Ponownie o zasadach nieoznaczoności:

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1 .. Ponownie o zasadach nieoznaczoności: Specjalne seminarium okolicznościowe CFT, 19 czerwca 2013 KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

2 Majorization Entropic Uncertainty Relations Karol Życzkowski in collaboration with Zbigniew Pucha la (Gliwice, Cracow) Lukasz Rudnicki (Warsaw, Freiburg) arxiv:quant-ph/ and J. Phys. A (2013) Okolicznościowe Seminarium CFT, Warszawa, 19 czerwca 2013 KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

3 Uncertainty Relations in Quantum Theory Heisenberg uncertainty relation (1927) for the product of variances of position and momentum ( = 1) A more general 2 x 2 p 1 4. formulation of Robertson (1929) for arbitrary operators A an B. Let 2 A = ψ A 2 ψ ψ A ψ 2 be the variance of an operator A. Then for any state ψ 2 A 2 B 1 ψ AB BA ψ 2 4 As [x, p] = xp px = i the latter form implies the former bound. KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

4 Entropic Uncertainty Relations (EUR) Continuous case Define continuous (Boltzmann Gibbs) entropies: and Then S(x) = dx ψ(x) 2 ln ψ(x) 2 S(p) = dp ψ(p) 2 ln ψ(p) 2. S(x) + S(p) ln(eπ). Bia lynicki-birula, Mycielski (1975) and Beckner, (1975) generalizations for Rényi entropies, ( ) S α (x) := 1 1 α ln dx ψ(x) 2α Bia lynicki-birula, (2006) = 2051 KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

5 Our paper is dedicated to Prof. Iwo Bialynicki Birula, (born June 14, 1933), on the occasion of his 80 th birthday KZ (CFT/UJ) Entropic uncertainty relations June 19, / 20

6 Our paper is dedicated to Prof. Iwo Bialynicki Birula, (born June 14, 1933), on the occasion of his 80 th birthday KZ (CFT/UJ) Entropic uncertainty relations June 19, / 20

7 Entropic Uncertainty Relations - N dimensional case State ψ = N i a i i = j b j β j is expanded in the eigenbases of operators A and B, related by a unitary matrix U ij = i β j. Let Shannon entropies in both expansion be S A (ψ) = N i=1 p i ln p i = S(p) with p i = a i 2, i p i = 1 and S B (ψ) = N j=1 q j ln q j = S(q) with q j = b j 2, j q j = 1. Let c(a, B) = max ij U ij. Then for any state ψ H N we have S A (ψ) + S B (ψ) 2 ln[(1 + c)/2] Deutsch, (1983), later improved S A (ψ) + S B (ψ) 2 ln c by Maassen, Uffink, (1988), KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

8 Entropic Uncertainty Relations - N dimensional case Example: the Fourier matrix F N Unitary matrix which defines the second basis U jk = (F N ) jk := 1 N exp(i 2πjk/N) with j, k = 0, 1,..., n 1. then c = max jk U jk = 1/ N. The bound of Maassen Uffink gives S(p) + S(q) 2 ln c = ln N If ψ = (1, 0,..., 0) then S A = 0 and S B = ln N so bound is saturated... The same bound holds for any unitary complex Hadamard matrix H, for which H ij 2 = 1/N for all i, j = 1,... N. see online Catalogue of Complex Hadamard matrices karol/hadamard KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

9 Warm up: a classical analogue of M U bounds Classical map: stochastic matrix T Let p be a probability vector of size N, and its image is q = Tp where T is a stochastic matrix, T ij 0 and i T ij = 1. we show that the sum of both entropies is bounded from below by a function of the largest entry of the transition matrix T, where κ = max ij T ji S(p) + S(Tp) ln κ, Proof is based on an inequality by S lomczyński (2002) concerning the mean entropy of a stochastic matrix T and a probability vector p, S (p) (T ) S(Tp) S (p) (T ) + H(p). Here S (p) (T ) = i p is(t i ), while S(T ) = i 1 N S(t i), KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

