scholarly journals The Markov gap for geometric reflected entropy

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Patrick Hayden ◽  
Onkar Parrikar ◽  
Jonathan Sorce
Keyword(s):  
Lower Bound ◽  
Cross Section ◽  
Multipartite Entanglement ◽  
Correlation Measure ◽  
Extremal Surface ◽  
Information Theoretic ◽  
Fixed Area ◽  
Quantum Mutual Information ◽  
Recovery Problem ◽  
Symmetric States

Abstract The reflected entropy SR(A : B) of a density matrix ρAB is a bipartite correlation measure lower-bounded by the quantum mutual information I(A : B). In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-N2 gap between SR and I. We provide an information-theoretic interpretation of this gap by observing that SR− I is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity SR− I the Markov gap. We then prove that for time-symmetric states in pure AdS3 gravity, the Markov gap is universally lower bounded by log(2)ℓAdS/2GN times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling SR− I using fixed area states. This analysis involves deriving a formula for the quantum fidelity — in fact, for all the sandwiched Rényi relative entropies — between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.

scholarly journals Reflected entropy for an evaporating black hole

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Tianyi Li ◽  
Jinwei Chu ◽  
Yang Zhou
Keyword(s):  
Black Hole ◽  
Cross Section ◽  
Early Time ◽  
Three Dimensions ◽  
Late Time ◽  
Mixed States ◽  
Correlation Measure ◽  
Extremal Surface ◽  
Brane Model ◽  
The Right

Abstract We study reflected entropy as a mixed state correlation measure in black hole evaporation. As a measure for bipartite mixed states, reflected entropy can be computed between black hole and radiation, radiation and radiation, and even black hole and black hole. We compute reflected entropy curves in three different models: 3-side wormhole model, End-of-the-World (EOW) brane model in three dimensions and two-dimensional eternal black hole plus CFT model. For 3-side wormhole model, we find that reflected entropy is dual to island cross section. The reflected entropy between radiation and black hole increases at early time and then decreases to zero, similar to Page curve, but with a later transition time. The reflected entropy between radiation and radiation first increases and then saturates. For the EOW brane model, similar behaviors of reflected entropy are found.We propose a quantum extremal surface for reflected entropy, which we call quantum extremal cross section. In the eternal black hole plus CFT model, we find a generalized formula for reflected entropy with island cross section as its area term by considering the right half as the canonical purification of the left. Interestingly, the reflected entropy curve between the left black hole and the left radiation is nothing but the Page curve. We also find that reflected entropy between the left black hole and the right black hole decreases and goes to zero at late time. The reflected entropy between radiation and radiation increases at early time and saturates at late time.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

scholarly journals m-Bonsai: A Practical Compact Dynamic Trie

2018 ◽  
Vol 29 (08) ◽  
pp. 1257-1278 ◽  
Author(s):  
Andreas Poyias ◽  
Simon J. Puglisi ◽  
Rajeev Raman
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Upper Bound ◽  
Hash Functions ◽  
Information Theoretic ◽  
Expected Time ◽  
Speed Performance ◽  
Practical Performance

We consider the problem of implementing a space-efficient dynamic trie, with an emphasis on good practical performance. For a trie with [Formula: see text] nodes with an alphabet of size [Formula: see text], the information-theoretic space lower bound is [Formula: see text] bits. The Bonsai data structure is a compact trie proposed by Darragh et al. (Softw. Pract. Exper. 23(3) (1993) 277–291). Its disadvantages include the user having to specify an upper bound [Formula: see text] on the trie size in advance (which cannot be changed easily after initalization), a space usage of [Formula: see text] (which is asymptotically non-optimal for smaller [Formula: see text] or if [Formula: see text]) and a lack of support for deletions. It supports traversal and update operations in [Formula: see text] expected time (based on assumptions about the behaviour of hash functions), where [Formula: see text] and has excellent speed performance in practice. We propose an alternative, m-Bonsai, that addresses the above problems, obtaining a trie that uses [Formula: see text] bits in expectation, and supports traversal and update operations in [Formula: see text] expected time and [Formula: see text] amortized expected time, for any user-specified parameter [Formula: see text] (again based on assumptions about the behaviour of hash functions). We give an implementation of m-Bonsai which uses considerably less memory and is slightly faster than the original Bonsai.

