Free probability and quantum information
|
|
- Georgina Hawkins
- 6 years ago
- Views:
Transcription
1 Free probability and quantum information Benoît Collins WPI-AIMR, Tohoku University & University of Ottawa Tokyo, Nov 8, 2013
2 Overview
3 Overview Plan: 1. Quantum Information theory: the additivity problem 2. A random matrix model 3. Free probability and norm convergence for random matrices 4. Violation of additivity
4 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C).
5 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C). A quantum channel is a linear completely positive trace preserving map Φ from M n (C) to M k (C).
6 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C). A quantum channel is a linear completely positive trace preserving map Φ from M n (C) to M k (C). The trace preservation condition: Φ maps D n to D k.
7 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C). A quantum channel is a linear completely positive trace preserving map Φ from M n (C) to M k (C). The trace preservation condition: Φ maps D n to D k. Complete positivity: d 1, Φ I d : M nd (C) M kd (C) is a positive map.
8 Quantum systems Stinespring theorem: Theorem A linear map Φ : M n (C) M k (C) is a quantum channel if and only if there exists a finite dimensional Hilbert space K = C d, a matrix Y D d and an unitary operation U U(nd) such that Φ(X ) = (id Tr) [U(X Y )U ], X M n (C). (1)
9 Rényi & Shannon Entropy
10 Rényi & Shannon Entropy For a positive real number p > 0, the Rényi entropy of order p of a probability vector x = (x 1,..., x d ) R d (x i 0, i x i = 1) is H p (x) = 1 d 1 p log x p i. i=1
11 Rényi & Shannon Entropy For a positive real number p > 0, the Rényi entropy of order p of a probability vector x = (x 1,..., x d ) R d (x i 0, i x i = 1) is H p (x) = 1 d 1 p log x p i. Since lim p 1 H p (x) exists, we define the Shannon entropy of x to be this limit: d H(x) = H 1 (x) = x i log x i. i=1 i=1
12 von Neumann Entropy, Minimum output entropy Rényi & von Neumann entropy: Extension to density matrices by functional calculus: H p (ρ) = 1 1 p log Tr ρp ; H(ρ) = H 1 (ρ) = Tr ρ log ρ.
13 von Neumann Entropy, Minimum output entropy Rényi & von Neumann entropy: Extension to density matrices by functional calculus: H p (ρ) = 1 1 p log Tr ρp ; H(ρ) = H 1 (ρ) = Tr ρ log ρ. p-minimum output entropy of a quantum channel Φ : M n (C) M k (C): H p min (Φ) = min A D n H p (Φ(A)).
14 Additivity problem
15 Additivity problem Most important problem in QIT until the last decade: For all quantum channels Φ and Φ, does one have H min (Φ Φ) = H min (Φ) + H min ( Φ)?
16 Additivity problem Most important problem in QIT until the last decade: For all quantum channels Φ and Φ, does one have H min (Φ Φ) = H min (Φ) + H min ( Φ)? Loosely speaking: can we transmit more classical data than expected with quantum channels (and entanglement)?
17 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure):
18 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure): For all quantum channels Φ and Φ, are the minimum output p-rényi entropies are additive p > 1, H p min (Φ Φ) = H p min (Φ) + Hp min ( Φ)?
19 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure): For all quantum channels Φ and Φ, are the minimum output p-rényi entropies are additive p > 1, H p min (Φ Φ) = H p min (Φ) + Hp min ( Φ)? Both problems have been proved false by Hayden, Winter, Hastings, and others.
20 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure): For all quantum channels Φ and Φ, are the minimum output p-rényi entropies are additive p > 1, H p min (Φ Φ) = H p min (Φ) + Hp min ( Φ)? Both problems have been proved false by Hayden, Winter, Hastings, and others. All counterexamples are random. We still don t have an explicit counterexample.
21 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random.
22 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate).
23 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate). Get a good lower bound on H p min (Φ) = Hp min ( Φ)
24 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate). Get a good lower bound on H p min (Φ) = Hp min ( Φ) Get a good upper bound on H p min (Φ Φ).
25 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate). Get a good lower bound on H p min (Φ) = Hp min ( Φ) Get a good upper bound on H p min (Φ Φ). In practice, the bound is given by an estimation of H p (Φ Φ(ρ)) for a well chosen ρ (Bell state, i.e. maximally entangled state).
