Walking Through MATHEMATICS Relevant for Quantum Mechanics: a Personal View

Walking through MATHEMATICS relevant for quantum mechanics: A personal view

Karol Zyczkowski˙

Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw CMS Meeting, Niagara Falls, December 3, 2016

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 1 / 76 Quantum Theory: basic notions

Quantum States (in a ﬁnite dimensional Hilbert space)

a) pure states: normalized elements of the complex Hilbert space N , 2 H ψ N with ψ = ψ ψ = 1, |deﬁnedi ∈ H up to a|| global|| phase,h | i ψ eiα ψ , | i ∼ | i thus the set ΩN of all pure states forms N−1 1 2 a complex projective space CP , e.g. Ω2 = CP = S b) mixed states: convex combinations of projectors onto pure states, P P ρ = i pi ψi ψ with i pi = 1, pi 0, so that the states are: Hermitian,| ρ ih= ρ|∗, positive, ρ 0 and≥ normalized, Trρ = 1. ≥ Unitary Quantum Dynamics a) pure states: ψ0 = U ψ , | i | i b) mixed states: ρ0 = UρU∗, where U U(N) is called a quantum gate. In the case∈ N = 2 it is called a single qubit quantum gate.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 2 / 76 Otton Nikodym & Stefan Banach,

talking mathematics in Planty Garden, Cracow, summer 1916

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 3 / 76 Otton Nikodym & Stefan Banach,

talking mathematics in Planty Garden, Cracow, summer 1916

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 3 / 76 Kye message: Quantum Theory provides interesting problems related to all ﬁelds of Mathematics.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 4 / 76 Mathematics: Part 1

diﬀerential geometry geometry of complex & real projective spaces topology group theory invariant theory representation theory

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 5 / 76 Pure states in a two dimensional complex Hilbert space 2 H Qubit = quantum bit; N = 2, ψ ψ = 1, ψ eiα ψ h | i | i ∼ | i Fubini-Study (geodesic) distance: DFS ( ψ , ϕ ) := arccos ψ ϕ | i | i |h | i|

1 2 Bloch sphere of N = 2 pure states, Ω2 = CP = S

qubit word geometry understood verbatim −→ KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 6 / 76 Pure states in a three dimensional Hilbert space 3 H

2 A qutrit - ψ 3, space of N = 3 pure states, Ω3 = CP | i ∈ H

Foliation CP2 into leaves of a ﬁxed distance to sphere S2 including 0 , 1 . | i | i KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 7 / 76 Pure states in a four dimensional Hilbert space 4 H separable pure state = product state, ψ = φA φB = φA, φB | i | i ⊗ | i | i other states are entangled – e.g. Bell state ψ+ = √1 ( 0, 0 + 1, 1 ) | i 2 | i | i A ququart two qubits, N = 4 = 2 2, ψ 4 2 2, ≈ ×3 | i ∈ H ≈ H ⊗ H Space of N = 4 pure states, Ω4 = CP ,

Foliation CP3 into leaves of the same entanglement = ﬁxed distance to Segre embedding CP1 CP1 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec.× 3, 2016 8 / 76 k Pure states of a k–qubit system, ψ k ⊗ | i ∈ H2 ≈ H2

Multipartite Entanglement: Open problems - stratiﬁcation of CP2k −1 - orbits SU(2)×k (local unitary) - orbits SL(2, C)×k (stochastic LOCC) - invariants - equivalence problem - topology of orbits -...

several open issues for k 4 ≥

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 9 / 76 Vistula river and Wawel castle in Cracow

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 10 / 76 Two great circles at the sphere do cross ! Equator, T 1 = S1, is non-displacable in S2 = CP1. ⇔

What known geometric theorem this ﬁgure illustrates?

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 11 / 76 What known geometric theorem this ﬁgure illustrates?

Two great circles at the sphere do cross ! Equator, T 1 = S1, is non-displacable in S2 = CP1. ⇔

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 11 / 76 Yes, a great K-torus T K is non–displacable in CPK , Cho (2004).

Crossing points marked X represent mutually unbiased states

(which exist for any two unitary bases!)

Projections of two 2–tori T 2 embedded in CP2: 6 crossing points marked X - see Korzekwa et al. (2014), Andersson, Bengtsson, (2015)

N 1 Non-displacable tori in CP −

A more general question: Do two great two–tori T 2 embedded in CP2 intersect?

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 12 / 76 N 1 Non-displacable tori in CP −

A more general question: Do two great two–tori T 2 embedded in CP2 intersect? Yes, a great K-torus T K is non–displacable in CPK , Cho (2004).

