On asymmetric quantum hypothesis testing
|
|
- Alison Lindsey
- 5 years ago
- Views:
Transcription
1 On asymmetric quantum hypothesis testing JMP, Vol 57, 6, / arxiv: Cambyse Rouzé (Cambridge) Joint with Nilanjana Datta (University of Cambridge) and Yan Pautrat (Paris-Saclay) Workshop: Quantum trajectories, parameter and state estimation Jan 2017, Toulouse January 25, / 31
2 Overview 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 2 / 31
3 Motivation: the quantum Stein lemma and its refinements Framework Let H finite dimensional Hilbert space, ρ, σ D(H) +, set of states on H Goal is to distinguish between: ρ (null hypothesis) and σ (alternative hypothesis). A test is a POVM {T, 1 T }, T B(H), 0 T 1 Errors when guessing are therefore given by: α(t ) = Tr(ρ(1 T )) β(t ) = Tr(σT ) type I error type II error Asymmetric case, minimize the type II error while controlling the type I error: β(ɛ) = inf {β(t ) α(t ) ɛ}. 0 T 1 3 / 31
4 Motivation: the quantum Stein lemma and its refinements Size-dependent hypothesis testing Let H n a sequence of Hilbert spaces, {ρ n} n N, {σ n} n N D(H n) +. Examples: the i.i.d. case: ρ n = ρ n vs. σ n = σ n. Gibbs states on a lattice Λ n of size Λ n = n d : e βρhρ Λn e βσ Hσ Λn ρ n := ( ) vs. σ n := ( ). Tr e βρhρ Λn Tr e βσ Hσ Λn Quasi-free fermions on a lattice etc. Question: Given a uniform bound ɛ on the type I errors, how quickly does β n(ɛ) decrease with n? 4 / 31
5 Motivation: the quantum Stein lemma and its refinements First-order asymptotics First proved in the i.i.d. setting Theorem (Quantum Stein s lemma Hiai&Petz91, Ogawa&Nagaoka00) When ρ n = ρ n, and σ n = σ n, 1 log βn(ɛ) D(ρ σ), ɛ (0, 1), where n D(ρ σ) = Tr(ρ(log ρ log σ)) Umegaki s relative entropy. r < D(ρ σ) lim 1 n n log min{α(tn) : β(tn) e nr } := sup 0<α<1 r > D(ρ σ) lim 1 n n log max{(1 α(tn)) : β(tn) e nr } = sup 1<α D α(ρ σ) : = 1 α 1 log Tr(ρα σ 1 α ) α 1 [r Dα(ρ σ)], α α 1 α [r D α(ρ σ)], where Rényi-α divergence, D α (ρ σ) : = 1 α 1 log Tr(ρ1/2 σ 1 α α ρ 1/2 ) α sandwiched Rényi-α divergence. 5 / 31
6 Motivation: the quantum Stein lemma and its refinements Interpretation of Stein s lemma α (1) 1 0 D(ρ σ) t 1 α (1) = inf {lim sup α(t n) : 0 T n 1 n lim inf n 1 n log β(tn) t 1}. Discontinuity of α (1) : manifestation of coarse-grained analysis. Question: can we better quantify the behaviour of the asymptotic type I error by adding a sub-exponential term for the type II error? 1 1 Lots of generalisations to the non i.i.d. framework: Bjelakovic&Siegmund-Schultze04, Bjelakovic,&Deuschel&Krueger&Seiler&Siegmund-Schultze&Szkola08, Hiai&Mosonyi&Ogawa08, Mosonyi&Hiai&Ogawa&Fannes08, Brandao&Plenio10, Jaksic&Ogata&Pillet&Seiringer12, etc. 6 / 31
7 Second-order asymptotics in Stein s lemma 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 7 / 31
8 Second-order asymptotics in Stein s lemma Second-order asymptotics in the i.i.d. case Tomamichel&Hayashi13 and Li14 proved the following second-order result: Theorem (Second-order asymptotics: the i.i.d. case Tomamichel&Hayashi13, Li14 ) When ρ n = ρ n and σ n = σ n, log β n(ɛ) = nd(ρ σ) + ns 1 (ɛ) + O(log n), s 1 (ɛ) := V (ρ σ)φ 1 (ɛ), where V (ρ σ) = Tr(ρ(log ρ log σ) 2 ) D(ρ σ) 2 quantum information variance, and Φ the cumulative distribution function of law N (0, 1). α (2) α (2) (t 2) = inf 0 Tn 1 { lim sup α(t n) lim inf n n 0 1 ( log βn(tn)+n D(ρ σ) ) } t 2 = Φ(t2 / n t 2 V (ρ σ)). 8 / 31
9 Second-order asymptotics in Stein s lemma Main result 1: Second-order asymptotics in the non-i.i.d. scenario For a given strictly increasing sequence of weights w n, under some conditions (see later), the quantum information variance rate is well-defined, and: Theorem (DPR16) Fix ɛ (0, 1). Define Then 1 v({ρ n}, {σ n}) = lim V (ρ n n σ n) w n t 2 (ɛ) = v({ρ n}, {σ n}) Φ 1 (ɛ). Optimality: t 2 > t 2 (ɛ) there exists a function ft 2 (x) = + o( x) such that n N and any sequence of tests T n: log β(t n) D(ρ n σ n) + w n t 2 + f t2 (w n), (1) Achievability: t 2 < t 2 (ɛ), f (x) = o( x), there exists a sequence of test T n such that + for n large enough: log β n(t n) D(ρ n σ n) + w n t 2 + f (w n). (2) The i.i.d. case of Tomamichel&Hayashi13 and Li14 follows (Bryc CLT). The Berry-Esseen theorem even provides information on third order: log β n(ɛ) = nd(ρ σ) + nv (ρ σ)φ 1 (ɛ) + O(log n), 9 / 31