26 Digression: Random matrices
27 Digression: Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties, usually as dimension.
28 Digression: Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties, usually as dimension. Discrete mathematics?
29 Digression: Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties, usually as dimension. Discrete mathematics? Yes, from a non-commutative point of view. We will adopt the point of view that a random matrix is a discrete non-commutative random variable, and that a quantum channel is a discrete non-commutative Markov operator.
30 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure).
31 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure). Classical RMT question: When does the (random) probability measure µ n have an interesting behaviour at n?
32 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure). Classical RMT question: When does the (random) probability measure µ n have an interesting behaviour at n? More recent RMT question: how about the (random) set supp(µ n )? (largest eigenvalue)
33 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure). Classical RMT question: When does the (random) probability measure µ n have an interesting behaviour at n? More recent RMT question: how about the (random) set supp(µ n )? (largest eigenvalue) More recent RMT question: how about the eigenvectors?
34 Back to our problem: a Random matrix Model We fix an integer k and a real number t (0, 1).
35 Back to our problem: a Random matrix Model We fix an integer k and a real number t (0, 1). Let n, N be such that n tnk. Let U n be a random partial isometry in M kn,n (C) (think of it as a truncated random Haar unitary matrix in M Nk (C))
36 Back to our problem: a Random matrix Model We fix an integer k and a real number t (0, 1). Let n, N be such that n tnk. Let U n be a random partial isometry in M kn,n (C) (think of it as a truncated random Haar unitary matrix in M Nk (C)) Our counterexample will be Φ n : M n (C) M k (C) given by Φ n (X ) = (id k Tr)(U n XU n).
37 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate.
38 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i
39 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C).
40 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C). As often in probability theory, it is a sequence of random objects. Unlike in usual random matrix theory, the size of the matrix doesn t change.
41 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C). As often in probability theory, it is a sequence of random objects. Unlike in usual random matrix theory, the size of the matrix doesn t change. The study of this random matrix will give us a good upper bound on H p min (Φ Φ)
42 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C). As often in probability theory, it is a sequence of random objects. Unlike in usual random matrix theory, the size of the matrix doesn t change. The study of this random matrix will give us a good upper bound on H p min (Φ Φ) (=H p min (Φ n Φ n ), for n large enough).
43 Weingarten calculus: integrating over compact groups The unitary Weingarten function Wg(n, σ): inverse of the function σ n #σ under the convolution for the symmetric group (#σ denotes the number of cycles of the permutation σ).
44 Weingarten calculus: integrating over compact groups The unitary Weingarten function Wg(n, σ): inverse of the function σ n #σ under the convolution for the symmetric group (#σ denotes the number of cycles of the permutation σ). Theorem Let n be a positive integer and i = (i 1,..., i p ), i = (i 1,..., i p), j = (j 1,..., j p ), j = (j 1,..., j p) be p-tuples of positive integers from {1, 2,..., n}. Then U(n) U i1 j 1 U ipj p U i 1 j 1 U i pj p du = σ,τ S p δ i1 i σ(1)... δ i pi σ(p) δ j 1 j τ(1)... δ j pj τ(p) Wg(n, τσ 1 ). (2)
45 Weingarten calculus: n limit Theorem For a permutation σ S p, let Cycles(σ) denote the set of cycles of σ. Then Wg(n, σ) = ( 1) n #σ Wg(n, c)(1 + O(n 2 )) (3) and c Cycles(σ) Wg(n, (1,..., d)) = ( 1) d 1 c d 1 where c i = (2i)! (i+1)! i! d+1 j d 1 is the i-th Catalan number. (n j) 1 (4)
46 Application of Weingarten calculus to QIT Theorem (Product channel) Almost surely, as n, the random matrix Φ n Φ n (X n ) = Z n M k 2(C) has non-zero eigenvalues converging towards γ (t) = t + 1 t, 1 t k 2 k 2,..., 1 t k 2 } {{ } k 2 1 times.
47 Application of Weingarten calculus to QIT Theorem (Product channel) Almost surely, as n, the random matrix Φ n Φ n (X n ) = Z n M k 2(C) has non-zero eigenvalues converging towards γ (t) = t + 1 t, 1 t k 2 k 2,..., 1 t k 2 } {{ } k 2 1 times. Consequence: get a good upper bound for H min (Φ Φ) for n large enough
48 Digression 2: Free probability
49 Digression 2: Free probability A non-commutative probability space : unital algebra A with tracial state ϕ. Elements: (non-commutative) random variables.