Crossing points marked X represent mutually unbiased states

(which exist for any two unitary bases!)

Projections of two 2–tori T 2 embedded in CP2: 6 crossing points marked X - see Korzekwa et al. (2014), Andersson, Bengtsson, (2015)

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 12 / 76 Yes, RPK is non–displacable in CPK , Tamarkin (2008).

Crossing points marked X represent mutually entangled states = entangled with respect to two splittings!

Projections of two sets of 2–qubit maximally entangled states 3 3 ∈ = U(2)/U(1) = RP embedded in CP E Open Problem: Is the set N = U(N)/U(1) of two-quNit maximally entangled states embeddedE in CPM2−1 non–displacable ?

Mutually Entangled States

Do two real projective spaces RP3 embedded in CP3 intersect?

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 13 / 76 Mutually Entangled States

Do two real projective spaces RP3 embedded in CP3 intersect? Yes, RPK is non–displacable in CPK , Tamarkin (2008).

Crossing points marked X represent mutually entangled states = entangled with respect to two splittings!

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 13 / 76 Wawel castle in Cracow

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 14 / 76 Ciesielski theorem

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 15 / 76 Ciesielski theorem: With probability 1 the bench Banach talked to Nikodym in 1916 was localized in η-neighbourhood− of the red arrow.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 16 / 76 Plate commemorating the discussion between Stefan Banach and Otton Nikodym (Krak´ow,summer 1916)

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 17 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 18 / 76 Mathematics: Part 2

linear algebra matrix theory operator theory geometry of convex sets probability theory random matrices

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 19 / 76 Mixed quantum states = density matrices

Set N of all mixed states of size N M † N := ρ : N N ; ρ = ρ , ρ 0, Trρ = 1 M { H → H ≥ } Example: N = 2, One–qubit states: 3 2 = B3 R - Bloch ball with all pure states at the boundary M ⊂ The set N is compact and M P convex: ρ = ai ψi ψi , iP where ai 0 and |ai =ih 1. | ≥ i It has N2 1 real − N2−1 dimensions, N R . M ⊂ What the set of all N = 3 mixed states looks like? An 8–dimensional convex set with only 4–dimensional subset of pure (extremal) states, which belong to its 7–dim boundary

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 20 / 76 The set N of quantum mixed states: M

What it looks like for (for N 3) ≥

An apophatic approach :

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 21 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 22 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 23 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 24 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 25 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 26 / 76 rotated edges of an equilateral triangle and its convex hull I. Bengtsson, S. Weiss, K. Z˙ (2012)

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 27 / 76 Some properties of the convex body of mixed states A body X has a constant height if a) every point of the boundary of X or area A is contained in a face tangent to the inscribed ball of radius r ⇐⇒ b) area/volume ratio is ﬁxed, rA/V = d = dim(X ) - Archimedean

Thus a body of a constant height can be decomposed into cones of the same height h !

N2−1 Set N R of quantum states of size N has a constant height M ⊂ St. Szarek, I. Bengtsson, K. Z.˙ , 2006,

Some other joint math.–physics papers on these issues (*) J. Grabowski, M. Ku´s,G. Marmo, Geometry of quantum systems: density states and entanglement, 2005 (*) A. Sawicki, A. Huckleberry, M. Ku´s, Symplectic geometry of entanglement, 2010 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 28 / 76 More can be found in I. Bengtsson and K. Z.˙ , Geometry of Quantum States: an Introduction to Quantum Entanglement (Cambridge 2006)

second, enlarged edition, CUP 2017

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 29 / 76 Some properties of the set of linear positive maps Positive map sends the set of mixed states into itself A map Φ is positive if for any ρ 0 its image is positive, ρ0 = Φ(ρ) 0. ≥ ≥ A map Ψ is completely positive if Ψ 1K is positive for any extension size K (Choi⊗1974).

Self-dual cone of completely positive maps is characterizedCP by Choi theorem. The cone of positive maps and its dual cone areP more difficult to describe... SP Some partial results: 2 – Størmer-Woronowicz 1976 (*) St. Szarek, E. Werner,P K. Z.˙ , 2008 A generic positive map acting on a high-dimensional system is not CP (*) L. Skowronek, E. Størmer, K. Z.˙ , 2009 Cones of positive maps and their duality relations, (*) L. Skowronek, 2016 Cone 3 of positive maps on 3 is not finitellyP generated. H KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 30 / 76 Cracow: Saint Mary’s Church & optical effects ...

