Tackling problems

in Quantum entanglement and channels via

factorisations of multivariable polynomials and super-operators

Ajit Iqbal Singh

INSA Honorary Scientist

The Indian National Science Academy, New Delhi

Problems and Recent Methods in Operator theoryIn memory of James E. Jamison

Dept of Mathematical Sciences, University of Memphis October 15 - October 16, 2015

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Entangled states constitute a powerful resource for quantum communication. In the simple setting of a finite tensor product of finite-dimensional Hilbert spaces, they are the vectors not expressible as a product of vectors. We shall present some new ones via multivariable polynomials, for instance, Bell polynomials and Vandermonde determinants. In the simplest setting, a quantum channel is a completely positive trace preserving unital map on a vector space of matrices to itself. We shall give an idea of their convex structure with emphasis on the asymptotic quantum Birkhoff property. It is related to factorization of such super-operators.

Abstract

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Latin squares and Hadamard matrices help to get such bases.

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Nobody likes delay in flights.But the one on Dec. 15,2013 from Delhi to Bengaluru

proved good for me.

That is when I first met Prof. James Jamison though I knew his work before. Prof. T.S.S.R.K. Rao had made it possible through phone from Bengaluru. James and I were both on our way to Indian Institute of Science in Bengaluru for the International Workshop on Operator Theory and Applications. We immediately got talking as if we were old friends. That was his super quality - so cool and yet so warm! We straightaway started on surjective isometries, Hermitian operators in Banach spaces and his work with Fernanda Botelho. His approach was so clear and concrete, every concept with illustration. I chipped in my spontaneous remarks and the discussion was so lively. And this was at the Airport Waiting Lounge, yet we were so comfortable!

I feel the same aura now. I suppose he is present here with his endearing smile and encouragement and I am sure I will not arrive at a contradiction!

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Quantum System

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Partial trace

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A state is a positive operator with trace one.

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A state with rank one is called a pure state.

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Physicists usually take inner product conjugate-linear in the first variable and linear in the second. Also at times they supress the symbol of tensor product. Further, they

follow Paul Dirac’s bra-ket notation. Let us join them!

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Pyramid for Progress and QualityV 5

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Examples of unextendable product bases from Bennett, DiVincenzo, Mor, Shor, Smolin and Terhal,Phys. Rev. Lett. 1999

ded

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Examples of UPB (contd.)

The orthogonal complement of a UPB contains no non-zero product vector, and, therefore, consists of some entangled vectors together with zero.

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Completely entangled spaces in general case

Nolan R. Wallach, AMS Contemp. Math. 305 (2002).K.R. Parthasarathy, Proc. Indian Acad. Sci. 2004.

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B.V.R. Bhat, Int. J. Quantum Inform. 2006

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Bicentenary of Karl Theodor Wilhelm Weierstrass, the "father of modern analysis” born on October 31, 1815,

At the General Assembly 2010 of The International Mathematical Union (IMU)in Bangalore, India, the Weierstrass Institute in Berlin, Germany (WIAS) was elected as the host institution of the permanent secretariat.

Weierstrass enhanced the significance of polynomials by proving that they are everywhere in the space of continuous functions on a closed bounded interval. The progress on its generalisations and applications is enormous.

And they help in determining if a vector is a product vector or an entangled one! Or a genuine entangled one!

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Role of multivariate polynomials

We can make the multivariable polynomial in k variables t-sub- i =product of the monomials t-sub-r raised to i-sub-r correspond to the vector e-sub-i. This enables us to identify script H-sup-(n) with the space of multivariable polynomials in k variables that are homogeneous of degree n with the restriction that the power of t-sub-j cannot exceed (d-sub-j )- 1. Further script H can be identified with such restricted multivariable polynomials of degree up to N. A product vector will then be a product of k polynomials, the j-th one being a polynomial in t-sub-j of degree not more than (d-sub-j )- 1. A vector in the space at n-th level of Bhat’s completely entangled space is such a homogeneous polynomial of degree n vanishing at (1,1,…,1). It occurred to me to utilise this observation, while listening to the excellent lectures by Hari Berkovici at The Operator Theory and Operator Algebras workshop and conference in December, 2014 at The Indian Statistical Institute, Bengaluru organised by Prof. T.S.S.R.K. Rao and his colleagues. Jai OTOA 2014! Jai Hari Bercovici!