50 Digression 2: Free probability A non-commutative probability space : unital algebra A with tracial state ϕ. Elements: (non-commutative) random variables. E.g. random matrices (M n (L (Ω, P)), E[n 1 Tr( )]) (discrete non-commutative random variables)
51 Digression 2: Free probability A non-commutative probability space : unital algebra A with tracial state ϕ. Elements: (non-commutative) random variables. E.g. random matrices (M n (L (Ω, P)), E[n 1 Tr( )]) (discrete non-commutative random variables) Let A 1,..., A k be subalgebras of A. They are freely independent if for all a i A ji (i = 1,..., k) such that ϕ(a i ) = 0, one has ϕ(a 1 a p ) = 0 as soon as j 1 j 2, j 2 j 3,..., j p 1 j p.
52 Free probability & non-commutative probability
53 Free probability & non-commutative probability The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its noncommutative moments.
54 Free probability & non-commutative probability The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its noncommutative moments. Convergence in distribution = pointwise convergence of moments.
55 Free probability & non-commutative probability The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its noncommutative moments. Convergence in distribution = pointwise convergence of moments. Sequences of random variables (a (n) 1 ) n,..., (a (n) k ) n are called asymptotically free as n iff the k-tuple (a (n) 1,..., a(n) k converges in distribution towards a family of free random variables. ) n
56 Free probability and random matrices Voiculescu provides the following crucial link between random matrix theory and free probability theory:
57 Free probability and random matrices Voiculescu provides the following crucial link between random matrix theory and free probability theory: Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a joint limit distribution for large n with respect to the state n 1 Tr.
58 Free probability and random matrices Voiculescu provides the following crucial link between random matrix theory and free probability theory: Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a joint limit distribution for large n with respect to the state n 1 Tr. Then, the family ((U1 n, Un 1 ),..., (Un k, Un k ),..., (W n i )) admits a limit distribution, and is asymptotically free with respect to n 1 Tr.
59 Free probability and random matrices: strong version Strong convergence in distribution = in addition to convergence in distribution, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense.
60 Free probability and random matrices: strong version Strong convergence in distribution = in addition to convergence in distribution, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense. Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a STRONG joint limit distribution for large n with respect to the state n 1 Tr.
61 Free probability and random matrices: strong version Strong convergence in distribution = in addition to convergence in distribution, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense. Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a STRONG joint limit distribution for large n with respect to the state n 1 Tr. Then the family ((U1 n, Un 1 ),..., (Un k, Un k ),..., (W n i )) admits a STRONG limit distribution, and is STRONGLY asymptotically free with respect to n 1 Tr.
62 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2.
63 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2. The non-trivial eigenvalues of P n P np n arcsine on [0, 1], and
64 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2. The non-trivial eigenvalues of P n P np n arcsine on [0, 1], and The eigenvalues P n + P n arcsine on [0, 2]
65 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2. The non-trivial eigenvalues of P n P np n arcsine on [0, 1], and The eigenvalues P n + P n arcsine on [0, 2] Same distribution modulo a scaling and trivial eigenvectors... no direct matrix / probability explanation; the only conceptual explanation is through free probability.
66 The t-norm Let A M k (C), P be a random projection of rank n tnk in M kn (C). Then the previous theorem implies that the operator norm of the random matrix P(A 1 N )P converges with probability one to a quantity A (t).
67 The t-norm Let A M k (C), P be a random projection of rank n tnk in M kn (C). Then the previous theorem implies that the operator norm of the random matrix P(A 1 N )P converges with probability one to a quantity A (t). It turns out to be a Banach norm on M k (C) which we call the t-norm (in free probability jargon: free compression norm).
68 Output space for a single random channel Let K n = Φ n (D n ) be the output set of the quantum channel. This is a random convex body.
69 Output space for a single random channel Let K n = Φ n (D n ) be the output set of the quantum channel. This is a random convex body. Theorem With probability one, as n, K n K, where K = {B D k, A D k, Tr(AB) A (t) }
70 Bound for the single channel Theorem For all p > 1, the minimizer of H p on K is attained at a point that does not depend on p. This point x is characterized up to conjugation by the following properties
71 Bound for the single channel Theorem For all p > 1, the minimizer of H p on K is attained at a point that does not depend on p. This point x is characterized up to conjugation by the following properties x K The eigenvalues of x are as follows a > b =... = b.
72 Violation of additivity With the results we stated so far, we obtain by inspection the following results:
73 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183.
74 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183. For any p > 1 the violation is asymptotically as bad as it can get with the Bell state.
75 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183. For any p > 1 the violation is asymptotically as bad as it can get with the Bell state. For p = 1 the violation can be as bad as log 2 (one bit...)
