Iso-entangled Mutually Unbiased Bases and mixed states t –designs Karol ˙ Zyczkowski Jagiellonian University, Cracow, & Polish Academy of Sciences, Warsaw in collaboration with Jakub Czartowski (Cracow) Dardo Goyeneche (Antofagasta) Markus Grassl (Erlangen) Quantum Walk , Dolomiti, July 16-19, 2019 K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 1 / 29
What is this talk about ? we analyze discrete structures in the finite Hilbert space H N . relevant for the standard Quantum Theory , K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 2 / 29
What is this talk about ? we analyze discrete structures in the finite Hilbert space H N . relevant for the standard Quantum Theory , for instance: Mutually Unbiased Bases ( MUB s) Symmetric Informationally Complete generalized quantum measurements ( SIC POVM s) Complex projective t-designs formed of pure quantum states and their generalizations: selected constellations of mixed states which form mixed states t-designs . Why we do it ? Beacause we a) do not fully understand these structures relevant for quantum theory ! b) wish to construct novel schemes of generalized measurements and c) design techniques averaging over the set of density matrices of size N K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 2 / 29
Mutually Unbiased Bases I Two orthogonal bases consisting of n vectors each in H N are called mutually unbiased (MUB) if |� φ i | ψ j �| 2 = 1 N , for i , j = 1 , . . . , N . Such bases provide maximally different quantum measurements. For a complex Hilbert space of dimension N there exist at most N + 1 Two unbiased bases in ❘ 2 such bases. Example N = 2: 3 eigenbases of σ x , σ y , σ y K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 3 / 29
Mutually Unbiased Bases & Hadamard matrices Full sets of ( N + 1) MUB’s are known if dimension is a power of prime , N = p k . For N = 6 = 2 × 3 only 3 < 7 MUB’s are known! A transition matrix H ij = � φ i | ψ j � from one unbiased basis to another forms a complex Hadamard matrix, which is a) unitary , H † = H − 1 , b) has ” unimodular ” entries, | H ij | 2 = 1 / N , i , j = 1 , . . . , N . Classification of all complex Hadamard matrices is complete for N = 2 , 3 , 4 , 5 only. ( Haagerup 1996) see Catalog of complex Hadamard matrices , at http://chaos.if.uj.edu.pl/ ∼ karol/hadamard K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 4 / 29
Standard set of 2-qubit MUBs consists of 3 separable bases + 2 maximally entangled bases in H 4 Reduced states ρ A and ρ B form 6 (doubly degenerated) vertices of the regular octahedron within the Bloch ball (eigenvectors of σ x , σ y , σ z = 3 MUBs for N = 2) and 8-fold degenerated maximally mixed state ✶ / 2 in the centre. K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 5 / 29
Symmetric Informationally Complete POVM Symmetric informationally complete (SIC) POVM is such a set of N 2 vectors {| ψ i �} in H N , that 1 |� ψ i | ψ j �| 2 = N + 1 Zauner (1999), Rennes, Blume- Kohout, Scott, Caves (2003) They may be thought as equiangular structures in the Hilbert space. SIC POVM are found analitically for N = 2 , . . . , 24 and numerically 4 pure states at the Bloch sphere up to 151 + some special cases: forming a SIC for N = 2. N = 844 Grassl & Scott (2017) K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 6 / 29
Complex projective t-designs Definition i =1 of pure states in H N is called complex projective Any ensemble | ψ i � M t-design if for any polynomial f t of degree at most t in both components of the states and their conjugates the average over the ensemble coincides with the average over the space C P N − 1 M 1 � � f t { ψ i } = C P N − 1 f t ( ψ ) d ψ FS . M i =1 with respect to the unitarily–invariant Fubini–Study measure d ψ FS . Complex projective t –designs are used for quantum state tomography, quantum fingerprinting and quantum cryptography. Examples of 2 -designs include maximal sets of mutually unbiased bases (MUB) and symmetric informationally complete (SIC) POVM. the larger t the better design approximates the set of states.. K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 7 / 29
To know more about these issues consult the book Cambridge University Press, I edition , 2006 II edition , 2017 (new chapters on multipartite entanglement & discrete structures in the Hilbert space), K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 8 / 29