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ProgressI have discussed it with a few colleagues over the time and we are planning a write-up on the topic. With Nikhil Srivastava and P. Branden, the discussion was on use of the theory of stable polynomials via e.mails. With Somshubhro Bandyopadhyay, we figured out how to develop the theory like that of Schmidt decomposition, quantum states, quantum channels during my visit to his institute, viz. , Bose Institute in Kolkata. With Rajat Mittal we looked at the relationship with semidefiniteprogramming and Computer Science techniques. With Aditi Sen De and Ujjwal Sen discussion was on the use of polynomial approach in the study of Resonance Valence bonds of P. Anderson and G. Baskaran at their institute, Harish-Chandra Institute (HRI). With Anoop Singh, a research student at HRI, we figured out uses of basic Algebraic Geometry, particularly for genuine entanglement. Discussion with Ajay Shukla took place after his talk at the International Conference on Special Functions and Applications at Amity University last month, the emphasis being on the role of Orthogonal polynomials in several variables. I thank all of them. We will give a brief account of some aspects.This multivariable viewpoint is taken by some to define the tensor product, for instance, see the recent

book by F. Hiai and D. Petz.

At times we will use the term poly for a multivariate polynomial.

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Binomial

• Let us begin with the simplest case of

k=2=d-sub-1=d-sub-2.

• Then any constant poly is a product poly. We omit this in further discussion. The poly 1+t-sub-1+t-sub-2+t-sub-1 t-sub-2 is a product poly but t-sub-1 +t-sub-2 is not. The first poly has zero set consisting of the union of t-sub-1=0 and t-sub-2=0, whereas the second one has t-sub-1+ t-sub-2=0, a hyperplane but not parallel to any co-ordinate one! For the entangled polynomial 1+t-sub-1 t-sub-2, the zero set contains no hyperplane at all.

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Binomial polynomials form a source of entangled polynomials

The catch is that we are sure from the point just noted that each non-constant polynomial from the lot gives rise to an entangled state and also expresses it as

A sum of product vectors coming from the family!

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Laguerre polynomials help too

Many orthogonal multivariate polynomials arise in this way..

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2013

It is common to use formal power series to define special functions.The one giving Bell polynomial s of E.T. Bell (Ann. Math. 1934) is here.

Explicit:

Summation extends over all tuples of non-negative integers satisfying

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Novelty in Cvijovic paper

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For applications in our context, we take n>k>0. First note that we can vary n and k keeping r=n-k+1 constant for studying an r-partite system. Then we note:(i) Bell polynomials with r=2 are all bipartite product vectors.(ii) Bell polynomials with r=3 are genuinely entangled.

MORE LATER!

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Stable polynomials

• Definition.

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Certain real stable polynomials are entangled

• Let p be a product poly with non-constant single variable factors q-sub-j(t-sub-i-sub-j), for 1 less than or equal to j less than or equal to r, say. Let Z-sub-j be the zero set of q-sub-j and Z their union. Then the zero set E of p is the union of hyperplanes t-sub-i-sub-j=a, a in Z-sub-j, 1 less than or equal to j less than or equal to r; we may say that E is infinite wall-roof.

• Then p is stable if and only if Im z is less than or equal to 0 for each z in Z. In particular, if p is real then p is stable if and only if Z is a subset of the set of real numbers; we may say that E is real infinite wall-roof. Thus, a non-constant real stable poly p is product poly if and only if its zero set is real infinite wall-roof. A real infinite wall-roof contains uncountably many points with all co-ordinates real.

• So certain real stable polys are entangled.• Basic Algebraic Geometry can help throw more light on the problem,

Fulton’s book is a good choice for that.

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Van der Monde determinant

Treating each as a variable, it is not separable

Consider the matrix

Its determinant

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Entanglement in Resonance Valence Bond

From a paper by Aditi Sen De et al

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Bipartite lattice

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RVB state

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Polynomial for RVB liquid

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Illustration

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Another way

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It is time to part with the polynomials

We quickly mention work related to the concepts introduced here by myself and my collaborators.1. (with Sengupta, Ritabrata and Arvind) Entanglement properties of positive

operators with ranges in completely entangled subspaces, Physical Review A,90(2014), 062323.