76 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183. For any p > 1 the violation is asymptotically as bad as it can get with the Bell state. For p = 1 the violation can be as bad as log 2 (one bit...) Known results before: violation for k 10 4, of order 10 3
77 Selected References (1) The strong asymptotic freeness of Haar and deterministic matrices math/arxiv: With C. Male. To appear in Annales Scientifiques de l ENS (2) Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product math/arxiv: With S. Belinschi and I. Nechita. Inventiones Mathematicae December 2012, Volume 190, Issue 3, pp (3) Almost one bit violation for the additivity of the minimum output entropy, arxiv: with S. Belinschi and I. Nechita.
Operator 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 informationOutput entropy of tensor products of random quantum channels 1
Output entropy of tensor products of random quantum channels 1 Motohisa Fukuda 1 [Collins, Fukuda and Nechita], [Collins, Fukuda and Nechita] 1 / 13 1 Introduction Aim and background Existence of one large
More informationGAUSSIANIZATION AND EIGENVALUE STATISTICS FOR RANDOM QUANTUM CHANNELS (III) BY BENOÎT COLLINS 1,2 AND ION NECHITA 2 University of Ottawa
The Annals of Applied Probability 2011, Vol. 21, No. 3, 1136 1179 DOI: 10.1214/10-AAP722 Institute of Mathematical Statistics, 2011 GAUSSIANIZATION AND EIGENVALUE STATISTICS FOR RANDOM QUANTUM CHANNELS
More informationum random g. channel & computation of / v QIT output min entropy free prob quantum groups
@ i e output free prob. µ @ um random g. channel / v QIT & computation of min. entropy ) quantum groups QIT LINK FP - : Belinski Collins Nechita 201282016 Invent math (2012) 190:647 697 DOI 10.1007/s00222-012-0386-3
More informationFactorizable completely positive maps and quantum information theory. Magdalena Musat. Sym Lecture Copenhagen, May 9, 2012
Factorizable completely positive maps and quantum information theory Magdalena Musat Sym Lecture Copenhagen, May 9, 2012 Joint work with Uffe Haagerup based on: U. Haagerup, M. Musat: Factorization and
More informationStructure of Unital Maps and the Asymptotic Quantum Birkhoff Conjecture. Peter Shor MIT Cambridge, MA Joint work with Anand Oza and Dimiter Ostrev
Structure of Unital Maps and the Asymptotic Quantum Birkhoff Conjecture Peter Shor MIT Cambridge, MA Joint work with Anand Oza and Dimiter Ostrev The Superposition Principle (Physicists): If a quantum
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 informationLecture III: Applications of Voiculescu s random matrix model to operator algebras
Lecture III: Applications of Voiculescu s random matrix model to operator algebras Steen Thorbjørnsen, University of Aarhus Voiculescu s Random Matrix Model Theorem [Voiculescu]. For each n in N, let X
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 informationRemarks on the Additivity Conjectures for Quantum Channels
Contemporary Mathematics Remarks on the Additivity Conjectures for Quantum Channels Christopher King Abstract. In this article we present the statements of the additivity conjectures for quantum channels,
More informationarxiv: v3 [quant-ph] 9 Nov 2012
The Annals of Applied Probability 2011, Vol. 21, No. 3, 1136 1179 DOI: 10.1214/10-AAP722 c Institute of Mathematical Statistics, 2011 arxiv:0910.1768v3 [quant-ph] 9 Nov 2012 GAUSSIANIZATION AND EIGENVALUE
More informationEnsembles and incomplete information
p. 1/32 Ensembles and incomplete information So far in this course, we have described quantum systems by states that are normalized vectors in a complex Hilbert space. This works so long as (a) the system
More information9th Sendai Workshop Infinite Dimensional Analysis and Quantum Probability
9th Sendai Workshop Infinite Dimensional Analysis and Quantum Probability September 11-12, 2009 Graduate School of Information Sciences Tohoku University PROGRAM September 11 (Fri) GSIS, Large Lecture
More informationFREE PROBABILITY THEORY AND RANDOM MATRICES
FREE PROBABILITY THEORY AND RANDOM MATRICES ROLAND SPEICHER Abstract. Free probability theory originated in the context of operator algebras, however, one of the main features of that theory is its connection
More informationFree Probability Theory and Non-crossing Partitions. Roland Speicher Queen s University Kingston, Canada