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 31 / 76 The set N of quantum mixed states for N 3 M ≥ A constructive approach:

Analysis of its structure with aid of operator theory notions like Numerical Range

The same tools are useful to investigate the structure of the subsets of N , namely sets of a) separable statesM and b) maximally entangled states.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 32 / 76 Operator theory: Numerical Range (Field of Values)

Deﬁnition

For any operator A acting on N one deﬁnes its NUMERICAL RANGE H (Wertevorrat) as a subset of the complex plane deﬁned by:

Λ(A) = x A x : x N , x x = 1 . (1) {h | | i | i ∈ H h | i }

Hermitian case † For any hermitian operator A = A with spectrum λ1 λ2 λN its numerical range forms an interval: the set of all possible≤ expectation≤ · · · ≤ values of the observable A among arbitrary pure states, Λ(A) = [λ1, λN ].

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 33 / 76 Numerical range and its properties

Compactness Λ(A) is a compact subset of C.

Convexity: Hausdorﬀ-Toeplitz theorem - Λ(A) is a convex subset of C.

Example Numerical range for random matrices of order N = 6 a) normal, b) generic (non-normal) Im Im æ 0.4 2 æ

æ 0.2 æ 1

æ Re -0.5 0.5 1.0

Re -2 -1 1 2 -0.2 æ

æ æ æ -0.4 -1

æ æ -0.6

-2 æ

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 34 / 76 Numerical range & quantum states:

Set of pure states of a ﬁnite size N: N−1 Let ΩN = CP denotes the set of all pure states in N . H 1 Example: for N = 2 one obtains Bloch sphere,Ω2 = CP .

Probability distributions on complex plane supported in Λ(A) - Fix an arbitrary operator A of size N, - Generate random pure states ψ of size N (with respect to unitary invariant,| Fubini–Studyi measure) and analyze their contribution ψ A ψ to the numerical range on the complex plane.... h | | i Probability distribution: (a ’numerical shadow’) Z PA(z) := dψ δ z ψ A ψ ΩN − h | | i is supported on Λ(A). How it looks like?

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 35 / 76 Normal case: a ’shadow’ of the classical simplex...

Normal matrices of size N = 2 with spectrum λ1, λ2 { } Distribution PA(z) covers uniformly the interval [λ1, λ2] on the complex plane,

Examples for diagonal matrices A of size two and three

Density plot of numerical range of operator [1, 0; 0, i] Density plot of numerical range of operator diag(1, i, i) 1.0 1.0 −

0.5 0.5

0.0 0.0

0.5 0.5 − −

1.0 1.0 − 1.0 0.5 0.0 0.5 1.0 − 1.0 0.5 0.0 0.5 1.0 − − − −

Normal matrices of order N = 3 with spectrum λ1, λ2, λ3 { } Distribution PA(z) covers uniformly the triangle ∆(λ1, λ2, λ3) on the complexKZ˙ (IF UJ/CFTplane. PAN ) Walk through mathematics Dec. 3, 2016 36 / 76 Normal case: a ’shadow’ of the classical simplex...

Normal matrices of size N = 4 with spectrum λ1, λ2, λ3, λ4 { } Distribution PA(z) covers the quadrangle (λ1, λ2, λ3, λ4) with the density determined by projection of the regular tetrahedron (3–simplex) on the complex plane

Density plot of numerical range of operator diag(1, exp(iπ/3), exp(i2π/3), 1) 1.0 −

0.5

0.0

0.5 −

1.0 − 1.0 0.5 0.0 0.5 1.0 − −

Examples for diagonal matrices A of size four. Observe shadow of the edges of the tetrahedron = the set of all classical mixed states for N = 4. KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 37 / 76 Numerical range for matrices of order N = 2. with spectrum λ1, λ2 { } a) normal matrix A Λ(A) = closed interval [λ1, λ2] ⇒ b) not normal matrix A Λ(A) = elliptical disk with λ1, λ2 as focal ⇒p † 2 2 points and minor axis, d = TrAA λ1 λ2 (Murnaghan, 1932; Li, 1996). − | | − | | 0 1 example Jordan matrix J = . Then its numerical range forms 0 0 a circular disk, Λ(J) = D(0, r = 1/2).

The set of N = 2 pure quantum states A projection of the Bloch sphere S2 = CP1 onto a plane forms an ellipse, (which could be degenerated to an interval).

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 38 / 76 Numerical shadow for N = 2

Non–normal matrices of size N = 2

Distribution PA(z) covers entire (elliptical) disk on the complex plane with non-uniform (!) density determined by projection of (empty!) Bloch sphere, S2 = CP1 on the complex plane.