10 Second-order asymptotics in Stein s lemma Key tools 1: the relative modular operator The relative modular operator is a generalisation of the Radon-Nikodym derivative for general von Neuman algebras. In finite dimensions, for two faithful states, it reduces to: ρ σ : B(H) B(H), A ρaσ 1. ρ σ is a positive operator, admits a spectral decomposition. Quantities of relevance can be rewritten in terms of ρ σ : D(ρ σ) = ρ 1/2, log( ρ σ )(ρ 1/2 ), Ψ s(ρ σ) := log Tr(ρ 1+s σ s ) = log ρ 1/2, s ρ σ (ρ1/2 ) where A, B = Tr(A B). Define X to be the classical random variable taking values on spec(log ρ σ ) such that for any measurable f : spec(log ρ σ ) R, ρ 1/2, f (log ρ σ )(ρ 1/2 ) E[f (X )]. f : x x E[X ] = D(ρ σ), f : x e sx log E[e sx ] = log ρ 1/2, e s log ρ σ (ρ 1/2 ) Ψ s(ρ σ), the cumulant-generating function. 10 / 31
11 Second-order asymptotics in Stein s lemma Key tools 2: Bryc s theorem The following theorem is a non i.i.d. generalization of the Central limit theorem: Theorem (Bryc s theorem Bryc93) Let (X n) n N be a sequence of random variables, and (w n) n N an increasing sequence of weights. If there exists r > 0 such that: ] n N, H n(z) := log E [e zxn is analytic in the complex open ball B C (0, r), 1 H(x) = lim n Hn(x) exists x ( r, r), wn 1 sup n N sup z BC (0,r) Hn(z) < +, wn then H is analytic on B C (0, r) and X n H n(0) wn d n N ( 0, H (0) ), 11 / 31
12 Second-order asymptotics in Stein s lemma The working condition Take X n associated to log ρn σn, ρn 1/2 H n(z) = Ψ z (ρ n σ n). The conditions of Bryc s theorem translate into: Condition Let {w n} an increasing sequence of weights. Assume there exists r > 0 such that n N, z Ψ z (ρ n σ n) is analytic in the complex open ball B C (0, r), 1 H(x) = lim n Ψz (ρn σn) exists x ( r, r), wn sup n N sup z BC (0,r) 1 Ψz (ρn σn) < +, wn Here H n (0) = D(ρn σn), By Bryc s theorem, the quantum information variance rate is well-defined. 1 v({ρ n}, {σ n}) = lim V (ρ n n σ n) H (0) w n 12 / 31
13 Second-order asymptotics in Stein s lemma Proof of achievability (2) We will need the following crucial technical lemma: Lemma (Li14, DPR16) Let ρ, σ D +(H). For all L > 0 there exists a test T such that Tr(ρ(1 T )) ρ 1/2, P (0,L) ( ρ σ )(ρ 1/2 ) and Tr(σT ) L 1. (3) Let L n := exp(d(ρ n σ n) + w nt 2 + f (w n)), f (x) = ( x). (3) log β(t n) D(ρ n σ n) + w n t 2 + f (w n) α(t n) ρ n 1/2, P (0,Ln)( ρn σ n ) ρ n 1/2, = ρ 1/2 1/2 n, P (,log Ln)(log ρn σn ) ρ n. ( Xn D(ρ n σ n) = P(X n log L n) = P t 2 + f (wn) ). wn wn Bryc s theorem RHS converges to Φ(t 2 / v({ρ n}, {σ n})) < ε for n large enough, α(t n) ε. Optimality (1) follows similarly from a lower bound on the total error provided by Jaksic&Ogata&Pillet&Seiringer / 31
14 Some examples 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 14 / 31
15 Some examples Quantum spin systems Lattice Zd. System prepared at each site: x, Hx = H. Example: particle (spin ± 21 ): Hx = C2. 15 / 31
16 Some examples X Zd, HX := x X Hx, Interaction between sites: Φ : X 7 ΦX Bsa (HX ). Φ Translation invariant: invariant on sets of same shape. Φ Finite range: R > 0 : diam(x ) > R ΦX = 0. Φ induces dynamics, with equilibrium states (Gibbs states): X HXΦ := ΦY Y X Φ ρφ,β := X e βhx Φ Tr(e βhx ). 16 / 31
17 Some examples Λ n := { n,..., n} d. Translation invariant, of finite range interactions Φ, Ψ. Suppose given one of two states ρ n := ρ Φ,β 1 Λ n or σ n := ρ Ψ,β 2 Λ n Tests performed on H Λn Proposition (DPR16) For β 1, β 2 small enough, ρ n and σ n satisfy the conditions for second order asymptotics w.r.t. w n = Λ n. 17 / 31
18 Some examples Free lattice fermions One-particle Hilbert space h := l 2 (Z d ), standard basis {e i : i Z d } (particle localized at site i). Construct the Fock space F(h) = n N n h, where x 1... x n = 1 n! σ S n sgn(σ)x σ(1)... x σ(n). Algebra of observables: C algebra generated by creation and anihilation operators on the Fock space, obeying the canonical anti-commutation relations: a(x)a(y) + a(y)a(x) = 0, a(x)a (y) + a (y)a(x) = x, y 1, x, y h. a (y): unique linear extension a (y) : F(h) F(h) of a (y)x 1... x n = y x 1... x n, a(y) = (a (y)) 18 / 31
19 Some examples Let H B(h) be a one-particle Hamiltonian, define Ĥ := dγ(h) the differential second quantization on F(h), closure of ( n ) H n(x 1... x n) = P x 1... Hx k... x n, k=1 where P is the projection onto the antisymmetric subspace. Define the Gibbs state: ρ H β = e βĥ ( ) Tr e βĥ Gibbs states are expressed as linear forms on CAR(h) as follows: Tr(ρ H β a (x 1 )...a (x n)a(y m)...a(y 1 )) = δ mn det( y i, Qx i ) i,j, Q := e βh 1 + e βh. ρ H β is called the quasi-free state with symbol Q. Define the shift operators T j : e i e i+j, i, j Z d. A symbol which commutes with all shift operators is said to be shift invariant. 19 / 31