Free Probability Theory and Non-crossing Partitions Roland Speicher Queen s University Kingston, Canada Freeness Definition [Voiculescu 1985]: Let (A, ϕ) be a non-commutative probability space, i.e. A
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 informationValerio Cappellini. References
CETER FOR THEORETICAL PHYSICS OF THE POLISH ACADEMY OF SCIECES WARSAW, POLAD RADOM DESITY MATRICES AD THEIR DETERMIATS 4 30 SEPTEMBER 5 TH SFB TR 1 MEETIG OF 006 I PRZEGORZAłY KRAKÓW Valerio Cappellini
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 informationSome Bipartite States Do Not Arise from Channels
Some Bipartite States Do Not Arise from Channels arxiv:quant-ph/0303141v3 16 Apr 003 Mary Beth Ruskai Department of Mathematics, Tufts University Medford, Massachusetts 0155 USA marybeth.ruskai@tufts.edu
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 informationHadamard matrices and Compact Quantum Groups
Hadamard matrices and Compact Quantum Groups Uwe Franz 18 février 2014 3ème journée FEMTO-LMB based in part on joint work with: Teodor Banica, Franz Lehner, Adam Skalski Uwe Franz (LMB) Hadamard & CQG
More informationK theory of C algebras
K theory of C algebras S.Sundar Institute of Mathematical Sciences,Chennai December 1, 2008 S.Sundar Institute of Mathematical Sciences,Chennai ()K theory of C algebras December 1, 2008 1 / 30 outline
More informationMaximal vectors in Hilbert space and quantum entanglement
Maximal vectors in Hilbert space and quantum entanglement William Arveson arveson@math.berkeley.edu UC Berkeley Summer 2008 arxiv:0712.4163 arxiv:0801.2531 arxiv:0804.1140 Overview Quantum Information
More informationAn Introduction to Quantum Computation and Quantum Information
An to and Graduate Group in Applied Math University of California, Davis March 13, 009 A bit of history Benioff 198 : First paper published mentioning quantum computing Feynman 198 : Use a quantum computer
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 informationFamily Feud Review. Linear Algebra. October 22, 2013
Review Linear Algebra October 22, 2013 Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns. Answer 1 Answer A is a 4 19 matrix, while
More informationDual Temperley Lieb basis, Quantum Weingarten and a conjecture of Jones
Dual Temperley Lieb basis, Quantum Weingarten and a conjecture of Jones Benoît Collins Kyoto University Sendai, August 2016 Overview Joint work with Mike Brannan (TAMU) (trailer... cf next Monday s arxiv)
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 informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationLimit Laws for Random Matrices from Traffic Probability
Limit Laws for Random Matrices from Traffic Probability arxiv:1601.02188 Slides available at math.berkeley.edu/ bensonau Benson Au UC Berkeley May 9th, 2016 Benson Au (UC Berkeley) Random Matrices from
More informationLUCK S THEOREM ALEX WRIGHT
LUCK S THEOREM ALEX WRIGHT Warning: These are the authors personal notes for a talk in a learning seminar (October 2015). There may be incorrect or misleading statements. Corrections welcome. 1. Convergence
More informationFree Probability Theory and Random Matrices. Roland Speicher Queen s University Kingston, Canada
Free Probability Theory and Random Matrices Roland Speicher Queen s University Kingston, Canada We are interested in the limiting eigenvalue distribution of N N random matrices for N. Usually, large N
More informationRECTANGULAR RANDOM MATRICES, ENTROPY, AND FISHER S INFORMATION
J. OPERATOR THEORY 00:0(XXXX), 00 00 Copyright by THETA, XXXX RECTANGULAR RANDOM MATRICES, ENTROPY, AND FISHER S INFORMATION FLORENT BENAYCH-GEORGES Communicated by Serban Stratila ABSTRACT. We prove that
More informationFRAMES IN QUANTUM AND CLASSICAL INFORMATION THEORY
FRAMES IN QUANTUM AND CLASSICAL INFORMATION THEORY Emina Soljanin Mathematical Sciences Research Center, Bell Labs April 16, 23 A FRAME 1 A sequence {x i } of vectors in a Hilbert space with the property
More informationConvex Sets Associated to C -Algebras
Convex Sets Associated to C -Algebras Scott Atkinson University of Virginia Joint Mathematics Meetings 2016 Overview Goal: Define and investigate invariants for a unital separable tracial C*-algebra A.