Interesting case – isoentangled SIC-POVM Averaging property implies a condition for the average entanglement (measured by the purity of partial trace) of vectors in a 2-design in H N ⊗ H N 2 N (Tr A | ψ i �� ψ i | ) 2 �� � � Tr = N 2 + 1 K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 9 / 29
Interesting case – isoentangled SIC-POVM Averaging property implies a condition for the average entanglement (measured by the purity of partial trace) of vectors in a 2-design in H N ⊗ H N 2 N (Tr A | ψ i �� ψ i | ) 2 �� � � Tr = N 2 + 1 Zhu & Englert (2011) found an interesting constelation of 4 2 = 16 states in H 2 ⊗ H 2 forming a SIC for two-qubit system, such that entanglement of all states is constant, = 4 (Tr A | ψ i �� ψ i | ) 2 � � Tr 5 , for i = 1 , . . . , 16 . Such a set of states can be obtained from a single fiducial state | φ 0 � by local unitary operations, | φ j � = U j ⊗ V j | φ 0 � . K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 9 / 29
Isoentangled MUBs for 2 qubits? Question: K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 10 / 29
Isoentangled MUBs for 2 qubits? Question: Is there a similar configuration for the full set of 5 iso-entangled MUBs for 2 qubits? the standard MUB solution for N = 4 consists of 3 separable bases and 2 maximally entangled ... K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 10 / 29
The answer is positive! | φ 0 � = 1 20( a + | 00 � − 10 i | 01 � + (8 i − 6) | 10 � + a − | 11 � ) , √ √ where a ± = − 7 ± 3 5 + i (1 ± 5) and other states are locally equivalent , | φ j � = U j ⊗ V j | φ 0 � Each of 5 × 4 = 20 pure states | ψ j � in H 2 ⊗ H 2 will be represented by its partial trace, ρ j = Tr B | ψ j �� ψ j | belonging to the Bloch ball of one-qubit mixed states. K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 11 / 29
Each basis is represented by a regular tetrahedron inside the Bloch ball. K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 11 / 29
Each basis is represented by a regular tetrahedron inside the Bloch ball. Each colour corresponds to a single basis. K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 11 / 29
Each basis is represented by a regular tetrahedron inside the Bloch ball. Each colour corresponds to a single basis. Entire five-color set forms a regular 5-tetrahedra compound . K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 11 / 29
Each basis is represented by a regular tetrahedron inside the Bloch ball. Each colour corresponds to a single basis. Entire five-color set forms a regular 5-tetrahedra compound . Its convex hull forms a regular dodecahedron , different from the one of Zimba and Penrose ... K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 11 / 29
Jakub Czartowski and his sculpture K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 12 / 29
Mixed states t-designs K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 13 / 29
Generalized designs In quantum theory one uses projective designs formed by pure states , | ψ i � ∈ H N unitary designs formed by unitary matrices , U i ∈ U ( N ) (which induce designes in the set of maximally entangled states, | φ j � = ( U j ⊗ ✶ ) | ψ + � spherical designs - sets of points evenly distributed at the sphere S k related notions, e.g. conical designs , Graydon & Appleby (2016), mixed designs by Brandsen, Dall’Arno, Szymusiak (2016) These examples for special case of a general construction of averaging sets by Seymour and Zaslavsky (1984) . It concerns a collection of M points x j from an arbitrary measurable set Ω with measure µ such that M 1 � � f t ( x i ) = f t ( x ) d µ ( x ) , M Ω i =1 where f t ( x ) denote selected continuous functions, e.g. f t ( x ) = x t . K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 14 / 29
Mixed states t-designs We apply this idea for a compact set of mixed states Ω N ⊂ ❘ N 2 − 1 endowed with the flat Hilbert-Schmidt measure d ρ HS Definition Any ensemble { ρ i } M i =1 of M density matrices of size N is called a mixed states t-design if for any polynomial g t of degree t in the eigenvalues λ j of the state ρ the average over the ensemble is equal to the mean value over the space of mixed states Ω N with respect to the Hilbert-Schmidt measure d ρ HS , M 1 � � g t ( ρ i ) = g t ( ρ ) d ρ HS . (1) M Ω N i =1 K ˙ Z (UJ/CFT) Iso-entangled MUBs & generalized t –designs 16.07.2019 15 / 29
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