Entropic Uncertainty Relations, Unbiased bases and Quantification of Quantum Entanglement

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1 Entropic Uncertainty Relations, Unbiased bases and Quantification of Quantum Entanglement Karol Życzkowski in collaboration with Lukasz Rudnicki (Warsaw) Pawe l Horodecki (Gdańsk) Phys. Rev. Lett. 107, (2011) IF UJ, Kraków, June 4, 2012 KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

2 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 = 1 the latter form implies the former. KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

3 Entropic Uncertainty Relations 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 q (x) := 1 1 q ln dx ψ(x) 2q Bia lynicki-birula, (2006) KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

4 Entropic Uncertainty Relations - N dimensional case State φ = N i a i α i = j b j β j is expanded in the eigenbases of operators A and B, related by a unitary matrix U ij = α i β j. Let Shannon entropies in both expansion be S A (ψ) = N i=1 a i 2 ln a i 2 and S B (ψ) = N j=1 b j 2 ln b j 2 and 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 lnc by Maassen, Uffink, (1988), who got more( general result for the Renyi entropies, Sq A := 1 1 q ln N ) i=1 a i 2q of order q and q = q/(2q 1) which reads Sq A (ψ) + Sq B (ψ) 2 lnc. KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

5 The result of Deutsch is based on the following bound obtained by variational calculus a i b j (1 + α i β j ) 2 = (1 + U ij ) 2 /4. Above bound can be generalized Assume that φ H and dim H = N. Select N orthonormal vectors: χ j H, j = 1,...,N, χ i χ j = δ ij Repeating the reasoning of Deutsch we show that for any pure state φ N φ χ j 2 N N j=1 Taking N = 2 and assuming orthogonality, α i β j = 0, we recover the previous result of Deutsch, as required... KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

6 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. (1) then C = max jk U jk = 1/ N. The bound of Maassen, Uffink gives S A (ψ) + S B (ψ) 2 ln C = ln N If ψ = (1, 0,...,0) then S A = 0 and S B = lnn so bound is saturated... Two bases which differ by a unitary matrix U such that C = max jk U jk = 1 N are called mutually unbiased KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

7 Complex Hadamard matrices A complex Hadamard matrix H satisfies H ij 2 = 1/N for all i, j = 1,...N. Such matrices exist for any N. Example: Fourier matrix, F (N) jk = 1 N exp(i 2πjk/N) For any complex Hadamard matrix one has C(H) = max jk H jk = 1/ N so the Maassen Uffink bound reads S A (ψ) + S B (ψ) ln N KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

8 Complex Hadamard matrices for N = 6 Classification of complex Hadamard matrices is complete up to N = 5 (there exists Fourier F (5) and equivalent matrices only... Example: complex Hadamard matrix C 6 of Björk (1995) 3 with an unimodular complex number d = i C 6 = d d 2 d 2 d 1 d 1 1 d 2 d 3 d d 2 d 2 1 d 2 d 2 1 d 2 d 3 d 2 1 d 1 d 1 d 2 d 2 d 1 1 More examples for N = 2,...,16 in Catalog of complex Hadamard matrices w. Tadej & K. Ż. OSID, 13, (2006) and its updated online version (W. Bruzda et al.) at karol/hadamard KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

9 Real Hadamard matrices A Real Hadamard matrix H satisfies H ij 2 = 1/N for all i, j = 1,...N with all entries real, H ij = ±1/ N. Real Hadamard matrices exist for N = 2, 4, 8, 12, 16,...,664. Hadamard conjectured that it is true for any multiple of four, N = 4k Example: N = 2 real Hadamard (Fourier) matrix H 2 = F (2) = 1 2 [ ]. (2) For N = 4 the tensor product does the job, H 4 = H 2 H 2 which differs from F (4) The same construction works for N = 8, since H 8 = H 2 H 2 H 2 but the case N = 12 is more diffcult... KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

10 Mutually unbiased bases (MUB) Let { α i } N i=1 and { β j } N j=1 be two orthonormal basis (e.g. given as eigenbasis of hermitian operators A and B). The bases are called maximally unbiased if all entries of the unitary transition matrix U have the same modulus, U ij 2 = α i β j 2 = 1 N, i.e. U is a complex Hadamard matrix. A set of d bases which satisfies this condition is called mutually unbiased Basic facts. 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 = p k (Ivanovic 1981; Wootters & Fields 1989) For any N 2 there exists at least a triple of MUBs, for N = 6 = 2 3 only 3 MUBs are known (so this is the first open case! as the upper bound is d = 7) a review on MUBs: T. Durt, B.-G. Englert, I. Bengtsson, K. Ż., Int.J. Quantum Information 8, (2010). KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

11 Entropic relation for two unbiased bases Let A = ½ and B be a complex Hadamard matrix which implies C 2 = max ij U ij 2 = 1 N so these two bases are maximally unbiased. Then the Maassen Uffink entropic uncertainty relation reads S A (ψ) + S B (ψ) lnn Entropic relations for several unbiased bases Let N = p k and {A 0,...,A N } form a complete set of d = N + 1 MUBs. Expanding ψ in m th basis A m = {αj m } N j=1 we get the entropy S(m). Then N+1 m=1 S (m) (ψ) (N + 1)[ln(N + 1) 1] N even N N+2 2 ln(n/2) + 2 ln[(n + 2)/2] N odd Ivanović 1992 and Sánchez-Ruiz 1993, 1998 generalization for mixed states, improvements for d < N + 1, Wu, Yu, Mølmer KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20.

