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Walking Through MATHEMATICS Relevant for Quantum Mechanics: a Personal View

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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 N with = = 1, definedj i 2 H up to ajj globaljj phase,h j i eiα , j i ∼ j 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 i with i pi = 1, pi 0, so that the states are: Hermitian,j ρ ih= ρj∗, positive, ρ 0 and≥ normalized, Trρ = 1. ≥ Unitary Quantum Dynamics a) pure states: 0 = U , j i j i b) mixed states: ρ0 = UρU∗, where U U(N) is called a quantum gate. In the case2 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, = 1, eiα h j i j i ∼ j i Fubini-Study (geodesic) distance: DFS ( ; ' ) := arccos ' j i j i jh j ij 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 - 3, space of N = 3 pure states, Ω3 = CP j i 2 H Foliation CP2 into leaves of a fixed distance to sphere S2 including 0 ; 1 . j i j 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, = φA φB = φA; φB j i j i ⊗ j i j i other states are entangled { e.g. Bell state + = p1 ( 0; 0 + 1; 1 ) j i 2 j i j i A ququart two qubits, N = 4 = 2 2, 4 2 2, ≈ ×3 j i 2 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, k ⊗ j i 2 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 2 = 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! Projections of two sets of 2{qubit maximally entangled states 3 3 2 = 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 ? 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 y N := ρ : N N ; ρ = ρ ; ρ 0; Trρ = 1 M f H !H ≥ g 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 i i , iP where ai 0 and jai =ih 1. j ≥ 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.
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