20 Some examples Assume now that we only have access to part of the lattice: denote Λ n := {0,..., n 1} d Z d, h n = l 2 (Λ n) h. Assume given two shift invariant symbols δ < Q, R < 1 δ. Denote Q n = P nqp n, R n = P nrp n, where P n orthogonal projection onto h n. The sequences of states that we want to distinguish between are then: {ρ n} associated with symbols {Q n} vs. {σ n} associated with symbols {R n}. Dierckx&Fannes&Pogorzelska08: ρ n, σ n can be written as: ρ n = det(1 Q n) dim h n k=0 k dim Qn h n σ n = det(1 R n) 1 Q n k=0 k Rn 1 R n. Proposition (DPR16) ρ n, and σ n satisfy the conditions for second order asymptotics w.r.t. w n = Λ n. 20 / 31
21 Finite sample size quantum hypothesis testing: the i.i.d. case 1 Motivation: the quantum Stein lemma and its refinements 2 Second-order asymptotics in Stein s lemma 3 Some examples 4 Finite sample size quantum hypothesis testing: the i.i.d. case 21 / 31
22 Finite sample size quantum hypothesis testing: the i.i.d. case Finite sample size bounds of Audenaert&Mosonyi&Verstraete12 Practical situations: finitely many copies available. Theorem (Audenaert&Mosonyi&Verstraete12) For ρ n := ρ n and σ n := σ n, D(ρ σ) f (ɛ) 1 g(ɛ) log βn(ɛ) D(ρ σ) +, n n n where f (ɛ) = 4 2 log η log(1 ɛ) 1, g(ɛ) = 4 2 log η log ɛ 1, and η := 1 + e 1/2D 3/2 (ρ σ) + e 1/2D 1/2 (ρ σ). The second order parts of the bounds scale as log ɛ 1, to compare with Φ 1 (ɛ) in the asymptotic case not tight. A better upper bound can be found by means of (non-commutative) martingale concentration inequalities. 22 / 31
23 Finite sample size quantum hypothesis testing: the i.i.d. case Non-commutative martingales A noncommutative probability space is a couple (M, τ), where M is a von Neumann algebra, and τ a normal tracial state on M. Let N be a von Neumann subalgebra of M. Then there exists a unique map E[. N ] : M N, called conditional expectation, such that E[1 N ] = 1, E[AXB N ] = AE[X N ]B, A, B N, X M, E [τ N ] = τ, A noncommutative filtration of M is an increasing sequence {M j } 1 j n of von Neumann subalgebras of M. A martingale is a sequence of noncommutative random variables {X j } 1 j n M n such that for each j, X j M j, τ( X j ) < E[X j+1 M j ] = X j (integrability), (martingale property). 23 / 31
24 Finite sample size quantum hypothesis testing: the i.i.d. case A noncommutative martingale concentration inequality Theorem (Noncommutative Azuma inequality, Sadeghi&Moslehian14) Let {X j, M j } 1 j n be a self-adjoint martingale. If d j X j+1 X j d j (d j > 0), ( ) α 2 τ(1 [α, ) (X n)) exp 2 n, α > 0. j=1 d2 j Idea (borrowed from Sason11): Take ρ n = ρ 1... ρ n, and σ n = σ 1... σ n. Then ρn σn = n n 1 ρj σ j log ρn σn = id j log ρj σ j id n j 1 j=1 Define algebras M k generated by id i log ρi σ i id n i 1, i k. ρ 1/2 n, (.) ρ 1/2 n is a tracial state on M n, and (log ρk σ k D(ρ k σ k ), M k ) is a martingale, where E[. M k ] := ρ k+1... ρ n, (.) ρ k+1... ρ n. j=0 24 / 31
25 Finite sample size quantum hypothesis testing: the i.i.d. case {log ρk σ k D(ρ k σ k )} 1 k n is a self-adjoint martingale, hence by NC Azuma: Recall: n, L n, T n: Hence, ρ 1/2 n, 1 (r, ) (log( ρn σn ) D(ρ n σ n))(ρn 1/2 r2 2 j ) e d2 j. α(t n) ρ 1/2 n, P (0,Ln)( ρn σ n )(ρ 1/2 n ) and β(t n) L 1 n. (log Ln D(ρn σn))2 α(t n) ρ 1/2 n, P (,log Ln)(log ρn σn )(ρn 1/2 2 j ) e d2 j ɛ log L n 2 dj 2 ln ɛ 1 + D(ρ n σ n) j β n(ɛ) β(t n) e D(ρn σn) 2 j d2 log ɛ j / 31
26 Finite sample size quantum hypothesis testing: the i.i.d. case Remember Audenaert&Mosonyi&Verstraete12 bound: log β n(ɛ) nd(ρ σ) + ng(ɛ), g(ɛ) := K ρ,σ log ɛ 1. Theorem (DR16) Suppose given states of the form n n ρ n = ρ j and σ n = σ j, j=1 j=1 where for each j, ρ j, σ j D(H j ) +. Then for each n N: n n log β n(ɛ) D( ρ j σ j ) + 2 log ɛ 1 dk 2, d k := log ρk σ k D( ρ k σ k ). j=1 k=1 In the i.i.d. case (ρ n = ρ n vs. σ n = σ n ), log β n(ɛ) nd(ρ σ) + n h(ɛ), h(ɛ) := 2 log ɛ 1 d / 31
27 Finite sample size quantum hypothesis testing: the i.i.d. case Comparison with Audenaert&Mosonyi&Verstraete12 Our error scales as: log ɛ 1, as opposed to log ɛ 1. We are getting closer to the asymptotic behavior in Φ 1 (ɛ). Rates ϵ g(ϵ) (ϵ) h s 1 (ϵ) -f(ϵ) / 31
28 Conclusion Summary and open questions We studied second order asymptotics as well as finite sample size hypothesis testing in the asymmetric scenario. We proved a theorem providing second order asymptotics for a large class of quantum systems, including the i.i.d. scenario of Tomamichel&Hayashi13 and Li14, as well as states of quantum spin systems and free fermions on a lattice. We found finite sample size bounds on the optimal type II error in the i.i.d. scenario using noncommutative martingale concentration inequalities. Open question: Does the martingale approach give bounds in physically relevant non i.i.d. examples? 28 / 31