scholarly journals A SIMPLE EFFICIENT INSTRUMENTAL VARIABLE ESTIMATOR FOR PANEL AR(p) MODELS WHEN BOTHNANDTARE LARGE

2009 ◽  
Vol 25 (3) ◽  
pp. 873-890 ◽  
Author(s):  
Kazuhiko Hayakawa
Keyword(s):  
Time Series ◽  
Lower Bound ◽  
Asymptotic Distribution ◽  
Cross Section ◽  
Simulation Study ◽  
Instrumental Variable ◽  
Asymptotic Variance ◽  
The Cross ◽  
Instrumental Variable Estimator ◽  
Normally Distributed

In this paper, we show that for panel AR(p) models, an instrumental variable (IV) estimator with instruments deviated from past means has the same asymptotic distribution as the infeasible optimal IV estimator when bothNandT, the dimensions of the cross section and time series, are large. If we assume that the errors are normally distributed, the asymptotic variance of the proposed IV estimator is shown to attain the lower bound when bothNandTare large. A simulation study is conducted to assess the estimator.

scholarly journals Heuristic for estimation of multiqubit genuine multipartite entanglement

2015 ◽  
Vol 13 (03) ◽  
pp. 1550023 ◽  
Author(s):  
Paulo E. M. F. Mendonça ◽  
Marcelo A. Marchiolli ◽  
Gerard J. Milburn
Keyword(s):  
Lower Bound ◽  
Density Matrix ◽  
Lower Bounds ◽  
Multipartite Entanglement ◽  
Zero Temperature ◽  
Mixed States ◽  
X State ◽  
Computational Basis ◽  
Genuine Multipartite Entanglement

For every N-qubit density matrix written in the computational basis, an associated "X-density matrix" can be obtained by vanishing all entries out of the main- and anti-diagonals. It is very simple to compute the genuine multipartite (GM) concurrence of this associated N-qubit X-state, which, moreover, lower bounds the GM-concurrence of the original (non-X) state. In this paper, we rely on these facts to introduce and benchmark a heuristic for estimating the GM-concurrence of an arbitrary multiqubit mixed state. By explicitly considering two classes of mixed states, we illustrate that our estimates are usually very close to the standard lower bound on the GM-concurrence, being significantly easier to compute. In addition, while evaluating the performance of our proposed heuristic, we provide the first characterization of GM-entanglement in the steady states of the driven Dicke model at zero temperature.

scholarly journals COMPLEMENTARITY IN GENERIC OPEN QUANTUM SYSTEMS

2010 ◽  
Vol 24 (24) ◽  
pp. 2485-2509 ◽  
Author(s):  
SUBHASHISH BANERJEE ◽  
R. SRIKANTH
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Atomic Systems ◽  
Information Theoretic ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Positive Operator Valued Measure

We develop a unified, information theoretic interpretation of the number-phase complementarity that is applicable both to finite-dimensional (atomic) and infinite-dimensional (oscillator) systems, with number treated as a discrete Hermitian observable and phase as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as a lower bound on entropy excess, X, the difference between the entropy of one variable, typically the number, and the knowledge of its complementary variable, typically the phase, where knowledge of a variable is defined as its relative entropy with respect to the uniform distribution. In the case of finite-dimensional systems, a weighting of phase knowledge by a factor μ (> 1) is necessary in order to make the bound tight, essentially on account of the POVM nature of phase as defined here. Numerical and analytical evidence suggests that μ tends to 1 as the system dimension becomes infinite. We study the effect of non-dissipative and dissipative noise on these complementary variables for an oscillator as well as atomic systems.

scholarly journals CORRELATIONS IN QUANTUM PHYSICS

2012 ◽  
Vol 27 (01n03) ◽  
pp. 1345017 ◽  
Author(s):  
ROSS DORNER ◽  
VLATKO VEDRAL
Keyword(s):  
Quantum Physics ◽  
Quantum Measurements ◽  
Harmonic Oscillators ◽  
Theoretic Analysis ◽  
Information Theoretic ◽  
Coupled Harmonic Oscillators ◽  
Quantum Mutual Information ◽  
Classical Correlations ◽  
The Difference ◽  
Thermodynamic Transformations

We provide a historical perspective of how the notion of correlations has evolved within quantum physics. We begin by reviewing Shannon's information theory and its first application in quantum physics, due to Everett, in explaining the information conveyed during a quantum measurement. This naturally leads us to Lindblad's information theoretic analysis of quantum measurements and his emphasis of the difference between the classical and quantum mutual information. After briefly summarizing the quantification of entanglement using these ideas, we arrive at the concept of quantum discord, which naturally captures the boundary between entanglement and classical correlations. Finally we discuss possible links between discord, which the generation of correlations in thermodynamic transformations of coupled harmonic oscillators.

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