12 Composed systems & entangled states bi-partite systems: H = H A H B separable pure states: ψ = φ A φ B entangled pure states: all states not of the above product form. Two qubit system: N = 2 2 = 4 ) Maximally entangled Bell state ϕ + := 1 2 ( Entanglement measures For any pure state ψ H A H B define its partial trace σ = Tr B ψ ψ. Definition: Entanglement entropy of ψ is equal to von Neumann entropy of the partial trace E( ψ ) := Tr σ lnσ The more mixed partial trace, the more entangled initial pure state... KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

13 Pure States Bi-partite systems: ψ H = H A H B Schmidt decomposition ψ = A ij i j = N m=1 λm m m Entanglement measures: functions of the Schmidt vector λ = (λ 1 λ 2 λ N ) Entanglement entropy E(φ) = S( λ) (Shannon entropy) Multipartite systems Geometric entanglement measure Brody, Hughston, 2001; Wei, Goldbart, 2003 G(ψ) = 1 max χsep ψ χ sep 2 a distance to the closest separable state χ sep = φ A φ B φ K KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

14 A collective entanglement measure for K qunits Maximal collectibility for a K partite pure state Let Ψ H = H A H B... H K, where all dimensions are equal, dim ( H J) = N. Select N separable pure states, χ sep j = aj A... ak j, where aj J HJ with j = 1,...,N and J = A,...,K. The states are mutually orthogonal, aj J aj k = δ jk for J = A,...,K. Define the maximal collectibility Y max [ Ψ ] := max χ sep N j=1 Ψ χ sep j 2. In Geometric entanglement measure G(Ψ) we look for the largest projection onto a single product state so the overlap Ψ χ sep 2 is smallest for a highly entangled state. In collectibility Y (Ψ) the product of N overlaps, Ψ χ sep 1 2 Ψ χ sep N 2 is largest for a highly entangled state Ψ. KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

15 Upper bound for collectibility For any pure state Ψ we show the following bound Y max [ Ψ ] N N. Setting Z max = lny max this relation takes the from Z max [Ψ] N ln N analogous to the entropic uncertainty relation. It is saturated for the maximally entangled state, Ψ + = 1 N i i, i (bi-partite case) and a generalized GHZ state GHZ K = 1 N i i A i K in K partite case. Collectibility for separable states For any separable state Ψ sep the following bound holds Y max [ Ψ sep ] N N K = A separability criterion: If Y max ( Ψ ) N NK then the state Ψ is entangled. KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

16 A partial collectibility for a general N K system To find Y max [ Ψ ] we need to optimize over a base consisting of N separable states χ sep j Start with a single optimization over the subspace H A, and define the N j=1 Ψ χsep j 2 partial collectibility, Y a [ Ψ ] := max a A parameterized by the set a of N product states aj B... aj K, with j = 1,...,N. By construction one has: max a Y a [ Ψ ] = Y max [ Ψ ]. Collectibility for two qubit system, K = 2 and N = 2 Y a [ Ψ ] = 1 ( G11 ) 2 G 22 + G 11 G 22 G 12 2, 4 where G jk = ϕ j ϕ k is a Gram matrix among projected states, so that ϕ j = a B j Ψ AB H A. KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

17 Collectibility for two qubit system, K = 2 and N = 2 Write a bi-partite pure state in its[ Schmidt ( ) form ) ] Ψ AB = (U A U B ) cos ψ sin( ψ 2 11 Direct optimization gives its collectibility Y max (ψ) = [1 + sin(ψ)] 2 /16. We get also Y min (ψ) = sin 2 (ψ)/4 and the mean value Ȳ (ψ) averaged over random position of the detector bases a1 B and ab Π 6 Π 3 Π 2 2Π 3 5Π Maximal (red), average (green) and minimal (blue) values of the rescaled (partial) collectibility, [16Y (ψ) 1]/3. Positive values identify entanglement. Π Ψ KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

18 Experimental setup: measuring G 12 2 with photons On the left side B the statistics of pairs of clicks after projections onto detectors are measured, on the right side A the Hong Ou Mandel interference is performed. The number G 12 2 is equal to the probability of the pair of the clicks at B multiplied by that of a double click at A. KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

19 Collectibility for three qubit system, K = 3 and N = 2 The collectibility is maximal for a a) GHZ state, GHZ := 1 2 ( ), and then Y max [ GHZ ] = 16/64=1/4, while for b) W state, W := 1 3 ( ) it reads Y max [ GHZ ] = 9/64. For a c) bi separable state, BS = Ψ AB φ C one has Y max [ BS ] = 4/64 = 1/16, while for d) separable state the collectibility reads, Y max [ Ψ sep ] = 1/64. Collectibility as a detector of the genuine entanglement Thus any measured value of Y max [ Ψ ] above 1/16 provides an evidence for genuine three party entanglement for the analyzed state Ψ! KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20

20 Concluding remarks 1 Entropic uncertainity relations can be formulated for various setups 2 In the case of N dimensional space H N and two measurements the bound for the sum of the two entropies is the largest if the eigenbases of both observables are unbiased (unitary transision matrix U ij is complex Hadamard). 3 Entropic inequality of Deutsch can be generalized for N orthogonal states living in H N. This allows us to introduce collectibility Y max (Ψ) a function of any pure state Ψ of a composed N K system which characterizes its entanglement. 4 An photonic scheme based on Hong Ou Mandel interferometry is proposed to measure collectibility in a two-qubit system. Entanglement can be characterized by coincidence measurements in a four-photon experiment. L. Rudnicki, P. Horodecki, K. Ż., Phys. Rev. Lett. 107, (2011). KŻ (IF UJ) Quantifying Quantum Entanglement June 4, / 20