29 Conclusion Thank you for your attention! 29 / 31
30 References References Bjelakovic&Siegmund-Schultze04: An ergodic theorem for quantum relative entropy. Comm. Math. Phys. 247, 697 (2004) Bjelakovic&Deuschel,&Krueger&Seiler&Siegmund-Schultze&Szkola08: Typical support and Sanov large deviations of correlated states. Comm. Math. Phys. 279, 559 (2008). Brandão&Plenio: A Generalization of Quantum Stein s Lemma. Commun. Math. Phys. 295: 791 (2010). Datta&Pautrat&Rouze16: Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. quantum lattice systems and more. Journal of Mathematical Physics, 57, 6, (2016) Audenaert&Koenraad&Verstraete12: Quantum state discrimination bounds for finite sample size. Journal of Mathematical Physics, Vol. 53, (2012). Hiai&Mosonyi&Ogawa08: Error exponents in hypothesis testing for correlated states on a spin chain. J. Math. Phys. 49, (2008). Hiai&Petz91: The proper formula for the relative entropy an its asymptotics in quantum probability. Comm. Math. Phys. 143, 99 (1991). 30 / 31
31 References Jaksic&Ogata&Piller&Seiringer12: Quantum hypothesis testing and non-equilibrium statistical mechanics. Rev. Math. Phys. 24, (2012). Li14: Second-order asymptotics for quantum hypothesis testing. Ann. Statist. Volume 42, Number 1, (2014). Mosonyi&Hiai&Ogawa&Fannes08: Asymptotic distinguishability measures for shiftinvariant quasi-free states of fermionic lattice systems. J. Math. Phys. 49, (2008). Ogawa&Nagaoka00: Strong Converse and Stein s Lemma in the Quantum Hypothesis Testing. IEEE Trans. Inf. Theo. 46, 2428 (2000). Sadghi&Moslehian14: Noncommutative martingale concentration inequalities. Illinois Journal of Mathematics, Volume 58, Number 2, (2014). Sason11: Moderate deviations analysis of binary hypothesis testing IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, pp (2012). Tomamichel&Hayashi13: A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks. IEEE Transactions on Information Theory, vol. 59, no. 11, pp (2013). 31 / 31
A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks Marco Tomamichel, Masahito Hayashi arxiv: 1208.1478 Also discussing results of: Second Order Asymptotics for Quantum
More informationCorrelation Detection and an Operational Interpretation of the Rényi Mutual Information
Correlation Detection and an Operational Interpretation of the Rényi Mutual Information Masahito Hayashi 1, Marco Tomamichel 2 1 Graduate School of Mathematics, Nagoya University, and Centre for Quantum
More informationStrong converse theorems using Rényi entropies
Strong converse theorems using Rényi entropies Felix Leditzky joint work with Mark M. Wilde and Nilanjana Datta arxiv:1506.02635 5 January 2016 Table of Contents 1 Weak vs. strong converse 2 Rényi entropies
More informationDifferent quantum f-divergences and the reversibility of quantum operations
Different quantum f-divergences and the reversibility of quantum operations Fumio Hiai Tohoku University 2016, September (at Budapest) Joint work with Milán Mosonyi 1 1 F.H. and M. Mosonyi, Different quantum
More informationOn Composite Quantum Hypothesis Testing
University of York 7 November 207 On Composite Quantum Hypothesis Testing Mario Berta Department of Computing with Fernando Brandão and Christoph Hirche arxiv:709.07268 Overview Introduction 2 Composite
More informationNullity of Measurement-induced Nonlocality. Yu Guo
Jul. 18-22, 2011, at Taiyuan. Nullity of Measurement-induced Nonlocality Yu Guo (Joint work with Pro. Jinchuan Hou) 1 1 27 Department of Mathematics Shanxi Datong University Datong, China guoyu3@yahoo.com.cn
More informationLecture 19 October 28, 2015
PHYS 7895: Quantum Information Theory Fall 2015 Prof. Mark M. Wilde Lecture 19 October 28, 2015 Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
More informationThe Phase Space in Quantum Field Theory
The Phase Space in Quantum Field Theory How Small? How Large? Martin Porrmann II. Institut für Theoretische Physik Universität Hamburg Seminar Quantum Field Theory and Mathematical Physics April 13, 2005
More informationQuantum Achievability Proof via Collision Relative Entropy
Quantum Achievability Proof via Collision Relative Entropy Salman Beigi Institute for Research in Fundamental Sciences (IPM) Tehran, Iran Setemper 8, 2014 Based on a joint work with Amin Gohari arxiv:1312.3822
More informationCapacity Estimates of TRO Channels
Capacity Estimates of TRO Channels arxiv: 1509.07294 and arxiv: 1609.08594 Li Gao University of Illinois at Urbana-Champaign QIP 2017, Microsoft Research, Seattle Joint work with Marius Junge and Nicholas
More informationPhenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 25-29, 2012
Phenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 25-29, 202 BOUNDS AND ASYMPTOTICS FOR FISHER INFORMATION IN THE CENTRAL LIMIT THEOREM
More informationStrong Converse and Stein s Lemma in the Quantum Hypothesis Testing
Strong Converse and Stein s Lemma in the Quantum Hypothesis Testing arxiv:uant-ph/9906090 v 24 Jun 999 Tomohiro Ogawa and Hiroshi Nagaoka Abstract The hypothesis testing problem of two uantum states is
More informationAlgebraic Theory of Entanglement
Algebraic Theory of (arxiv: 1205.2882) 1 (in collaboration with T.R. Govindarajan, A. Queiroz and A.F. Reyes-Lega) 1 Physics Department, Syracuse University, Syracuse, N.Y. and The Institute of Mathematical
More informationOn Third-Order Asymptotics for DMCs
On Third-Order Asymptotics for DMCs Vincent Y. F. Tan Institute for Infocomm Research (I R) National University of Singapore (NUS) January 0, 013 Vincent Tan (I R and NUS) Third-Order Asymptotics for DMCs
More informationvon Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai)
von Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai) Lecture 3 at IIT Mumbai, April 24th, 2007 Finite-dimensional C -algebras: Recall: Definition: A linear functional tr
More informationQuantum Symmetric States
Quantum Symmetric States Ken Dykema Department of Mathematics Texas A&M University College Station, TX, USA. Free Probability and the Large N limit IV, Berkeley, March 2014 [DK] K. Dykema, C. Köstler,
More informationFréchet differentiability of the norm of L p -spaces associated with arbitrary von Neumann algebras
Fréchet differentiability of the norm of L p -spaces associated with arbitrary von Neumann algebras (Part I) Fedor Sukochev (joint work with D. Potapov, A. Tomskova and D. Zanin) University of NSW, AUSTRALIA
More informationBosonic quadratic Hamiltonians. diagonalization
and their diagonalization P. T. Nam 1 M. Napiórkowski 2 J. P. Solovej 3 1 Masaryk University Brno 2 University of Warsaw 3 University of Copenhagen Macroscopic Limits of Quantum Systems 70th Birthday of
More informationOn representations of finite type
Proc. Natl. Acad. Sci. USA Vol. 95, pp. 13392 13396, November 1998 Mathematics On representations of finite type Richard V. Kadison Mathematics Department, University of Pennsylvania, Philadelphia, PA
More informationThe general reason for approach to thermal equilibrium of macroscopic quantu
The general reason for approach to thermal equilibrium of macroscopic quantum systems 10 May 2011 Joint work with S. Goldstein, J. L. Lebowitz, C. Mastrodonato, and N. Zanghì. Claim macroscopic quantum
More informationFree probability and quantum information
Free probability and quantum information Benoît Collins WPI-AIMR, Tohoku University & University of Ottawa Tokyo, Nov 8, 2013 Overview Overview Plan: 1. Quantum Information theory: the additivity problem
More informationMarch 1, Florida State University. Concentration Inequalities: Martingale. Approach and Entropy Method. Lizhe Sun and Boning Yang.
Florida State University March 1, 2018 Framework 1. (Lizhe) Basic inequalities Chernoff bounding Review for STA 6448 2. (Lizhe) Discrete-time martingales inequalities via martingale approach 3. (Boning)
More informationSTAT 200C: High-dimensional Statistics
STAT 200C: High-dimensional Statistics Arash A. Amini May 30, 2018 1 / 59 Classical case: n d. Asymptotic assumption: d is fixed and n. Basic tools: LLN and CLT. High-dimensional setting: n d, e.g. n/d
More informationMod-φ convergence I: examples and probabilistic estimates
Mod-φ convergence I: examples and probabilistic estimates Valentin Féray (joint work with Pierre-Loïc Méliot and Ashkan Nikeghbali) Institut für Mathematik, Universität Zürich Summer school in Villa Volpi,
More informationModular theory and Bekenstein s bound
Modular theory and Bekenstein s bound Roberto Longo Okinawa, OIST, Strings 2018, June 2018 Partly based on a joint work with Feng Xu Thermal equilibrium states A primary role in thermodynamics is played
More informationA complete criterion for convex-gaussian states detection
A complete criterion for convex-gaussian states detection Anna Vershynina Institute for Quantum Information, RWTH Aachen, Germany joint work with B. Terhal NSF/CBMS conference Quantum Spin Systems The
More informationMean-field equations for higher-order quantum statistical models : an information geometric approach
Mean-field equations for higher-order quantum statistical models : an information geometric approach N Yapage Department of Mathematics University of Ruhuna, Matara Sri Lanka. arxiv:1202.5726v1 [quant-ph]
More informationOn positive maps in quantum information.