More informationChapter 5. Density matrix formalism
Chapter 5 Density matrix formalism In chap we formulated quantum mechanics for isolated systems. In practice systems interect with their environnement and we need a description that takes this feature
More informationSecond Order Freeness and Random Orthogonal Matrices
Second Order Freeness and Random Orthogonal Matrices Jamie Mingo (Queen s University) (joint work with Mihai Popa and Emily Redelmeier) AMS San Diego Meeting, January 11, 2013 1 / 15 Random Matrices X
More informationConcentration of Measure Effects in Quantum Information. Patrick Hayden (McGill University)
Concentration of Measure Effects in Quantum Information Patrick Hayden (McGill University) Overview Superdense coding Random states and random subspaces Superdense coding of quantum states Quantum mechanical
More informationLecture: Quantum Information
Lecture: Quantum Information Transcribed by: Crystal Noel and Da An (Chi Chi) November 10, 016 1 Final Proect Information Find an issue related to class you are interested in and either: read some papers
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 informationQuantum Information Types
qitd181 Quantum Information Types Robert B. Griffiths Version of 6 February 2012 References: R. B. Griffiths, Types of Quantum Information, Phys. Rev. A 76 (2007) 062320; arxiv:0707.3752 Contents 1 Introduction
More informationUnitary t-designs. Artem Kaznatcheev. February 13, McGill University
Unitary t-designs Artem Kaznatcheev McGill University February 13, 2010 Artem Kaznatcheev (McGill University) Unitary t-designs February 13, 2010 0 / 16 Preliminaries Basics of quantum mechanics A particle
More informationQuantum Entanglement and the Bell Matrix
Quantum Entanglement and the Bell Matrix Marco Pedicini (Roma Tre University) in collaboration with Anna Chiara Lai and Silvia Rognone (La Sapienza University of Rome) SIMAI2018 - MS27: Discrete Mathematics,
More informationFREE PROBABILITY THEORY
FREE PROBABILITY THEORY ROLAND SPEICHER Lecture 3 Freeness and Random Matrices Now we come to one of the most important and inspiring realizations of freeness. Up to now we have realized free random variables
More informationQuantum Symmetric States on Free Product C -Algebras
Quantum Symmetric States on Free Product C -Algebras School of Mathematical Sciences University College Cork Joint work with Ken Dykema & John Williams arxiv:1305.7293 WIMCS-LMS Conference Classifying
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationPhysics 239/139 Spring 2018 Assignment 6
University of California at San Diego Department of Physics Prof. John McGreevy Physics 239/139 Spring 2018 Assignment 6 Due 12:30pm Monday, May 14, 2018 1. Brainwarmers on Kraus operators. (a) Check that
More informationInduced Ginibre Ensemble and random operations
Induced Ginibre Ensemble and random operations Karol Życzkowski in collaboration with Wojciech Bruzda, Marek Smaczyński, Wojciech Roga Jonith Fischmann, Boris Khoruzhenko (London), Ion Nechita (Touluse),
More informationThe Free Central Limit Theorem: A Combinatorial Approach
The Free Central Limit Theorem: A Combinatorial Approach by Dennis Stauffer A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honour s Seminar)
More informationApplications and fundamental results on random Vandermon
Applications and fundamental results on random Vandermonde matrices May 2008 Some important concepts from classical probability Random variables are functions (i.e. they commute w.r.t. multiplication)
More informationUCLA UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Approximations in Operator Theory and Free Probability Permalink https://escholarship.org/uc/item/9wv6f1z6 Author Skoufranis, Paul Daniel Publication
More informationPreliminaries on von Neumann algebras and operator spaces. Magdalena Musat University of Copenhagen. Copenhagen, January 25, 2010
Preliminaries on von Neumann algebras and operator spaces Magdalena Musat University of Copenhagen Copenhagen, January 25, 2010 1 Von Neumann algebras were introduced by John von Neumann in 1929-1930 as
More informationClassical and Quantum Channel Simulations
Classical and Quantum Channel Simulations Mario Berta (based on joint work with Fernando Brandão, Matthias Christandl, Renato Renner, Joseph Renes, Stephanie Wehner, Mark Wilde) Outline Classical Shannon
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 informationParadigms of Probabilistic Modelling
Paradigms of Probabilistic Modelling Hermann G. Matthies Brunswick, Germany wire@tu-bs.de http://www.wire.tu-bs.de abstract RV-measure.tex,v 4.5 2017/07/06 01:56:46 hgm Exp Overview 2 1. Motivation challenges
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationMathematical Methods for Quantum Information Theory. Part I: Matrix Analysis. Koenraad Audenaert (RHUL, UK)
Mathematical Methods for Quantum Information Theory Part I: Matrix Analysis Koenraad Audenaert (RHUL, UK) September 14, 2008 Preface Books on Matrix Analysis: R. Bhatia, Matrix Analysis, Springer, 1997.
More informationQubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable,
Qubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable, A qubit: a sphere of values, which is spanned in projective sense by two quantum
More informationQuantum Statistics -First Steps
Quantum Statistics -First Steps Michael Nussbaum 1 November 30, 2007 Abstract We will try an elementary introduction to quantum probability and statistics, bypassing the physics in a rapid first glance.
More informationMultiplicativity of Maximal p Norms in Werner Holevo Channels for 1 < p 2
Multiplicativity of Maximal p Norms in Werner Holevo Channels for 1 < p 2 arxiv:quant-ph/0410063v1 8 Oct 2004 Nilanjana Datta Statistical Laboratory Centre for Mathematical Sciences University of Cambridge
More informationEntanglement Manipulation
Entanglement Manipulation Steven T. Flammia 1 1 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada (Dated: 22 March 2010) These are notes for my RIT tutorial lecture at the
More informationOperational extreme points and Cuntz s canonical endomorphism
arxiv:1611.03929v1 [math.oa] 12 Nov 2016 Operational extreme points and Cuntz s canonical endomorphism Marie Choda Osaka Kyoiku University, Osaka, Japan marie@cc.osaka-kyoiku.ac.jp Abstract Based on the
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationQuantum decoherence. Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, Quantum decoherence p. 1/2
Quantum decoherence p. 1/2 Quantum decoherence Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, 2007 Quantum decoherence p. 2/2 Outline Quantum decoherence: 1. Basics of quantum
More informationFunctional Analysis II held by Prof. Dr. Moritz Weber in summer 18
Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18 General information on organisation Tutorials and admission for the final exam To take part in the final exam of this course, 50 % of
More informationOrthogonal Pure States in Operator Theory
Orthogonal Pure States in Operator Theory arxiv:math/0211202v2 [math.oa] 5 Jun 2003 Jan Hamhalter Abstract: We summarize and deepen existing results on systems of orthogonal pure states in the context
More informationFactorization of unitary representations of adele groups Paul Garrett garrett/
(February 19, 2005) Factorization of unitary representations of adele groups Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The result sketched here is of fundamental importance in
More informationCUT-OFF FOR QUANTUM RANDOM WALKS. 1. Convergence of quantum random walks
CUT-OFF FOR QUANTUM RANDOM WALKS AMAURY FRESLON Abstract. We give an introduction to the cut-off phenomenon for random walks on classical and quantum compact groups. We give an example involving random
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationPerron-Frobenius theorem
Perron-Frobenius theorem V.S. Sunder Institute of Mathematical Sciences sunder@imsc.res.in Unity in Mathematics Lecture Chennai Math Institute December 18, 2009 Overview The aim of the talk is to describe
More informationPrivate quantum subsystems and error correction
Private quantum subsystems and error correction Sarah Plosker Department of Mathematics and Computer Science Brandon University September 26, 2014 Outline 1 Classical Versus Quantum Setting Classical Setting
More informationAlgebraic reformulation of Connes embedding problem and the free group algebra.
Algebraic reformulation of Connes embedding problem and the free group algebra. Kate Juschenko, Stanislav Popovych Abstract We give a modification of I. Klep and M. Schweighofer algebraic reformulation
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
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 informationRandom Matrix Theory in Quantum Information
Random Matrix Theory in Quantum Information Some generic aspects of entanglement Université Claude Bernard Lyon 1 & Universitat Autònoma de Barcelona Cécilia Lancien BMC & BAMC 2015 - Cambridge Cécilia
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 informationFree Probability Theory
Noname manuscript No. (will be inserted by the editor) Free Probability Theory and its avatars in representation theory, random matrices, and operator algebras; also featuring: non-commutative distributions
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 informationBASIC VON NEUMANN ALGEBRA THEORY
BASIC VON NEUMANN ALGEBRA THEORY FARBOD SHOKRIEH Contents 1. Introduction 1 2. von Neumann algebras and factors 1 3. von Neumann trace 2 4. von Neumann dimension 2 5. Tensor products 3 6. von Neumann algebras
More informationQUANTUM FIELD THEORY: THE WIGHTMAN AXIOMS AND THE HAAG-KASTLER AXIOMS
QUANTUM FIELD THEORY: THE WIGHTMAN AXIOMS AND THE HAAG-KASTLER AXIOMS LOUIS-HADRIEN ROBERT The aim of the Quantum field theory is to offer a compromise between quantum mechanics and relativity. The fact
More informationAlgebraic Properties of Riccati equations. Ruth Curtain University of Groningen, The Netherlands
Algebraic Properties of Riccati equations Ruth Curtain University of Groningen, The Netherlands Special Recognition Peter Falb Jan Willems Tony Pritchard Hans Zwart Riccati equation Ṗ(t) + A(t) P(t) +
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More informationI will view the Kadison Singer question in the following form. Let M be the von Neumann algebra direct sum of complex matrix algebras of all sizes,
THE KADISON-SINGER QUESTION FOR TYPE II 1 FACTORS AND RELATED QUESTIONS by Charles Akemann, with help from Sorin Popa, David Sherman, Joel Anderson, Rob Archbold, Nik weaver, Betul Tanbay (and no doubt
More informationFree Probability and Random Matrices: from isomorphisms to universality
Free Probability and Random Matrices: from isomorphisms to universality Alice Guionnet MIT TexAMP, November 21, 2014 Joint works with F. Bekerman, Y. Dabrowski, A.Figalli, E. Maurel-Segala, J. Novak, D.
