Uncertainty Relations, Unbiased bases and Quantification of Quantum Entanglement Karol Życzkowski in collaboration with Lukasz Rudnicki (Warsaw) Pawe l Horodecki (Gdańsk) Jagiellonian University, Cracow, & Academy of Sciences, Warsaw see preprint arxiv:1106.2018 43 SMP, Toruń, June 21, 2011 KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 1 / 23
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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 2 / 23
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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 3 / 23
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) an S B (ψ) 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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 4 / 23
Complex Hadamard matrices A complex Hadamard matrix H satisfies H ij 2 = 1/N for all i, j = 1,...N. Example: Fourier matrix, F (N) jk = 1 N exp(i2πjk/n) Classification of complex Hadamard matrices is complete up to N = 5 (Fourier F (5) only), while the case N = 6 is still open! Less trivial example: the complex Hadamard matrix H B of Björk (1995) 3 with an unimodular complex number d = 1 3 2 + i 2 1 1 1 1 1 1 1 1 d d 2 d 2 d H B = 1 d 1 1 d 2 d 3 d 2 1 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 KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 5 / 23
Exemplary 1-d family of complex Hadamard matrices 1 1 1 1 1 1 1 1 1/x y y 1/x B (1) 6 (y) = 1 x 1 y 1/z 1/t 1 1/y 1/y 1 1/t 1/t 1 1/y z t 1 1/x 1 x t t x 1 by Beauchamp & Nicoara (2006). Here y = e iφ is a free parameter while x(y) = 1 + 2y + y2 ± 2 1 + 2y + 2y 3 + y 4 1 + 2y y 2 z(y) = 1 + 2y y2 y( 1 + 2y + y 2 ) ; t(y) = xyz More examples for N = 2,...,16 in Catalog of complex Hadamard matrices Tadej & Życzkowski OSID, 13, 133-177 (2006) and its updated online version at http://chaos.if.uj.edu.pl/ karol/hadamard KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 6 / 23
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) see a review on MUBs T. Durt et al. Int.J Quantum Information 8, 535-640 (2010). KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 7 / 23
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, KŻ (IF UJ/CFT PAN ) Wu, Quantifying Yu, Mølmer Quantum Entanglement 2009. June 21, 2011 8 / 23.
Open Systems & Information Dynamics Editorial Board Meeting - this afternoon! KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 9 / 23
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 ( 00 + 11 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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 10 / 23
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) generalized - - E q (φ) = S q ( λ) (Renyi/Tsallis entropies) concurrence C = 2(1 m λ2 m) (function of E 2 (or purity) Symmetric functions, e.g. µ 2 = m n λ mλ n. Multipartite systems Geometric entanglement measure ( 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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 11 / 23
Entanglement of two real qubits Entanglement entropy at the thetrahedron of N = 4 real pure states KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 12 / 23
Mixed states Definition of Entanglement separable mixed states: ρ sep = j p j ρ A j ρ B j entangled mixed states: all states not of the above product form. Entanglement measures for mixed states: generalization of a measure M from pure states to mixed states: M(ρ) := min E i p im( ψ i ) (convex roof) where ensemble E = {p i, ψ i } such that ρ = i p i ψ i ψ i. Examples: Entropy of formation, E(ρ) = ; min E i p ie( ψ i ) Concurrence of formation, C(ρ) = min E i p ic( ψ i ) KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 13 / 23
Measurable entanglement measures measuring purity Trρ 2 two copies in a coincidence experiment higher moments: Trρ k k copies in a coincidence experiment P. Horodecki, A. Ekert, 2002 Measurable bounds for concurrence purity difference (Mintert, Buchleitner, 2007) C(ρ) Trρ 2 Trρ 2 A and C(ρ) Trρ 2 Trρ 2 B where ρ A = Tr B ρ and ρ B = Tr A ρ are partial traces... Generalization for other maps by Augusiak, Lewenstein 2009 Entanglement witness State ρ is entangled if there exists an observable W (entanglement witness) such that for any separable σ one has TrWσ 0 and w := TrWρ < 0 In general, the smaller w (more negative), the larger entanglement (but this approach depends on the choice of W) KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 14 / 23
Reports on Mathematical Physics Editorial Board Meeting - this afternoon! KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 15 / 23
A collective entanglement measure 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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 16 / 23
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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 17 / 23
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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 18 / 23
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 2 00 + 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 2. 1 0.8 0.6 0.4 0.2 0.2 Π 6 Π 3 Π 2 2Π 3 5Π 6 0.4 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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 19 / 23
The number G 12 2 is equal to the probability of the pair of the KŻ (IFclicks UJ/CFT PAN at ) B multiplied Quantifyingby Quantum thatentanglement of a double clickjune at21, A. 2011 20 / 23 Experimental setup with photons: measurement of G 12 2 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.
Quantum network to measure G 12 2 exploiting two copies of an analyzed state Ψ AB, a control qubit c initially in state 0, controlled SWAP gate and two Hadamard gates. Mean value of Pauli σ z matrix of the controlled qubit c is measured under the condition that the chosen pair (i, j) of results is obtained in measurement of the same observables performed on both qubits B. Purity assumption for Ψ AB may be dropped at a price of performing two variants of the experiment in two complementary settings... KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 21 / 23
Collectibility for three qubit system, K = 3 and N = 2 The collectibility is maximal for a a) GHZ state, GHZ := 1 2 ( 000 + 111 ), and then Y max [ GHZ ] = 16/64=1/4, while for b) W state, W := 1 3 ( 001 + 010 + 100 ) 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/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 22 / 23
Concluding remarks 1 We introduced collectibility Y max (Ψ) as a function of any pure state Ψ of a composed N K system, 2 Collectibility satisfies inequalities analogous to entropic uncertainty relations 3 The partial collectibility Y a (Ψ) labeled by parameters describing positions of detectors a B j,..., a K j is experimentally accessible for any K qubit system, 4 An experimental photonic scheme based on Hong Ou Mandel interferometry is proposed to measure collectibility in a two-qubit system. 5 Results for collectibility, presented here for pure states only, can also be generalized for non-ideal pure states with purity less then one, Trρ 2 > 1 ǫ. 6 Sufficiently large value of Y a (Ψ) detects entanglement in a bi-partite system and genuine three party entanglement for an analyzed state Ψ of a three partite system. KŻ (IF UJ/CFT PAN ) Quantifying Quantum Entanglement June 21, 2011 23 / 23