Density plot of numerical range of operator [0, 1; 0, 1/2] Density plot of numerical range of operator [0, 1; 0, 1] 1.0 2.0

1.5

0.5 1.0

0.5

0.0 0.0

0.5 −

0.5 1.0 − −

1.5 −

1.0 2.0 − 1.0 0.5 0.0 0.5 1.0 − 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − − − − − −

Examples for non-normal matrices A of size two. Note the ’shadow’ of the projective space CP1 = S2 = set of quantum pure states for N = 2. KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 39 / 76 Shadows of three dimensional objects...

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 40 / 76 Quantum States and Numerical Range & Shadow

Classical States & normal matrices

Proposition 1.Let N denote the set of classical states of size N, which C N−1 forms the regular simplex ∆N−1 in R . Then the set of similar images of orthogonal projections of N on a 2–plane is equivalent to the set of all possible numerical ranges Λ(CA) of all normal matrices A of order N (such that AA∗ = A∗A).

Quantum States & non–normal matrices

Proposition 2.Let N denotes the set of quantum states size N embedded in RN2−1Mwith respect to Euclidean geometry induced by Hilbert-Schmidt distance. Then the set of similar images of orthogonal projections N on a 2-plane is equivalent to the set of all possible numerical rangesM Λ(A) of all matrices A of order N.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 41 / 76 Normal case: a ’shadow’ of the 4-D simplex...

Normal matrices of size N = 5 with spectrum (λ1, λ2, λ3, λ4, λ5)

Distribution PA(z) covers the pentagon exp(2πj/5) , j = 0,..., 4 with the density determined by projection of the{ 4-D simplex} on the complex plane

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 42 / 76 How complex / real projective space looks like?

Not normal matrices of size N = 3

Distribution PA(z) covers the numerical range Λ(A) on the complex plane with a non-uniform density

Density plot of numerical range of operator O5 subject to complex states Density plot of numerical range of operator O5 subject to real states

1.0 1.0

0.5 0.5 im im

0.0 0.0

0.5 0.5 − −

1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 − − re − − re 2 2 ’shadow’ of the projective space Ω3 = CP (left) and RP (right) for a N = 3 non-normal matrix A = [0, 1, 1; 0, i, 1; 0, 0, 1] − KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 43 / 76 Numerical range for matrices of order N = 3. Classification by Keeler, Rodman, Spitkovsky 1997 Numerical range Λ of a 3 3 matrix A forms: × a) Λ(A) is a compact set of an ’ovular’ shape (which contains three eigenvalues!) – the generic case, b) a compact set with one flat part (e.g. convex hull of a cardioid), c) a compact set with two flat parts (e.g. convex hull of an ellipse and a point outside it), d) triangle of eigenvalues, Λ(A) = ∆(λ1, λ2, λ3) for any normal matrix A.

These four cases describe the shape of possible projections of the complex projective space CP2 (embedded into R8) onto a 2–plane.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 44 / 76 Shadows of typical matrices of size N = 3

correspond to the four classes of N = 3 numerical ranges by Keeler et al. KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 45 / 76 Joint Numerical Range (JNR) of m operators

m Λ(A1, A2,..., Am) = ( ψ A1 ψ , ψ A2 ψ ,..., ψ Am ψ ) R . h | | i h | | i h | | i ⊂ For m = 2, decompose A = AH + iAA into its Hermitian and anti–Hermitian part. Then Λ(A) = Λ(AH , AA)

Joint numerical range of 3 hermitian operators of size N = 3 3 Λ(A1, A2, A3) R gives a projection of the 8D set 3 of mixed states of a qutrit into⊂3D. Examples: M

Classiﬁcation of possible shapes for N = 3: Szyma´nski,Weis, K.Z.˙ 2016

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 46 / 76 Higher rank numerical range Numerical range of rank k of a matrix X reads Wk (X ) = λ C : Pk XPk = λPk { ∈ } for some rank–k projector Pk Choi, Kribs, K.Z.˙ , Higher–rank numerical ranges and compression problems, Lin. Alg. Appl. 418, 828 (2006)

Some other results on numerical ranges (*) Puchala, Gawron, Miszczak, Skowronek, Choi, K.Z.˙ , Product numerical range in a space with tensor product structure, Linear Algebra Applications 434, 327-342 (2011). (**) Dunkl, Gawron, Holbrook, Miszczak, Puchala, K.Z.˙ , Numerical shadow and geometry of quantum states, J. Phys. A 44, 335301 (2011). (***) Eugene Gutkin, K.Z.˙ , Joint numerical ranges and quantum maps, Linear Algebra Applications, (2013).