On positive maps in quantum information. Wladyslaw Adam Majewski Instytut Fizyki Teoretycznej i Astrofizyki, UG ul. Wita Stwosza 57, 80-952 Gdańsk, Poland e-mail: fizwam@univ.gda.pl IFTiA Gdańsk University
More informationApproximate reversal of quantum Gaussian dynamics
Approximate reversal of quantum Gaussian dynamics L. Lami, S. Das, and M.M. Wilde arxiv:1702.04737 or how to compute Petz maps associated with Gaussian states and channels. Motivation: data processing
More informationSelf-normalized Cramér-Type Large Deviations for Independent Random Variables
Self-normalized Cramér-Type Large Deviations for Independent Random Variables Qi-Man Shao National University of Singapore and University of Oregon qmshao@darkwing.uoregon.edu 1. Introduction Let X, X
More informationAQI: Advanced Quantum Information Lecture 6 (Module 2): Distinguishing Quantum States January 28, 2013
AQI: Advanced Quantum Information Lecture 6 (Module 2): Distinguishing Quantum States January 28, 2013 Lecturer: Dr. Mark Tame Introduction With the emergence of new types of information, in this case
More informationhere, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional
15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:
More informationLAPLACIANS COMPACT METRIC SPACES. Sponsoring. Jean BELLISSARD a. Collaboration:
LAPLACIANS on Sponsoring COMPACT METRIC SPACES Jean BELLISSARD a Georgia Institute of Technology, Atlanta School of Mathematics & School of Physics Collaboration: I. PALMER (Georgia Tech, Atlanta) a e-mail:
More informationEntanglement: concept, measures and open problems
Entanglement: concept, measures and open problems Division of Mathematical Physics Lund University June 2013 Project in Quantum information. Supervisor: Peter Samuelsson Outline 1 Motivation for study
More informationMultivariate Trace Inequalities
Multivariate Trace Inequalities Mario Berta arxiv:1604.03023 with Sutter and Tomamichel (to appear in CMP) arxiv:1512.02615 with Fawzi and Tomamichel QMath13 - October 8, 2016 Mario Berta (Caltech) Multivariate
More informationConsistency of the maximum likelihood estimator for general hidden Markov models
Consistency of the maximum likelihood estimator for general hidden Markov models Jimmy Olsson Centre for Mathematical Sciences Lund University Nordstat 2012 Umeå, Sweden Collaborators Hidden Markov models
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationFREE PROBABILITY THEORY
FREE PROBABILITY THEORY ROLAND SPEICHER Lecture 4 Applications of Freeness to Operator Algebras Now we want to see what kind of information the idea can yield that free group factors can be realized by
More informationStein Couplings for Concentration of Measure
Stein Couplings for Concentration of Measure Jay Bartroff, Subhankar Ghosh, Larry Goldstein and Ümit Işlak University of Southern California [arxiv:0906.3886] [arxiv:1304.5001] [arxiv:1402.6769] Borchard
More informationConverse bounds for private communication over quantum channels
Converse bounds for private communication over quantum channels Mark M. Wilde (LSU) joint work with Mario Berta (Caltech) and Marco Tomamichel ( Univ. Sydney + Univ. of Technology, Sydney ) arxiv:1602.08898
More informationFermionic coherent states in infinite dimensions
Fermionic coherent states in infinite dimensions Robert Oeckl Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Morelia, Mexico Coherent States and their Applications CIRM, Marseille,
More informationQuantum Fisher Information. Shunlong Luo Beijing, Aug , 2006
Quantum Fisher Information Shunlong Luo luosl@amt.ac.cn Beijing, Aug. 10-12, 2006 Outline 1. Classical Fisher Information 2. Quantum Fisher Information, Logarithm 3. Quantum Fisher Information, Square
More informationOn the classification and modular extendability of E 0 -semigroups on factors Joint work with Daniel Markiewicz
On the classification and modular extendability of E 0 -semigroups on factors Joint work with Daniel Markiewicz Panchugopal Bikram Ben-Gurion University of the Nagev Beer Sheva, Israel pg.math@gmail.com
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationTrace Class Operators and Lidskii s Theorem
Trace Class Operators and Lidskii s Theorem Tom Phelan Semester 2 2009 1 Introduction The purpose of this paper is to provide the reader with a self-contained derivation of the celebrated Lidskii Trace
More informationA Holevo-type bound for a Hilbert Schmidt distance measure
Journal of Quantum Information Science, 205, *,** Published Online **** 204 in SciRes. http://www.scirp.org/journal/**** http://dx.doi.org/0.4236/****.204.***** A Holevo-type bound for a Hilbert Schmidt
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationDecay of correlations in 2d quantum systems
Decay of correlations in 2d quantum systems Costanza Benassi University of Warwick Quantissima in the Serenissima II, 25th August 2017 Costanza Benassi (University of Warwick) Decay of correlations in
More informationSchur-Weyl duality, quantum data compression, tomography
PHYSICS 491: Symmetry and Quantum Information April 25, 2017 Schur-Weyl duality, quantum data compression, tomography Lecture 7 Michael Walter, Stanford University These lecture notes are not proof-read
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationSpectral Continuity Properties of Graph Laplacians
Spectral Continuity Properties of Graph Laplacians David Jekel May 24, 2017 Overview Spectral invariants of the graph Laplacian depend continuously on the graph. We consider triples (G, x, T ), where G
More informationLecture 11: Quantum Information III - Source Coding
CSCI5370 Quantum Computing November 25, 203 Lecture : Quantum Information III - Source Coding Lecturer: Shengyu Zhang Scribe: Hing Yin Tsang. Holevo s bound Suppose Alice has an information source X that
More informationPropagation bounds for quantum dynamics and applications 1
1 Quantum Systems, Chennai, August 14-18, 2010 Propagation bounds for quantum dynamics and applications 1 Bruno Nachtergaele (UC Davis) based on joint work with Eman Hamza, Spyridon Michalakis, Yoshiko
More informationGaussian vectors and central limit theorem
Gaussian vectors and central limit theorem Samy Tindel Purdue University Probability Theory 2 - MA 539 Samy T. Gaussian vectors & CLT Probability Theory 1 / 86 Outline 1 Real Gaussian random variables
More informationTHE ALTERNATIVE DUNFORD-PETTIS PROPERTY FOR SUBSPACES OF THE COMPACT OPERATORS
THE ALTERNATIVE DUNFORD-PETTIS PROPERTY FOR SUBSPACES OF THE COMPACT OPERATORS MARÍA D. ACOSTA AND ANTONIO M. PERALTA Abstract. A Banach space X has the alternative Dunford-Pettis property if for every
More informationUpper triangular forms for some classes of infinite dimensional operators
Upper triangular forms for some classes of infinite dimensional operators Ken Dykema, 1 Fedor Sukochev, 2 Dmitriy Zanin 2 1 Department of Mathematics Texas A&M University College Station, TX, USA. 2 School
More informationMean-Field Theory: HF and BCS
Mean-Field Theory: HF and BCS Erik Koch Institute for Advanced Simulation, Jülich N (x) = p N! ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )........ ' (x N ) ' 2 (x N ) ' N (x N ) Slater determinant
More informationABSTRACT CONDITIONAL EXPECTATION IN L 2
ABSTRACT CONDITIONAL EXPECTATION IN L 2 Abstract. We prove that conditional expecations exist in the L 2 case. The L 2 treatment also gives us a geometric interpretation for conditional expectation. 1.