More informationCompression, Matrix Range and Completely Positive Map
Compression, Matrix Range and Completely Positive Map Iowa State University Iowa-Nebraska Functional Analysis Seminar November 5, 2016 Definitions and notations H, K : Hilbert space. If dim H = n
More informationQuantum Symmetries in Free Probability Theory. Roland Speicher Queen s University Kingston, Canada
Quantum Symmetries in Free Probability Theory Roland Speicher Queen s University Kingston, Canada Quantum Groups are generalizations of groups G (actually, of C(G)) are supposed to describe non-classical
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 informationEntropy in Classical and Quantum Information Theory
Entropy in Classical and Quantum Information Theory William Fedus Physics Department, University of California, San Diego. Entropy is a central concept in both classical and quantum information theory,
More informationExample Linear Algebra Competency Test
Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,
More information2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2
1 A Good Spectral Theorem c1996, Paul Garrett, garrett@math.umn.edu version February 12, 1996 1 Measurable Hilbert bundles Measurable Banach bundles Direct integrals of Hilbert spaces Trivializing Hilbert
More informationThe norm of polynomials in large random matrices
The norm of polynomials in large random matrices Camille Mâle, École Normale Supérieure de Lyon, Ph.D. Student under the direction of Alice Guionnet. with a significant contribution by Dimitri Shlyakhtenko.
More informationUnitary-antiunitary symmetries
Unitary-antiunitary symmetries György Pál Gehér University of Reading, UK Preservers: Modern Aspects and New Directions 18 June 2018, Belfast Some classical results H a complex (or real) Hilbert space,
More informationBoolean Inner-Product Spaces and Boolean Matrices
Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver
More informationFree Probability Theory and Random Matrices
Free Probability Theory and Random Matrices R. Speicher Department of Mathematics and Statistics Queen s University, Kingston ON K7L 3N6, Canada speicher@mast.queensu.ca Summary. Free probability theory
More informationLinear and Multilinear Algebra. Linear maps preserving rank of tensor products of matrices
Linear maps preserving rank of tensor products of matrices Journal: Manuscript ID: GLMA-0-0 Manuscript Type: Original Article Date Submitted by the Author: -Aug-0 Complete List of Authors: Zheng, Baodong;
More informationASYMPTOTIC EIGENVALUE DISTRIBUTIONS OF BLOCK-TRANSPOSED WISHART MATRICES
ASYMPTOTIC EIGENVALUE DISTRIBUTIONS OF BLOCK-TRANSPOSED WISHART MATRICES TEODOR BANICA AND ION NECHITA Abstract. We study the partial transposition W Γ = (id t)w M dn (C) of a Wishart matrix W M dn (C)
More informationarxiv: v1 [math-ph] 19 Oct 2018
A geometrization of quantum mutual information Davide Pastorello Dept. of Mathematics, University of Trento Trento Institute for Fundamental Physics and Applications (TIFPA-INFN) via Sommarive 14, Povo
More informationAlmost periodic functionals
Almost periodic functionals Matthew Daws Leeds Warsaw, July 2013 Matthew Daws (Leeds) Almost periodic functionals Warsaw, July 2013 1 / 22 Dual Banach algebras; personal history A Banach algebra A Banach
More informationEntropy, mixing, and independence
Entropy, mixing, and independence David Kerr Texas A&M University Joint work with Hanfeng Li Let (X, µ) be a probability space. Two sets A, B X are independent if µ(a B) = µ(a)µ(b). Suppose that we have
More information