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 47 / 76 How a generic cross-section of a n–cube looks like??

Dvoretzky theorem A. Dvoretzky, Some results on convex bodies and Banach spaces (1961) (Under some technical assumptions) a generic cross-section of a compact, high dimensional, convex set almost surely forms a disk ! Quand les cubes deviennent ronds Webpage by Guillaume Auburn and Jos Leys

Convex sets in high dimensions

How a cross-section of a3 –cube looks like??

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 48 / 76 Dvoretzky theorem A. Dvoretzky, Some results on convex bodies and Banach spaces (1961) (Under some technical assumptions) a generic cross-section of a compact, high dimensional, convex set almost surely forms a disk ! Quand les cubes deviennent ronds Webpage by Guillaume Auburn and Jos Leys

Convex sets in high dimensions

How a cross-section of a3 –cube looks like??

How a generic cross-section of a n–cube looks like??

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 48 / 76 Convex sets in high dimensions

How a cross-section of a3 –cube looks like??

How a generic cross-section of a n–cube looks like??

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 48 / 76 Numerical range of random matrices

Non–hermitian Ginibre matrices of size N normalized as TrGG ∗ = N

√2 √2 √2 1 1 1

= 0 = 0 = 0 r 1 1 1 √−2 √−2 √−2 − − − √2 1 0 1 √2 √2 1 0 1 √2 √2 1 0 1 √2 − − − − − − < < < N = 10 N = 100 N = 1000 Numerical range and spectrum of random Ginibre matrices of size N. Note circular disk of eigenvalues of Girko and the non–normality belt.

Result: In the limit N the numerical range of a random Ginibre matrix G converges→ (in ∞ the Hausdorﬀ distance) to the disk of radius √2.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 49 / 76 Random matrices & quantum states for a large size N

Theorem (Collins, Gawron, Litvak,KZ,˙ 2014)

Let R > 0 and let XN N be a sequence of complex random matrices or order N such that{ for every} θ R with probability one ∈ iθ lim Re e XN = R N→∞ || || Then with probability one lim dH Λ(XN ), D(0, R) = 0. N→∞ Example: For random Ginibre matrix G the operator norm . does not depend on the phase θ and (upon the normalization used) converges|| || to R = √2. Hence the numerical range of G almost surely forms a disk of radius R. Relation to the Dvoretzky theorem ! = a generic projection of the set N of mixed states on a plane ⇒ is close a disk (for largeMN). KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 50 / 76 Cracow: Market Square KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 51 / 76 number theory

Mathematics: Part 3

discrete mathematics combinatorics graph theory dynamical systems & fractals game theory ...

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 52 / 76 Mathematics: Part 3

discrete mathematics combinatorics graph theory dynamical systems & fractals game theory ... number theory

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 52 / 76 A chess problem on a N N chessboard ×

Take NM chess rooks with M N/2 and ﬁx an integer K M, ≤ ≤ Try to place all the rooks on the board such that: a) there are exactly M rooks on each row and each column of the board, and b) for any pair of rows or columns, there are exactly K pairs of mutually attacking rooks.

Example 1: Arbitrary N and M = 1 and K = 0: Identity or a permutation matrix does the job (no attacking rocks !)

X ...   . X ..   . X ..   .. X .  N = 4,   ,    .. X .   ... X  ... X X ...

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 53 / 76 ﬁnite plane of Fano

‘Symmetric balanced incomplete block designs’ (BIBD) Colbourn & Dinitz, Handbook of Combinatorial Designs (2007)

Example 2. Let K = 1, N = 7 and M = 3

XXX ....  X .. XX ..    X .... XX     . X . X .. X     . X .. XX .     .. X . X . X  .. XX . X .

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 54 / 76 Example 2. Let K = 1, N = 7 and M = 3

XXX ....  X .. XX ..    X .... XX     . X . X .. X     . X .. XX .     .. X . X . X  .. XX . X . ﬁnite plane of Fano

‘Symmetric balanced incomplete block designs’ (BIBD) Colbourn & Dinitz, Handbook of Combinatorial Designs (2007)

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 54 / 76 the Paley biplane

Example 3. Let K = 2, N = 11 and M = 5  . X . XXX ... X .   .. X . XXX ... X     X .. X . XXX ...     . X .. X . XXX ..       .. X .. X . XXX .     ... X .. X . XXX     X ... X .. X . XX     XX ... X .. X . X     XXX ... X .. X .     . XXX ... X .. X  X . XXX ... X ..