More informationInformation Theory. Lecture 5 Entropy rate and Markov sources STEFAN HÖST
Information Theory Lecture 5 Entropy rate and Markov sources STEFAN HÖST Universal Source Coding Huffman coding is optimal, what is the problem? In the previous coding schemes (Huffman and Shannon-Fano)it
More informationEinstein-Hilbert action on Connes-Landi noncommutative manifolds
Einstein-Hilbert action on Connes-Landi noncommutative manifolds Yang Liu MPIM, Bonn Analysis, Noncommutative Geometry, Operator Algebras Workshop June 2017 Motivations and History Motivation: Explore
More informationAlmost sure invariance principle for sequential and non-stationary dynamical systems (see arxiv )
Almost sure invariance principle for sequential and non-stationary dynamical systems (see paper @ arxiv ) Nicolai Haydn, University of Southern California Matthew Nicol, University of Houston Andrew Török,
More informationNon commutative Khintchine inequalities and Grothendieck s theo
Non commutative Khintchine inequalities and Grothendieck s theorem Nankai, 2007 Plan Non-commutative Khintchine inequalities 1 Non-commutative Khintchine inequalities 2 µ = Uniform probability on the set
More informationarxiv:quant-ph/ v1 27 May 2004
arxiv:quant-ph/0405166v1 27 May 2004 Separability condition for the states of fermion lattice systems and its characterization Hajime Moriya Abstract We introduce a separability condition for the states
More informationarxiv:math/ v1 [math.oa] 3 Jun 2003
arxiv:math/0306061v1 [math.oa] 3 Jun 2003 A NOTE ON COCYCLE-CONJUGATE ENDOMORPHISMS OF VON NEUMANN ALGEBRAS REMUS FLORICEL Abstract. We show that two cocycle-conjugate endomorphisms of an arbitrary von
More informationarxiv: v1 [math.oa] 23 Jul 2014
PERTURBATIONS OF VON NEUMANN SUBALGEBRAS WITH FINITE INDEX SHOJI INO arxiv:1407.6097v1 [math.oa] 23 Jul 2014 Abstract. In this paper, we study uniform perturbations of von Neumann subalgebras of a von
More informationare Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication
7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =
More informationAsymptotic Pure State Transformations
Asymptotic Pure State Transformations PHYS 500 - Southern Illinois University April 18, 2017 PHYS 500 - Southern Illinois University Asymptotic Pure State Transformations April 18, 2017 1 / 15 Entanglement
More informationA Tight Upper Bound on the Second-Order Coding Rate of Parallel Gaussian Channels with Feedback
A Tight Upper Bound on the Second-Order Coding Rate of Parallel Gaussian Channels with Feedback Vincent Y. F. Tan (NUS) Joint work with Silas L. Fong (Toronto) 2017 Information Theory Workshop, Kaohsiung,
More informationLecture 1 Operator spaces and their duality. David Blecher, University of Houston
Lecture 1 Operator spaces and their duality David Blecher, University of Houston July 28, 2006 1 I. Introduction. In noncommutative analysis we replace scalar valued functions by operators. functions operators
More informationIntroduction to Quantum Information Hermann Kampermann
Introduction to Quantum Information Hermann Kampermann Heinrich-Heine-Universität Düsseldorf Theoretische Physik III Summer school Bleubeuren July 014 Contents 1 Quantum Mechanics...........................
More informationOperator norm convergence for sequence of matrices and application to QIT
Operator norm convergence for sequence of matrices and application to QIT Benoît Collins University of Ottawa & AIMR, Tohoku University Cambridge, INI, October 15, 2013 Overview Overview Plan: 1. Norm
More informationNumerical Simulation of Spin Dynamics
Numerical Simulation of Spin Dynamics Marie Kubinova MATH 789R: Advanced Numerical Linear Algebra Methods with Applications November 18, 2014 Introduction Discretization in time Computing the subpropagators
More informationSecond-Order Asymptotics in Information Theory
Second-Order Asymptotics in Information Theory Vincent Y. F. Tan (vtan@nus.edu.sg) Dept. of ECE and Dept. of Mathematics National University of Singapore (NUS) National Taiwan University November 2015
More informationStrong converse theorems using Rényi entropies
Strong converse theorems using Rényi entropies Felix Leditzky a, Mark M. Wilde b, and Nilanjana Datta a a Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge C3
More informationNEW FUNCTIONAL INEQUALITIES
1 / 29 NEW FUNCTIONAL INEQUALITIES VIA STEIN S METHOD Giovanni Peccati (Luxembourg University) IMA, Minneapolis: April 28, 2015 2 / 29 INTRODUCTION Based on two joint works: (1) Nourdin, Peccati and Swan
More informationStochastic domination in space-time for the supercritical contact process
Stochastic domination in space-time for the supercritical contact process Stein Andreas Bethuelsen joint work with Rob van den Berg (CWI and VU Amsterdam) Workshop on Genealogies of Interacting Particle
More informationQuantum Macrostates, Equivalence of Ensembles and an H Theorem
Quantum Macrostates, Equivalence of Ensembles and an H Theorem Wojciech De Roeck, Christian Maes 2 Instituut voor Theoretische Fysica, K.U.Leuven Karel Netočný 3 Institute of Physics AS CR, Prague Dedicated
More informationTHE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS
THE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS. STATEMENT Let (X, µ, A) be a probability space, and let T : X X be an ergodic measure-preserving transformation. Given a measurable map A : X GL(d, R),
More informationAsymptotic results for empirical measures of weighted sums of independent random variables
Asymptotic results for empirical measures of weighted sums of independent random variables B. Bercu and W. Bryc University Bordeaux 1, France Workshop on Limit Theorems, University Paris 1 Paris, January
More informationLecture I: Asymptotics for large GUE random matrices
Lecture I: Asymptotics for large GUE random matrices Steen Thorbjørnsen, University of Aarhus andom Matrices Definition. Let (Ω, F, P) be a probability space and let n be a positive integer. Then a random
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
More informationOn large deviations for combinatorial sums
arxiv:1901.0444v1 [math.pr] 14 Jan 019 On large deviations for combinatorial sums Andrei N. Frolov Dept. of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia E-mail address:
More informationEECS 750. Hypothesis Testing with Communication Constraints
EECS 750 Hypothesis Testing with Communication Constraints Name: Dinesh Krithivasan Abstract In this report, we study a modification of the classical statistical problem of bivariate hypothesis testing.
