Patterned matrix derivatives

Tracy, Derrick S.; Jinadasa, K.G.

doi:10.2307/3314938

Cited by 8 publications

(6 citation statements)

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“…We must employ a careful strategy for taking derivatives with respect to a matrix if there are any elements in that matrix which are (possibly constant) functions of the other elements. Such an approach was developed in [70][71][72] and we summarize it here for a differentiable function G(A, A * ) of a complex patterned matrix A:…”

Section: Discussionmentioning

confidence: 99%

“…In this appendix we recount the calculus of complex, patterned matrices developed in [47,[70][71][72] and summarized in [41]. This calculus was used in [41] to rigorously compute the derivative of a scalar function with respect to a Hermitian argument (20), and also provides a rigourous derivation of our result on patterned Hessians (36).…”

Section: Appendix B: Calculus Of Complex Patterned Matricesmentioning

confidence: 99%

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Perturbative expansion of entanglement negativity using patterned matrix calculus

2019

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Negativity is an entanglement monotone frequently used to quantify entanglement in bipartite states. Because negativity is a non-analytic function of a density matrix, existing methods used in the physics literature are insufficient to compute its derivatives. To this end we develop techniques in the calculus of complex, patterned matrices and use them to conduct a perturbative analysis of negativity in terms of arbitrary variations of the density operator. The result is an easy-to-implement expansion that can be carried out to all orders. On the way we provide convenient representations of the partial transposition map appearing in the definition of negativity. Our methods are well-suited to study the growth and decay of entanglement in a wide range of physical systems, including the generic linear growth of entanglement in many-body systems, and have broad relevance to many functions of quantum states and observables.1 By most definitions, entanglement measures ought to reduce to the entanglement entropy [17,24] in the pure state case; negativity does not. Nevertheless, we alternate between calling it a monotone and measure depending on the subsystem dimensions.arXiv:1809.07772v2 [quant-ph]

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Section: Discussionmentioning

confidence: 99%

Section: Appendix B: Calculus Of Complex Patterned Matricesmentioning

confidence: 99%

Perturbative expansion of entanglement negativity using patterned matrix calculus

2019

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“…First, we recount the calculus of complex, patterned matrices developed in [267,283,284,285] and summarized in [261]. This calculus was used in [261] to rigorously compute the derivative of a scalar function with respect to a Hermitian argument (6.19), and also provides a rigourous derivation of our result on patterned Hessians (6.36).…”

Section: When Is Patterned Matrix Calculus Required?mentioning

confidence: 99%

“…We must employ a careful strategy for taking derivatives with respect to a matrix if there are any elements in that matrix which are (possibly constant) functions of the other elements. Such an approach was developed in [283,284,285] and we summarize it here for a differentiable function G(A, A * ) of a complex patterned matrix A:…”

Section: Formal Calculus Of Complex Patterned Matricesmentioning

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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“…Tracy & Jinadasa (1988) [157] have shown that when we have one observation y t ∼ MV N(0, V t distribution the Fisher information matrix for the covariance matrix is I F (…”

Section: Expected Information Matrixmentioning

confidence: 99%

High dimensional time-varying covariance matrices with applications in finance

Plataniotis¹,

Πλατανιώτης²

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Patterned matrix derivatives

Cited by 8 publications

References 11 publications

Perturbative expansion of entanglement negativity using patterned matrix calculus

Perturbative expansion of entanglement negativity using patterned matrix calculus

Quantum Information Approaches to Quantum Gravity

High dimensional time-varying covariance matrices with applications in finance

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