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 55 / 76 Example 3. Let K = 2, N = 11 and M = 5  . X . XXX ... X .   .. X . XXX ... X     X .. X . XXX ...     . X .. X . XXX ..       .. X .. X . XXX .     ... X .. X . XXX     X ... X .. X . XX     XX ... X .. X . X     XXX ... X .. X .     . XXX ... X .. X  the Paley biplane X . XXX ... X ..

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 55 / 76 Hadamard matrices real quantum gates ⇒ Deﬁnition (Sylvester 1867) mutually orthogonal row and columns,

∗ HH = 1 , Hij = 1. (2) ± simplest example of order N = 2 unitary Hadamard gate ⇒ 1 1 1 1 H = so that U = √1 is orthogonal. 2 1 1 2 2 1 1 The most often− used gate in Quantum Information− Theory, as it forms a quantum superposition U2 0 = ( 0 + 1 )/√2; | i | i | i and U2 1 = ( 0 1 )/√2. | i | i − | i Hadamard conjecture (real) Hadamard matrices do exist for N = 2 and 4k for any k = 1, 2, ... After a discovery of N = 428 Hadamard matrix( Kharaghani and Tayfeh-Razaie, 2005) this conjecture is known to hold up to N = 664 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 56 / 76 Almost-Hadamard matrices for other N

Real orthogonal matrix A with ’almost equal’ moduli P Formally: Orthogonal matrix with maximal 1–norm, A 1 = ij Aij . For N = 2, 4, 8, 12, 16, ... any (rescaled) Hadamard matrix|| || is also Almost| | Hadamard with one–norm equal to N√N.

 1 2 2  1 − N = 3, A3 = 3  2 1 2  for which A3 1 = 5 < 3√3 5.196 2− 2 1 || || ≈ −  3 2 2 2 2  −  2 3 2 2 2  1  −  N = 5, A5 =  2 2 3 2 2  5  −   2 2 2 3 2  2 2 2− 2 3 − T. Banica, I. Nechita, K. Z.˙ , Almost Hadamard matrices: 2012

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 57 / 76 N = 7. Incidence matrix of the Fano plane (x = 2 4√2, y = 2 + 3√2): x x x y y y y − x y y x x y y   x y y y y x x 1   A7 = √ y x y x y y x 2 7   y x y y x x y   y y x y x y x y y x x y x y N = 11. Incidence matrix of the Paley biplane (x = 6 12√3,   y = 6 + 10√3): yxyxxxyyyxy−  yyxyxxxyyyx     xyyxyxxxyyy     yxyyxyxxxyy       yyxyyxyxxxy  A = √1  yyyxyyxyxxx  11 6 11      xyyyxyyxyxx     xxyyyxyyxyx     xxxyyyxyyxy     yxxxyyyxyyx  KZ˙ (IF UJ/CFT PAN ) xyxxxyyyxyyWalk through mathematics Dec. 3, 2016 58 / 76 Queensland, Australia

another picture from Cracow... KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 59 / 76 another picture from Cracow... Queensland, Australia KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 59 / 76 other option: Complex Hadamard matrices

∗ HH = N 1 and Hij = 1, | | hence Hij = exp(iφij ) with an arbitrary complex phase.

Complex Hadamard matrices do exist for any N example: the Fourier matrix

(FN )jk := exp(ijk2π/N) with j, k = 0, 1,..., N 1. − special case : N = 4

 1 1 1 1   1 i 1 i  F4 =  − −   1 1 1 1  1 −i 1 −i − −

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 60 / 76 Equivalent Hadamard matrices, H0 H ∼ iﬀ there exist permutation matrices P1 and P2 and diagonal unitary matrices D1 and D2 such that

0 H = D1P1 HP2D2 .

N = 2 all complex Hadamard matrices are equivalent to real Hadamard (Fourier) matrix H2 = F2

N = 3 all complex Hadamard matrices are equivalent to the Fourier matrix

 1 1 1  2 2πi/3 F3 =  1 w w  , w = e . 1 w 2 w

U. Haagerup, Orthogonal maximal abelian -subalgebras of the N N matrices and cyclic N∗–roots, 1996. KZ˙ (IF UJ/CFT PAN ) × Walk through mathematics Dec. 3, 2016 61 / 76 Classiﬁcation of Complex Hadamard matrices II

N = 4 Lemma (Haagerup). For N = 4 all complex Hadamard matrices are equivalent to one of the matrices from the following 1–d orbit, w = i

 1 1 1 1  1 2 3 (1)  1 w exp(i a) w w exp(i a)  F (a) =  · · · ·  , a [0, π]. 4  1 w 2 1 w 2  ∈ 1 w 3 exp(i a) w 2 w 1 exp(i a) · · · · N = 5 All N = 5 complex Hadamard matrices are equivalent to the Fourier matrix F5 (Haagerup 1996).