10 Back to quantum case... The Maassen Uffink bounds are strong, but not optimal. How to improve them?? A useful tool: the Rényi entropy S α (p) = 1 1 α ln( pi α ), where α 0 is a free parameter. In the limit α 1 the Rényi entropy tends to the Shannon entropy, lim α 1 S α (p) = S(p) =: S 1 (p). The Rényi entropy S α (p) is a monotonously decreasing function of the Renyi parameter α. A simple relation S (p) = ln(p max ) allows us to rewrite the Maassen Uffink bound: S 1 (p) + S 1 (q) max j [ S (v j ) ], where columns of the transition matrix read v j = T jm = U jm 2, j = 1,..., N KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20 i

11 A first observation a) Take a base vector ψ = k, so that p = (0,.., 1, 0,..0) and S(p) = 0. b) Then U ψ = j U ij j, so q j = β j ψ 2 = U jk 2 and q = ( U 1k 2,..., U N,k 2 ) = v k c) In this case one has : S(p) + S(q) = 0 + S(q) = S(v k ) A first attempt (to use several elements of unitary U) The above observation sugests one to [ analyze a relation (with α = 1) S 1 (p) + S 1 (q) max i S1 (v i ), S 1 (v i )], We use here the columns v i = T ij, and the rows, v i = T ji, of the unistochastic transition matrix T ij = U ij 2 This relation is false, but its weaker form [ with α = 2 S 1 (p) + S 1 (q) max i S2 (v i ), S 2 (v i )], holds for N = 2 and N = 3 (numerical results) However, it fails for N 4! Grudka, Horodecki 3, K lobus, Pankowski (2012) arxiv: KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

12 Another approach: Key ingredients used A) An algebraic tool: Majorization Consider two probability vectors of length N ordered decreasingly, x = (x 1 x 2... x N 0) and y = (y 1 y 2... y N 0). The vector x is called to be majorized by y, written x y, if m i=1 x i m i=1 y i, for m = 1,... N 1 Majorization x y implies inequalities for Renyi entropies S α (x) S α (y) (and other Schur concave functions) B) Bi-entropy and product probability vectors Let p q = (p 1 q 1, p 1 q 2,..., p 1 q N,... p N q N ) denotes a product probability vector of size N 2. Then the bientropy reads S α (p) + S α (q) = S α (p q). To arrive at an entropic uncertainty relation we need to find a vector R majorizing p q. KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

13 First step, (k = 1): The bound of Deutsch reused Deriving his uncertainty relation Deutsch proved in 1983 that max p iq j = max ψ,i,j ψ,i,j where as before c = max ij U ij. A direct implication The Rényi entropies are Schur-concave, so ( ) pi + q 2 ( ) j 1 + c 2 = =: R 1, 2 2 (p q) (R 1, 1 R 1, 0,... 0) =: (Q 1, Q 2,..., Q N 2). which implies a first bound for the entire family of Rényi entropies S α S α (p) + S α (q) S α (Q) = 1 1 α ln [Rα 1 + (1 R 1 ) α ]. For α = 1 we get a stronger result than the original bound of Deutsch. KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

14 Example: Bounds for N = 2 orthogonal matrix O For an orthogonal ( matrix ) of size N = 2 cos θ sin θ O(θ) = we plot sin θ cos θ bounds for the Rényi binetropy S α (p) + S α (q) B = S α (R 1, 1 R 1 ) Bounds for equivalent unitary matrices are equal Two unitary matrices U and V are equivalent, U V, if there exist permutation matrices P 1, P 2 and diagonal unitary matrices D 1, D 2, such that V = P 1 D 1 UD 2 P 2 Lemma 1. Massen-Uffink and Majorization bounds for equivalent unitaries are the same. Lemma 2. Any N = 2 unitary matrix is equivalent to an orthogonal matrix, U(2) U O(θ). KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

15 Second step (k = 2) k = 2: norms of 2-subvectors of unitary U To improve the former result we look for stronger majorization of the type (p q) (R 1, R 2 R 1, 1 R 2, 0,... 0). (1) Consider the { longest 2 sub vector of unitary U and denote its norm by } s 2 = max max i,j1,j 2 Uij1 2 + U ij2 2, max i1,i 2,j Ui1 j 2 + U i2 j 2 Our main theorem implies that the above majorization relation with R 2 = ( 1+s 2 ) 2 2 holds! Example: On orthogonal matrix U U(4) with entries truncated to two decimal digits KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

