Asymptotic properties of quantum states and channels Ion Nechita, Łukasz Pawela , Zbigniew Puchała, Karol Życzkowski Institute of Theoretical and Applied Informatics, Polish Academy of Sciences 18 June 2017 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 1 / 31
Outline Random matrix theory and free probability crash course 1 Random states and channels 2 Eigenvalues of random quatnum states and channels 3 Distances between random quantum states 4 The diamond norm 5 Final remarks 6 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 2 / 31
Random matrix theory and free probability crash course Contents Random matrix theory and free probability crash course 1 Random states and channels 2 Eigenvalues of random quatnum states and channels 3 Distances between random quantum states 4 The diamond norm 5 Final remarks 6 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 3 / 31
Random matrix theory and free probability crash course Some useful random matrix ensembles Ginibre matrices Matrices G such that G ij ∼ N C ( 0 , 1 / 2 ) Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 4 / 31
Random matrix theory and free probability crash course Some useful random matrix ensembles Ginibre matrices Matrices G such that G ij ∼ N C ( 0 , 1 / 2 ) Wishart matrices W = GG † Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 4 / 31
Random matrix theory and free probability crash course Free probability introduction Why free probability Let us consider two sequences A N and B N of selfadjoint N × N matrices Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31
Random matrix theory and free probability crash course Free probability introduction Why free probability Let us consider two sequences A N and B N of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f ( A N , B N ) for some non-trivial selfadjoint function f. Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31
Random matrix theory and free probability crash course Free probability introduction Why free probability Let us consider two sequences A N and B N of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f ( A N , B N ) for some non-trivial selfadjoint function f. Freenes N tr [ p 1 ( A n ) q 1 ( B N ) p 2 ( A N ) q 2 ( B N ) . . . ] → 0 . 1 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31
Random matrix theory and free probability crash course Free probability introduction Why free probability Let us consider two sequences A N and B N of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f ( A N , B N ) for some non-trivial selfadjoint function f. Freenes N tr [ p 1 ( A n ) q 1 ( B N ) p 2 ( A N ) q 2 ( B N ) . . . ] → 0 . 1 If A N and B N have almost surely an asymptotic eigenvalue distribution for N → ∞ ; Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31
Random matrix theory and free probability crash course Free probability introduction Why free probability Let us consider two sequences A N and B N of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f ( A N , B N ) for some non-trivial selfadjoint function f. Freenes N tr [ p 1 ( A n ) q 1 ( B N ) p 2 ( A N ) q 2 ( B N ) . . . ] → 0 . 1 If A N and B N have almost surely an asymptotic eigenvalue distribution for N → ∞ ; A N and B N are independent; Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31
Random matrix theory and free probability crash course Free probability introduction Why free probability Let us consider two sequences A N and B N of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f ( A N , B N ) for some non-trivial selfadjoint function f. Freenes N tr [ p 1 ( A n ) q 1 ( B N ) p 2 ( A N ) q 2 ( B N ) . . . ] → 0 . 1 If A N and B N have almost surely an asymptotic eigenvalue distribution for N → ∞ ; A N and B N are independent; B N is a unitarily invariant ensemble. Then, A N and B N are almost surely asymptotically free. Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31
Random matrix theory and free probability crash course Free convolution If x , y are free then moments of x + y are uniquely determined by the moments of x and y . Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31
Random matrix theory and free probability crash course Free convolution If x , y are free then moments of x + y are uniquely determined by the moments of x and y . Notation We say that x + y is the free convolution of the distribution of x and distribution of y . We write: (1) µ x + y = µ x ⊞ µ y Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31