N 6 ≥ Several orbits of Complex Hadamard matrices are known, but the problem of their complete classiﬁcation remains open!

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 62 / 76 Zoo of N = 6 complex Hadamard matrices

(1) B6 (y): 1-d family of Beauchamp & Nicoara, 2006  1 1 1 1 1 1   1 1 1/x y y 1/x   − − −  (1)  1 x 1 y 1/z 1/t  B (y) =  − −  6  1 1/y 1/y 1 1/t 1/t   − − −   1 1/y z t 1 1/x  1 x t− t x − 1 − − − where y = exp(i α) is a free parameter, α [0, 2π) and √ ∈ 1+2y+y 2± 2√1+2y+2y 3+y 4 x(y) = 1+2y−y 2 1+2y−y 2 z(y) = y(−1+2y+y 2) ; t(y) = xyz

Several CHM for n = 6..16 are listed in a Catalog of CHM, see W. Tadej & K.Z.˙ OSID, 13, 133-177 (2006) and its updated online version at http://chaos.if.uj.edu.pl/ karol/hadamard ∼ KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 63 / 76 Complex Hadamard matrices: some applications

1 –subalgebras of finite dimensional von Neuman algebras, Haagerup,∗ Popa, de la Harpe & Jones, Munemasa & Watatani, 2 Bi-unimodular sequences and cyclic n–roots, Björk,Fröberg, Saffari 3 Equiangular lines, Godsil & Roy

4 quantum optics: symmetric multiports, Jex, Stenholm & Zeilinger 5 quantum designs, Zauner 1999 6 quantum information: bases of unitary operators, Werner 2001. 7 Mutually Unbiased Bases (MUB)

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 64 / 76 Mutually Unbiased Bases & Complex Hadamards

Mutually unbiased bases (MUB) in ﬁnite dimensions N N Let αi i=1 and βj j=1 be two orthonormal basis N (e.g.{| given} as eigenbasis{| i} of hermitian operators A andH B). The bases are called maximally unbiased if all entries of the unitary 2 2 1 transition matrix U have the same modulus, Uij = αi βj = N , i.e. U is a complex Hadamard matrix| (CHM).| |h | i| A set of d bases which satisﬁes this condition is called mutually unbiased ∗ Two CHM’s H1 and H2 are unbiased if H1 H2/√N is also a CHM.

Basic facts and ... prime numbers ... For any N there exists not more then d = N + 1 MUBs, ∗ This upper bound is saturated for prime dimensions and for powers of ∗ primes, N = pk (Ivanovic 1981; Wootters & Fields 1989) For any N 2 there exists at least a triple of MUBs, ∗ for N = 6 =≥ 2 3 only3 MUBs are known ∗ (so this is the× ﬁrst open case ! as the upper bound is d = 7) KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 65 / 76 MUB’s and some number theory

Standard MUB solution for a prime dimension, N = p (1) Choose Fourier matrix of order N and set H = FN . ∗ 2π 2 Take diagonal unitary [EN ]jk := δjk exp i (j 1) , ∗ · N − Deﬁne set of N complex Hadamard matrices H(r) N where ∗ { }r=1 (r) r−1 (1) H := (EN ) H for r = 1, 2,..., N

and check that they are mutually unbiased (s) ∗ (r) since Xr−s := (H ) H /√N is also a complex Hadamard.

This is ﬁne for any prime N = p. But why this construction does not work for composite dimensions ??

Analyze diagonal elements of the matrices (Xr ) for any (composite) N...

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 66 / 76 A prime distinguishing function

Consider a function of an integer argument N > 1 N−1 N 1 X 1 X i· 2π r(j−1)2 g(N) = e N 1 . N 1 √N − − r=1 j=1 Use the Gauss sum to show that it vanishes for any prime N = p 2, p−1 p i· 2π r(j−1)2 p−1 ≥ g(p) = 1 P √1 P e p 1 = 1 P √1 √p 1 = 0 p−1 r=1 p j=1 − p−1 r=1 p −

Rescaled prime distinguishing function g(N) T.Durt, B.Englert, I.Bengtsson, K.Z.˙ , 2010 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 67 / 76 KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 68 / 76 Dynamical Systems & Fractals Classical Dynamical Systems

