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arXiv:1812.09251 · Physical Review A

Separability gap and large-deviation entanglement criterion

Jakub Czartowski, Konrad Szymański, Bartłomiej Gardas, Yan V. Fyodorov, Karol Życzkowski

Published Oct 28, 2019• 6 citations
arXivHTMLPDF

Abstract

For a given Hamiltonian $H$ on a multipartite quantum system, one is interested in finding the energy ${E}_{0}$ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}$ of $H$ among all product states. For several concrete model Hamiltonians, we investigate the difference ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}\ensuremath{-}{E}_{0}$, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.

DOI
10.1103/physreva.100.042326

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For a given Hamiltonian HHH on a multipartite quantum system, one is interested in finding the energy E0{E}_{0}E0​ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value \ensuremathλmin\ensuremath⨂{\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}\ensuremathλmin\ensuremath⨂​ of HHH among all product states. For several concrete model Hamiltonians, we investigate the difference \ensuremathλmin\ensuremath⨂\ensuremath−E0{\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}\ensuremath{-}{E}_{0}\ensuremathλmin\ensuremath⨂​\ensuremath−E0​, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.
In a previous work [Shchukin, Phys. Rev. A 84, 052325 (2011)] a lower bound was found for the multipartite uncertainty product of the position and momentum observables over all separable states. In this work we are trying to minimize this uncertainty product over a broader class of states to find the fundamental limits imposed by nature on the observable quantities. We show that it is necessary to consider pure states only and find the infimum of the uncertainty product over a special class of pure states (states with spherically symmetric wave functions). It is shown that this infimum is not attained. We also explicitly construct a parametrized family of states that approaches the infimum by varying the parameter. Since the constructed states beat the lower bound for separable states, they are entangled. We thus show that there is a gap that separates the values of a simple measurable quantity for separable states from entangled ones and we also try to find the size of this gap.
In a seminal paper [Phys. Rev. Lett. 71, 1291 (1993)], Page conjectured that the entanglement entropy of typical pure states is Styp≃ln⁡DA−(1/2)DA2/DS_{\rm typ}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}Styp​≃lnDA​−(1/2)DA2​/D, where DA{\cal D}_{\rm A}DA​ and D{\cal D}D are the Hilbert space dimensions of the subsystem and the system, respectively, and typicality is with respect to the Haar measure. Typical pure states are hence (nearly) maximally entangled. We prove that S1+=ln⁡DA−[1/(2ln⁡2)](ln⁡DA)2/ln⁡DS^+_1=\ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}S1+​=lnDA​−[1/(2ln2)](lnDA​)2/lnD is an upper bound for the average entanglement entropy of all many-body fermionic eigenstates of a class of translationally invariant quadratic Hamiltonians in one dimension, and that the variance vanishes with increasing system size. Consequently, whenever the size of the subsystem is not a vanishing fraction of the size of the system, typical eigenstates of such Hamiltonians have a volume-law entanglement but do not become maximally entangled as D→∞{\cal D}\rightarrow\inftyD→∞, i.e., they are qualitatively different from typical pure states. We also prove that, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal.
A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum ppp-spin glasses, random Hamiltonians whose interactions act on ppp-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement. We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with nnn, we prove that, for all sufficiently large fixed ppp, any circuit preparing an nnn-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth Ωp(log⁡n)Ω_p(\log n)Ωp​(logn). In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed DDD, a sufficiently large degree prefactor rules out depth-DDD preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits. Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.
Abstract Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state ∣ψ⟩|\psi \rangle ∣ψ⟩ | ψ ⟩ . Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform ∣ψ⟩|\psi \rangle ∣ψ⟩ | ψ ⟩ into another state ∣ϕ⟩|\phi \rangle ∣ϕ⟩ | ϕ ⟩ . Famously, Nielsen’s theorem [28] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the entanglement spectrum of ∣ϕ⟩|\phi \rangle ∣ϕ⟩ | ϕ ⟩ should majorise the entanglement spectrum of ∣ψ⟩|\psi \rangle ∣ψ⟩ | ψ ⟩ ). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states ∣ψ⟩,∣ϕ⟩|\psi \rangle , |\phi \rangle ∣ψ⟩,∣ϕ⟩ | ψ ⟩ , | ϕ ⟩ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states ∣ψ⟩,∣ϕ⟩|\psi \rangle , |\phi \rangle ∣ψ⟩,∣ϕ⟩ | ψ ⟩ , | ϕ ⟩ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen’s theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart–Laguerre ensemble . Concretely, let X and Y be independent n×mn \times mn×m n × m random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen’s conjecture says that the probability that the spectrum of XX†/tr⁡(XX†)X X^\dagger / \operatorname {tr}(X X^\dagger )XX†/tr(XX†) X X † / tr ( X X † ) majorises the spectrum of YY†/tr⁡(YY†)Y Y^\dagger / \operatorname {tr}(Y Y^\dagger )YY†/tr(YY†) Y Y † / tr ( Y Y † ) tends to zero as both n and m grow large. We prove this conjecture, and we also confirm some related predictions of Cunden et al. [12].
It was recently proven that, in contrast to their matrix analogues, the moments of a real Gaussian tensor of size N do not in general factorize over their connected components in the asymptotic large N limit. While the original proof of this rather surprising result was not constructive, explicit examples of non-factorizing moments, which are expectation values of trace-invariants, have since then been discovered. We explore further aspects of this problem, with a focus on Haar-distributed (or Gaussian) complex random tensors, which are more directly relevant to quantum information. We start out by exhibiting an explicit example of non-factorizing trace-invariant, thereby filling a gap in the recent literature. We then turn to the opposite question: that of finding interesting families of trace-invariants that do in fact factorize at large N. We establish three main theorems in this regard. The first one provides a sufficient combinatorial bound ensuring large N factorization, that is also simple enough to be applicable to various cases of practical relevance. Our second main result shows that the expectation value of any compatible trace-invariant is dominated by certain tree-like combinatorial structures at large N, which we refer to as tree-like dominant pairings. Our third main theorem establishes that any trace-invariant admitting tree-like dominant pairings does actually factorize at large N. In this way, we are able to prove that various families of trace-invariants that have been previously studied in the literature do factorize at large N. We apply our findings to the theory of multipartite quantum entanglement: to any trace-invariant is associated a multipartite generalization of Rényi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.