Perturbative expansion of entanglement negativity using patterned matrix calculus

Cresswell, Jesse C.; Tzitrin, Ilan; Goldberg, Aaron Z.

doi:10.1103/physreva.99.012322

Cited by 6 publications

(8 citation statements)

References 84 publications

(199 reference statements)

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“…One must be careful with the derivatives of the logarithmic negativity due to its non-analyticity derived from the use of the trace norm. See Ref [103]. for important discussion on taking derivatives of the negativity (in particular the trace norm).…”

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confidence: 99%

The negativity contour: a quasi-local measure of entanglement for mixed states

2020

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In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the negativity contour, a local measure of entanglement that determines which real-space degrees of freedom in a subregion are contributing to the logarithmic negativity and with what magnitude. We construct an explicit contour function for Gaussian states using the fermionic partial-transpose. We generalize this contour function to generic many-body systems using a natural combination of derivatives of the logarithmic negativity. Though the latter negativity contour function is not strictly positive for all quantum systems, it is simple to compute and produces reasonable and interesting results. In particular, it rigorously satisfies the positivity condition for all holographic states and those obeying the quasi-particle picture. We apply this formalism to quantum field theories with a Fermi surface, contrasting the entanglement structure of Fermi liquids and holographic (hyperscale violating) non-Fermi liquids. The analysis of non-Fermi liquids show anomalous temperature dependence of the negativity depending on the dynamical critical exponent. We further compute the negativity contour following a quantum quench and discuss how this may clarify certain aspects of thermalization.1

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confidence: 99%

The negativity contour: a quasi-local measure of entanglement for mixed states

2020

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“…maximizing the fidelity over the set of unitaries U on H B . This function is bounded between 0 and 1, with maximally entangled states achieving the upper bound regardless of U. Perturbative expansions of the trace norm appearing in (8) for variations in U or Σ can be carried out using recently-developed methods in patterned-matrix calculus [39]. The operational symmetries enjoyed by fully entangled states motivate new ways to quanti fy quantum entanglement, a burgeoning area of research [40][41][42][43][44][45][46][47].…”

Section: Proof First Let Us Use the Singular Value Decompositionmentioning

confidence: 99%

“…1. The functions m (ρ) and E S (ρ) are distinct from both the entanglement entropy S (ρ A ) = −Tr (ρ A log ρ A ) and the entanglement negativity N (ρ) ∝ Tr ρ TA − 1, where a superscript T i denotes the partial transpose with respect to subsystem i [37,38]. For pure states, entanglement entropy and negativity reduce to −Tr Σ 2 log Σ 2 , and [Tr (Σ)] 2 − 1 respectively, and share similar behaviours with our measures as demonstrated in Fig.…”

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Operational symmetries of entangled states

Tzitrin¹,

Goldberg²,

Cresswell³

2020

J. Phys. A: Math. Theor.

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Quantum entanglement obscures the notion of local operations; there exist quantum states for which all local actions on one subsystem can be equivalently realized by actions on another. We characterize the states for which this fundamental property of entanglement does and does not hold, including multipartite and mixed states. Our results lead to a method for quantifying entanglement based on operational symmetries and has connections to quantum steering, envariance, the Reeh-Schlieder theorem, and classical entanglement.

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“…In particular, one of the most commonly used entanglement measures, called negativity, does not conform to the growth behaviour described above. In Chapter 6 we conduct an independent perturbative analysis of negativity and find that additional mathematical tools are required [122]. These new tools are extended from an underdeveloped branch of mathematics known as patterned matrix calculus.…”

Section: Outlinementioning

confidence: 99%

Quantum Information Approaches to Quantum Gravity

Cresswell¹

2019

Preprint

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In this thesis we apply techniques from quantum information theory to study quantum gravity within the framework of the anti-de Sitter / conformal field theory correspondence (AdS/CFT). A great deal of interest has arisen around how quantum information ideas in CFT translate to geometric features of the quantum gravitational theory in AdS. Through AdS/CFT, progress has been made in understanding the structure of entanglement in quantum field theories, and in how gravitational physics can emerge from these structures. However, this understanding is far from complete and will require the development of new tools to quantify correlations in CFT.This thesis presents refinements of a duality between operator product expansion (OPE) blocks in the CFT, giving the contribution of a conformal family to the OPE, and geodesic integrated fields in AdS which are diffeomorphism invariant quantities. This duality was originally discovered in the maximally symmetric setting of pure AdS dual to the CFT ground state. In less symmetric states the duality must be modified. Working with excited states within AdS 3 /CFT 2 , this thesis shows how the OPE block decomposes into more fine-grained CFT observables that are dual to AdS fields integrated over non-minimal geodesics. These constructions are presented for several classes of asymptotically AdS spacetimes.Additionally, this thesis contains results on the dynamics of entanglement measures for general quantum systems, not necessarily confined to quantum gravity. The quantification of quantum correlations is the main objective of quantum information theory, and it is crucial to understand how they are generated dynamically. Results are presented for the family of quantum Rényi entropies and entanglement negativity. Rényi entropies are studied for general dynamics by imposing special initial conditions. Around pure, separable initial states, all Rényi entropies grow with the same timescale at leading, and next-to-leading order. For negativity, mathematical tools are developed for the differentiation of non-analytic matrix functions with respect to constrained arguments. These tools are used to construct analytic expressions for derivatives of negativity. We establish bounds on the rate of change of state distinguishability under arbitrary dynamics, and the rate of entanglement growth for closed systems.

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Perturbative expansion of entanglement negativity using patterned matrix calculus

Cited by 6 publications

References 84 publications

The negativity contour: a quasi-local measure of entanglement for mixed states

The negativity contour: a quasi-local measure of entanglement for mixed states

Operational symmetries of entangled states

Quantum Information Approaches to Quantum Gravity

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