Parabolic maps on the torus x0 1 + AA/B x = y 0 AB 1 A y − − | mod 1 Description of the structure of the invariant set P.Ashwin, X.Fu, T.Nishikawa, K.Z˙ , 2000

Dynamical Systems with Noise Basis Markov Partitions Take a classical map: x0 = f (x) + ξ. For certain noise profiles P(ξ) the corresponding Markov operator, can be represented by a finite matrix of size M. In the deterministic limit M . E. Bollt, P. Góra, A.Ostruszka,→ K.Z. ∞˙ 2008. KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 69 / 76 Iterated Function Systems & Quantum analogues

Classical IFS IFS: collection of k functions and k probabilities, d d k fj : R R ; pj (x) . { → }j=1 Example : multi–fractal invariant measure on the Cantnor set, for which d = 1, k = 2, f1 = x/3, f2 = (x + 2)/3. W. Slomczy´nski,J. Kwapie´n,K. Z.˙ , 2000

Quantum IFS QIFS: collection of k functions and k k probabilities, Fj : N N ; pj (φ) . { H → H }j=1 Example tartan: invariant measures on (Cantor set)2 for classical and quantum IFS. A.Lozi´nski, W. Slomczy´nski,K. Z.˙ , 2003.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 70 / 76 Random Matrices & Free Random Variables Tensor products of random unitary matrices Nearest neighbour statistics P(s) for spectra of tensor products of two Haar random unitary matrices, U1 U2 of size M. ⊗ For large M the statistics becomes Poissonian (level clustering) T. Tkocz, O. Zeituni, M. Smaczy´nski, M.Ku´s,K.Z˙ , Random Matrices, 2012

Graph Random States & Fuss–Catalan distribution Ensembles of Random mixed states of size N corresponding to a graph: for a class of graphs and N the spectral density is described→ by ∞ the Fuss–Catalan distribution. B. Collins, I. Nechita, K.Z˙ , 2010

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 71 / 76 Stefan Banach sitting at his bench close to the Wawel Castle

Sculpture: Stefan Dousa Fot. Andrzej Kobos

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 72 / 76 The presence of physics in contemporary pure mathematics is undeniable and has presented the community with both opportunities and challenges.

Michael Atiyah et al. Geometry and physics, Phil. Trans. R. Soc. A 368, 913 (2010).

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 73 / 76 almost all ... A reasonable theoretical physicist is able to formulate his problem in a way accessible for a mathematician (= comprehensible without studying physics ...)

Jumping with a chess knight through various branches of mathematics:

Which ﬁelds of mathematics are interesting for a quantum physicist ?

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 74 / 76 Jumping with a chess knight through various branches of mathematics:

Which ﬁelds of mathematics are interesting for a quantum physicist ?

almost all ... A reasonable theoretical physicist is able to formulate his problem in a way accessible for a mathematician (= comprehensible without studying physics ...)

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 74 / 76 Conjecture 2 For the subset of ”reasonable mathematicians” and the subset of ”reasonable physicists” the probability deﬁned above behaves as p = 1 , where 1. −

Conjecture 3

This number will asymptotically tend to zero, limt→∞ (t) = 0.

Concluding Remarks

Theorem 1 A randomly taken pair (M, P) of scientists, consisting of a mathematician and a theoretical physicist, with a probability p > 0 can eﬃciently work together on problems interesting to both parties.

Proof: By demonstation of concrete cases, as mentioned above...

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 75 / 76 Conjecture 3

This number will asymptotically tend to zero, limt→∞ (t) = 0.

Concluding Remarks

Proof: By demonstation of concrete cases, as mentioned above...

Conjecture 2 For the subset of ”reasonable mathematicians” and the subset of ”reasonable physicists” the probability deﬁned above behaves as p = 1 , where 1. −

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 75 / 76 Concluding Remarks

Proof: By demonstation of concrete cases, as mentioned above...

Conjecture 2 For the subset of ”reasonable mathematicians” and the subset of ”reasonable physicists” the probability deﬁned above behaves as p = 1 , where 1. −

Conjecture 3

This number will asymptotically tend to zero, limt→∞ (t) = 0.

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 75 / 76 Bench commemorating the discussion between Stefan Banach and Otton Nikodym (Krak´ow,summer 1916)

Sculpture: Stefan Dousa Fot. Andrzej Kobos opened in Planty Garden, Cracow, Oct. 14, 2016

KZ˙ (IF UJ/CFT PAN ) Walk through mathematics Dec. 3, 2016 76 / 76