More informationMatematica e Fisica al crocevia
Matematica e Fisica al crocevia Roberto Longo Colloquium di Macroarea, parte I Roma, 9 Maggio 2017 The cycle Phys-Math-Math-Phys Nessuna humana investigazione si può dimandara vera scienzia s essa non
More informationConformal Blocks, Entanglement Entropy & Heavy States
Conformal Blocks, Entanglement Entropy & Heavy States Pinaki Banerjee The Institute of Mathematical Sciences, Chennai April 25, 2016 arxiv : 1601.06794 Higher-point conformal blocks & entanglement entropy
More informationCOUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE. Nigel Higson. Unpublished Note, 1999
COUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE Nigel Higson Unpublished Note, 1999 1. Introduction Let X be a discrete, bounded geometry metric space. 1 Associated to X is a C -algebra C (X) which
More informationReal Analysis, 2nd Edition, G.B.Folland Signed Measures and Differentiation
Real Analysis, 2nd dition, G.B.Folland Chapter 3 Signed Measures and Differentiation Yung-Hsiang Huang 3. Signed Measures. Proof. The first part is proved by using addivitiy and consider F j = j j, 0 =.
More informationMean-Field Limits for Large Particle Systems Lecture 2: From Schrödinger to Hartree
for Large Particle Systems Lecture 2: From Schrödinger to Hartree CMLS, École polytechnique & CNRS, Université Paris-Saclay FRUMAM, Marseilles, March 13-15th 2017 A CRASH COURSE ON QUANTUM N-PARTICLE DYNAMICS
More informationON THE ZERO-ONE LAW AND THE LAW OF LARGE NUMBERS FOR RANDOM WALK IN MIXING RAN- DOM ENVIRONMENT
Elect. Comm. in Probab. 10 (2005), 36 44 ELECTRONIC COMMUNICATIONS in PROBABILITY ON THE ZERO-ONE LAW AND THE LAW OF LARGE NUMBERS FOR RANDOM WALK IN MIXING RAN- DOM ENVIRONMENT FIRAS RASSOUL AGHA Department
More informationDISSIPATIVE TRANSPORT APERIODIC SOLIDS KUBO S FORMULA. and. Jean BELLISSARD 1 2. Collaborations:
Vienna February 1st 2005 1 DISSIPATIVE TRANSPORT and KUBO S FORMULA in APERIODIC SOLIDS Jean BELLISSARD 1 2 Georgia Institute of Technology, Atlanta, & Institut Universitaire de France Collaborations:
More informationA D VA N C E D P R O B A B I L - I T Y
A N D R E W T U L L O C H A D VA N C E D P R O B A B I L - I T Y T R I N I T Y C O L L E G E T H E U N I V E R S I T Y O F C A M B R I D G E Contents 1 Conditional Expectation 5 1.1 Discrete Case 6 1.2
More informationDEFINABLE OPERATORS ON HILBERT SPACES
DEFINABLE OPERATORS ON HILBERT SPACES ISAAC GOLDBRING Abstract. Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize
More informationFinite-dimensional representations of free product C*-algebras
Finite-dimensional representations of free product C*-algebras Ruy Exel Terry A. Loring October 28, 2008 preliminary version Department of Mathematics and Statistics University of New Mexico Albuquerque,
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationOn the uniform Opial property
We consider the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that
More informationRisi Kondor, The University of Chicago
Risi Kondor, The University of Chicago Data: {(x 1, y 1 ),, (x m, y m )} algorithm Hypothesis: f : x y 2 2/53 {(x 1, y 1 ),, (x m, y m )} {(ϕ(x 1 ), y 1 ),, (ϕ(x m ), y m )} algorithm Hypothesis: f : ϕ(x)
More informationDiagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations 1
Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations 1 P. T. Nam 1 M. Napiórkowski 1 J. P. Solovej 2 1 Institute of Science and Technology Austria 2 Department of Mathematics,
More information4. Conditional risk measures and their robust representation
4. Conditional risk measures and their robust representation We consider a discrete-time information structure given by a filtration (F t ) t=0,...,t on our probability space (Ω, F, P ). The time horizon
More informationIntroduction to Ergodic Theory
Introduction to Ergodic Theory Marius Lemm May 20, 2010 Contents 1 Ergodic stochastic processes 2 1.1 Canonical probability space.......................... 2 1.2 Shift operators.................................
More informationarxiv: v1 [math-ph] 15 May 2017
REDUCTION OF QUANTUM SYSTEMS AND THE LOCAL GAUSS LAW arxiv:1705.05259v1 [math-ph] 15 May 2017 RUBEN STIENSTRA AND WALTER D. VAN SUIJLEOM Abstract. We give an operator-algebraic interpretation of the notion
More information