Random matrix theory and free probability crash course Free convolution If x , y are free then moments of x + y are uniquely determined by the moments of x and y . Notation We say that x + y is the free convolution of the distribution of x and distribution of y . We write: (1) µ x + y = µ x ⊞ µ y R transform Consider a random variable x and its Cauchy transform G ( z ) . Then 1 (2) G ( z ) + R [ G ( z )] = z Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31
Random matrix theory and free probability crash course Free convolution If x , y are free then moments of x + y are uniquely determined by the moments of x and y . Notation We say that x + y is the free convolution of the distribution of x and distribution of y . We write: (1) µ x + y = µ x ⊞ µ y R transform Consider a random variable x and its Cauchy transform G ( z ) . Then 1 (2) G ( z ) + R [ G ( z )] = z (3) R x + y ( z ) = R x ( z ) + R y ( z ) Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31
Random states and channels Contents Random matrix theory and free probability crash course 1 Random states and channels 2 Eigenvalues of random quatnum states and channels 3 Distances between random quantum states 4 The diamond norm 5 Final remarks 6 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 7 / 31
Random states and channels Random states in Ω N From pure states Consider a random pure state | φ � ∈ X ⊗ Y . 1 Trace out one of the systems ρ = tr Y | φ �� φ | . 2 If dim ( X ) = dim ( Y ) , we get the Hilbert-Schmidt distribution of ρ . 3 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 8 / 31
Random states and channels Random states in Ω N From pure states Consider a random pure state | φ � ∈ X ⊗ Y . 1 Trace out one of the systems ρ = tr Y | φ �� φ | . 2 If dim ( X ) = dim ( Y ) , we get the Hilbert-Schmidt distribution of ρ . 3 From Ginibre matrices Let G be a N × K Ginibre matrix (independent normal complex entries). Then, the matrix ρ = GG † (4) tr GG † , is a random mixed state. If N = K we recover the flat Hilbert-Schmidt distribution. Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 8 / 31
Random states and channels Geometry of the set Ω N Ω 2 ⊂ R 3 - Bloch ball, pure states reside on the sphere Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 9 / 31
Random states and channels Geometry of the set Ω N Ω 2 ⊂ R 3 - Bloch ball, pure states reside on the sphere For N > 2, Ω N is not a Ball! Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 9 / 31
Random states and channels Geometry of the set Ω N Ω 2 ⊂ R 3 - Bloch ball, pure states reside on the sphere For N > 2, Ω N is not a Ball! ∂ Ω 3 Ω 3 Figure: Sketch of a qutrit Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 9 / 31
Eigenvalues of random quatnum states and channels Contents Random matrix theory and free probability crash course 1 Random states and channels 2 Eigenvalues of random quatnum states and channels 3 Distances between random quantum states 4 The diamond norm 5 Final remarks 6 Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 10 / 31
Eigenvalues of random quatnum states and channels Marchenko-Pastur distribution GG † Consider ρ = tr GG † , where G is N × K . Then, the asymptotic eigenvalue x − ( 1 − √ c ) 2 � ( 1 + √ c ) 2 − x , where x = N λ 1 � distribution is MP c ( x ) = 2 π x and c = N / K . √ 4 / x − 1 When c = 1: MP 1 ( x ) = 2 π Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 11 / 31
Eigenvalues of random quatnum states and channels Symmetrized Marchenko-Pastur distribution Trace distance D tr ( ρ, σ ) = 1 2 � ρ − σ � 1 = 1 (5) 2 tr | ρ − σ | We need the eigenvalue distribution of ρ − σ . Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 12 / 31
Eigenvalues of random quatnum states and channels Symmetrized Marchenko-Pastur distribution Trace distance D tr ( ρ, σ ) = 1 2 � ρ − σ � 1 = 1 (5) 2 tr | ρ − σ | We need the eigenvalue distribution of ρ − σ . Free additive convolution Symmetrized Marchenko-Pastur distribution SMP c ( x ) = MP c ( x ) ⊞ MP c ( − x ) . In the case of HS measure ( c = 1), we get: � √ �� 2 / 3 − 1 − 3 x 2 + � 3 + 33 x 2 − 3 x 4 + 6 x 1 + 3 x (6) SMP 1 ( x ) = . √ � √ �� 1 / 3 � 3 + 33 x 2 − 3 x 4 + 6 x 2 3 π x 1 + 3 x Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 12 / 31
Recommend
More recommend