2. Quantum dynamical semigroups involving separable and entangled states, arXiv :1201.0250v3 [math:DS]. Revised for possible publication in Electronic J.

Linear Algebra, Special Issue in honour of R.B. Bapat.3.(with Sibasish Ghosh)Invariants for maximally entangled vectors and unitary bases.

arXiv: 1401.0099 [quant-ph].4. (with S. Chaturvedi, Sibasish Ghosh and K. R. Parthasarathy), Optimal quantum state

determination by constrained elementary measurements, arXiv: 1411.0152 [quant-ph].

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Polytopes and corners

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Doubly stochastic also called bistochastic

• , the matrix associate

• permutation (123).

matrix associated to the permutation (123).

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Birkhoff’s Theorem

A point of a convex set is called an extreme point if it is not an interior point of a line segment contained in the set. It may be remarked that an extreme point of a subset of a set may not be an extreme point of the set, just take a triangle and a median! Krein -Milman Theorem says that a bounded closed convex set in a Euclidean space (result holds for much more general set-ups) is the closed convex hull of the set of its extreme points and the convex hull of a closed set in a Euclidean space (no such analogue for infinite dimensional spaces) is closed.

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From classical to quantum

• So an alternative form of the Birkhoff ‘s Theorem is : The set of bistochastic matrices is the convex hull of the set of permutation matrices.

• Both the forms have their quantum counterparts asked and answered! Even some asymptotic versions are relevant and have been questioned and answered! I give a window, present the basics of the quantum set-up and indicate what more can be done.

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Birkhoff’s Theorem for Panstochastic matricesDean Alvis and Michael Kanyon, Amer. Math. Monthly, 2001

• A doubly stochastic matrix is PANSTOCHASTIC if the sum of the entries along any downward diagonal or upward diagonal, either broken or unbroken, is equal to 1.

• Theorem. A 5 x 5 real matrix is panstochastic if and only if it is a convex combination of panstochastic permutation matrices.

• Theorem. If n>1and n not equal to 5, then there is some n x n panstochastic matrix that is not a convex combination of panstochastic permutation matrices.

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BIRKHOFF’S POLYTOPE AND UNISTOCHASTIC MATRICES

N=3 AND N=4

• arXiv:math/0402325 v 3

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• all elements equal.

having all elements equal.

The Birkhoff polytope is the set of bistochasticmatrices (of a given order, say, N) with the van der Waerden matrix at the centre and distance D given as follows.

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Quantum set-up

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Wigner’s Theorem

It is a matter of encouragement for our young researchers that L. Molnar, soon after his Ph.D., collected all the properties (spreading over more than 200 papers) which force a map to be of the above type in a Springer Lecture Notes in Math. Vol. 1800 (2007) and the cover page has just U*AU except for the usual necessities. The list continues to grow! Peter Semrl shows that it is a 2 x 2 affair even for infinite-dimensional Hilbert space set-up for some of them!

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Quantum Birkhoff question

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Landau and Streater’s results

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Diagonal maps or the maps given by Schur i.e. Hadamard product

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Diagonal maps (continued)

• In fact there is a good amount of research

on these maps under the name of Schur maps .

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Operator algebra set-up

• Birkhoff Theorem type study has been carried out in the set-up of operator algebras. We mention

• K.R. Parthasarathy, Infinite DiemensionalAnalysis, Quantum Probability and Related Topics, 1998, and

• more recent work by U. Haagerup and M. Musat. They also solved the Asymptotic Quantum Birkhoff conjecture in the negative.

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Asymptotic Quantum Birkhoff Conjecturelooked at by U. Haagerup and M. Musat 2010

Haagerup had a knack for looking at difficult problems, solving themand presenting them with each and every detail. Every paper by him

is a masterpiece and a treasure. His early tragic death in an accident a few months ago is a great loss for Mathematics and us all too.

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Uffe Haagerup and Magdalena Musat 2010Use of Factorization in AQBC not true

Here, T-sub-B s the Schur multiplication operator by B.57

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• There is a wonderful talk by M. Musat about two years ago freely available.

• And there has been more progress on the relationship (or the lack of it!)with ConnesEmbedding problem, which we shall not go into here.

• THANK YOU.

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