16 Example: Bounds for a family of N = 3 unitary matrices Consider an N = 3 permutation matrix P = EUR for unitary matrices P β, where β [0, 1] for Shannon binetropy: S(p) + S(q) S(R 1, R 2 R 1, 1 R 2 ) S(R 1, 1 R 1 ). Majorization EUR: a comparison with Maassen - Uffink Bistochastic triangle B = Convhull(P, P 2, P 3 = I). Corresponding unitary matrices, U ij 2 = B ij, belong to 3-hypocycloid. Blue region: the MU bound is weaker than majorization bound. KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

17 Third step, (k = 3) k = 3: spectral norms of submatrices of unitary U For any matrix X we define its spectral norm by the largest singular value, X = σ max (X ) = max[ λ(xx ]. Let A m,n denote the maximal m n submatrix of U (with largest norm). Define s 3 := max { A 1,3, A 2,2, A 3,1 } and R 3 := ( 1+s 3 2 Then a stronger majorization relation holds (p q) (R 1, R 2 R 1, R 3 R 2, 1 R 3, 0,... 0). (2) Example: the same orthogonal matrix U U(4) with entries truncated to two decimal digits ) 2. KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

18 Example: Bounds for a family of N = 4 unitary matrices ( ) Consider an N = 4 permutation matrix P 4 = Entropic Uncertainty Relations for unitary matrices P β 4, where β [0, 1] for Shannon binetropy: S(p) + S(q) S(R 1, R 2 R 1, R 3 R 2, 1 R 3 ) S(R 1, R 2 R 1, 1 R 2 ) S(R 1, 1 R 1 ). ( ) 2 with R k = 1+sk 2 KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

19 The main theorem: Majorization EUR Let k = 1,... N 1: spectral norms of all submatrices of unitary U Let A m,n denote the maximal m n submatrix of U. Define s k := max { A 1,k, A 2,k 1,..., A k 1,2, A k,1 }. ) 2 We have s k s k 1 and R k := Rk 1. Theorem: For any unitary U of order N the following majorization relation holds: ( 1+sk 2 (p q) (R 1, R 2 R 1,..., R N 1 R N 1, 1 R N 1 ) =: Q. (3) This implies an explicit majorization entropic uncertainty relation S α (p) + S α (q) S α (Q) = 1 1 α Qi α. ln N2 i=1 KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

20 The main theorem: sketch of the proof Bound of sum R k of k largest components of vector Q majorizing p q, which contains terms (p 1 q 1, p 1 q 2, p 2 q 1,,..., p N q N ) ordered decreasingly ( ) 2 k m m k m max p i 1 m q j max p i + q j. (4) 1 m k 1 1 m k 1 4 i=1 j=1 i=1 j=1 and use the fact xy (x + y)/2. To complete the proof we use an algebraic lemma Let 1, 2,..., m and a 1, a 2... a n be two orthonormal sets of vectors, then m n i ψ 2 + a j ψ 2 = 1 + σ 1 (A), max ψ i=1 where σ 1 (A) is the leading singular value of rectangular matrix A = {A ij } m,n i=1,j=1 for A ij = a i j. One can find ψ such that both terms above are equal = x = y and bound (4) can be saturated. j=1 KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

21 Concluding remarks A classical analogue of Maassen-Uffink bound is derived Main result: Majorization Entropic Uncertainty Relation derived for any unitary U U(N) in terms of spectral norms of its maximal submatrices. For a random unitary matrix U the majorization EUR are stronger than Maassen Uffink relations with probability for N = 2 and 0.971, 0.972, 0.983, 0.990, for N = 3, 4, 5, 6 respectively. Majorization technique allows us to derive inequalities for Rényi entropies and other Schur-concave functions. Majorization EUR for equivalent unitary matrices are equal. Possible applications: pure states entanglement criteria of Gühne Lewenstein (2004) Analogous results obtained recently by Friedland, Gheorghiu and Gour, preprint arxiv: KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

22 Foto: A. Kobos KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20

23 Foto: A. Kobos Panie Profesorze: Sto lat! (lub jeszcze wiecej!) KŻ (CFT/UJ) Entropic uncertainty relations June 19, / 20