of bipartite pure states of less-than-full Schmidt rank, the states of lower rank are less prevalent than those of higher rank. The causal inference methods used in Sect. 5 will do better for this case.

  27. An observation about the impracticality of arranging a similar evolution was made in a different context by [31]. They consider sending a spin-1/2 particle in a \(\sigma _x\) eigenstate through a Stern–Gerlach device set to measure \(\sigma _z\), which will split the incoming beam into a superposition of the two eigenstates of \(\sigma _z\). They investigate the degree of precision with which an experimenter would need to control the magnetic field in the Stern–Gerlach device to ensure that the spin-1/2 particle returns to its original \(\sigma _x\) eigenstate after passing through the Stern–Gerlach device. They show that an exact return to the original \(\sigma _x\) eigenstate is unattainable in practice and that even to reproduce the original state with 99% accuracy would require the ability to control the gradient of the macroscopic magnetic field in the Stern–Gerlach device to at least 5 decimal places.

  28. Sklar expresses a similar dissatisfaction with such candidate explanations of why subsystems of a larger system obey the second law: “\(\ldots\) we would simply posit an initial state that gives rise to parallel entropic increase in branch systems with each other and with the main system. But to characterise the state in that way would, of course, not be offering us an explanation of the sort we expected. It would be one thing to be able to characterize the initial state in some simple way\(\ldots\) and be able to derive the Second Law from that. But to derive the Second Law from a bald assertion that “initial conditions were such that they would lead to Second Law behavior” hardly seems of much interest” [89, p. 330].

  29. Although Maudlin doesn’t dwell on it, Ref. [104] has given an extended and enlightening examination of the very similar principle that forms the foundation of the causal inference methods I describe in Sect. 4.

  30. The restriction to classical statistical variables is important because a joint probability distribution for non-commuting variables, like conjugate observables in quantum mechanics, is generally not well-defined. If one restricts to quantum observables that commute then one can define a joint probability distribution over their possible values. See [36] for a review and connection to hidden variable theories.

  31. See, for example, [72, chapter 6.5], [91, chapter 3], or [70, chapter 2.4], where the faithfulness condition is called “stability”.

  32. Faithfulness is often motivated by the fact that the set of parameter values quantifying causal influence that produce this type of cancellation are measure zero in the set of all parameter values [60, 91, theorem 3.2]. For a clarifying discussion of alternative justifications for imposing faithfulness, see [100].

  33. This is roughly a combination of two different examples from [72, chapter 2] and [51].

  34. I very loosely follow [47, section 2.1] here. See [55] for a textbook introduction.

  35. In algorithmic information theory, equalities are generally only equalities up to a constant that is independent of the object itself, but may depend on the alphabet or programming language chosen for the encoding or the particular universal Turing machine being considered. For example, a program to output Hamlet may be shorter when written in Python than in FORTRAN. This tells us something about Python and FORTRAN, but nothing about the algorithmic information content of Hamlet itself. This “equality up to a constant” is denoted by \({\mathop {=}\limits ^{+}}\)

  36. Note that this has the somewhat counterintuitive consequence that a binary sequence that is completely random has maximal algorithmic information.

  37. The equation of algorithmic entropy with K(s) by [51] assumes that the microstate s is perfectly known to the observer. More generally, one defines the algorithmic entropy of a microstate s as the sum of the algorithmic information and the thermodynamic entropy \({\mathcal{S}}(s) = K(s) + H(s)\); see [107] or [55, chapter 8].

  38. This means that \(|\langle {\phi }|{\mathcal{C}}|{0}\rangle |^{2} \ge 1 - \varepsilon\).

  39. The two limiting cases may be clarifying. Let \(\varepsilon \rightarrow 1\); then the “patch” is the entire sphere of pure states and \(K^{\varepsilon =1}_Q(|\phi \rangle )\) quantifies the information required to construct a circuit \({\mathcal{C}}\) that will act on \(|0\rangle\) to put it into any pure state inside \({\mathcal{H}}_N\). In this limit \(K^{\varepsilon =1}_Q\) is zero for all states, as it obviously should be: \(|0\rangle\) is already such a state, so just leave it alone. Now let \(\varepsilon \rightarrow 0\); then the “patch” approaches the single state \(|\phi \rangle\) and \(K^{\varepsilon =0}_Q(|\phi \rangle )\) quantifies the information required to prepare exactly \(|\phi \rangle\) from \(|0\rangle\). This obviously diverges at the limit; for example, it would require preparing an amplitude to be exactly \(\frac{1}{\sqrt{2}}\) rather than the closest rational approximation to \(\frac{1}{\sqrt{2}}\).

  40. See [68, section 4.5.4] for how the number of patches required scales with the fidelity \(\varepsilon\) and system size N.

  41. I am presuming a preferred factorization of the big Hilbert space \({\mathcal{H}}\) into tensor factors. In general, entanglement measures for multipartite systems are sensitive to different choices of partition; this is part of the difficulty of extending such measures beyond bipartite systems. Where a preferred partition isn’t available, one could more generally define the Schmidt measure as the minimum value of r over all possible partitions. See [106, 21], or [14] for different proposals for identifying preferred factorizations in certain contexts.

  42. Since the algorithmic information of a quantum state is the classical algorithmic information of the circuit \({\mathcal{C}}\) that prepares it, this is a straightforward adaptation of the standard proof that the relative frequency of compressible strings in the set of all n-bit strings goes to zero as \(n\rightarrow \infty\).

  43. Time evolution for a large class of Hamiltonians can be well-approximated as a quantum circuit \({\mathcal{C}}_{U_t}\); see [68, section 4.7].

  44. More precisely, this the volume occupied by the equilibrium macrostate for any energy shell in phase space containing microstates with energies in the range \(E + \Delta E\).

  45. See [98, chapter 9] for a discussion of decoherence and temporal asymmetry.

  46. In language that may be more familiar: typically one splits the multiparticle system into two and calls the subsystem(s) of interest “the system” and the rest of the particles “the environment”. The entanglement between the subsystems of interest is then decohered by their respective entanglements with the environment.

  47. As I mentioned in Sect. 1, the relationship between entanglement and the direction of time has received extensive exploration.

  48. See also [28, p. 914]:

    “\(\ldots\) metaphysical accounts have provided essentially no guidance for methods of discovery because it remains unclear how they could be operationalized into discovery procedures that do not depend on the availability of causal knowledge in the first place. Epistemological headway was made by a completely different strategy that largely ignored metaphysical considerations.”

    I would only emphasize that I think that once one has made this epistemological headway, it can be valuable to use it as a basis for circling back to address some of the metaphysical considerations that one initially set aside to make epistemic progress.

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Acknowledgements

I would like to thank John Dougherty, Casey McCoy, Michael Miller, Siddarth Muthukrishnan, and an audience at Oxford University for helpful discussion. I would like to thank David Albert, Naftali Weinberg, Jim Woodward, and two very helpful referees for valuable and constructive feedback on a previous draft.

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    Porter Williams

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Williams, P. Entanglement, Complexity, and Causal Asymmetry in Quantum Theories. Found Phys 52, 47 (2022). https://doi.org/10.1007/s10701-022-00562-0

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