Valerio Cappellini. References

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1 CETER FOR THEORETICAL PHYSICS OF THE POLISH ACADEMY OF SCIECES WARSAW, POLAD RADOM DESITY MATRICES AD THEIR DETERMIATS 4 30 SEPTEMBER 5 TH SFB TR 1 MEETIG OF 006 I PRZEGORZAłY KRAKÓW Valerio Cappellini References [1] V. Cappellini, H. J. Sommers and K. Życzkowski, Distribution of G concurrence of random pure states, quant-ph/ (006). [] I. Bengtsson and K. Życzkowski, Geometry of quantum states: An introduction to quantum entanglement, Cambridge University Press, Cambridge, 006. [3] K. Życzkowski and H. J. Sommers, Induced measures in the space of mixed quantum states, J. Phys. A 34, (001). Center for Theoretical Physics PAS, Al. Lotników 3/ Warsaw, Poland E mail: valerio@cft.edu.pl

2 OUTLIE(1) 1) Density operators acting on H, purification, distances ) Measure in the set of pure quantum states ) Algorithms for producing random pure states a) Hurwitz parametrization. b) Rescaling random gaussian variables 3) Measures in the set of mixed quantum states a) Measures related to metric a 1 ) Hilbert Schmidt measure a ) Bures measure b) Measure induced by random vectors b 1 ) Unitary product measure b ) Orthogonal product measure c) Measure induced by partial tracing of composite systems 3 ) Algorithms for producing density matrices HS distributed 4) Pure state entanglement & entanglement monotones 5) Mean values: averaging entropy... a)... on the Hilbert Smidth measure b)... on the Bures measure c)... on the Induced measure

3 OUTLIE() 6) Determinants of reduced ρ A and their th roots a) Determinant (D) b) G concurrence (G) 7) Average moments of G concurrence 8) HS Probability distribution P (β) (G) 8 ) Asymptotic behavior of P C (G) for D 0 8 ) Asymptotic behavior of P R (G) for D 0 8 ) Asymptotic behavior of P (β) (G) for D (1/) 9) Asymptotic behavior for P (β) (G) at large 10) Induced distributions P (β) l 1,l (G) at large 11) Concentration of measure and Levy s Lemma OUTLOOK & PERSPECTIVE 3

4 1. Density operators acting on H Define the set M of all density operators ρ which act in an dimensional Hilbert space H and satisfy: :: Hermiticity (ρ = ρ); :: normalization (Tr ρ = 1); :: positivity (ρ 0); Pure states. Are those ρ such that Tr ρ = 1, or equivalently such that ρ = Ψ Ψ Purification. Every mixed state ρ A acting on a finite dimensional Hilbert space H A can be viewed as the reduced state of some pure state living in an extended bipartite Hilbert space H A H B Theorem [Purification]. Let ρ A be a density matrix acting on a Hilbert space H A of finite dimension, then there exist a Hilbert space H B and a pure state Ψ H A H B such that the partial trace of Ψ Ψ with respect to H B Proof: Tr B [ Ψ Ψ ] = ρ A A density matrix is by definition positive semidefinite. square root factorization ρ A = MM. So ρ A has Let H B be another copy of the dimensional Hilbert space and consider the pure state Ψ, living on the bipartite system H = H A H B, represented in a product basis as Ψ = i=1 M i j i A j B, j=1 4

5 Direct calculation gives Tr B [ Ψ Ψ ] = ( ) MM i A A l = ρ A il i,l=1 Since square root decompositions of a positive semidefinite matrix are not unique, neither are purifications. Distances. For any ρ, σ M R 1 one defines the following distances: a) Hilbert Smidth distance, D HS (ρ, σ) = Tr [ (ρ σ) ] ; b) Bures distance, D B (ρ, σ) = Tr ρ 1/ σρ 1/. Example: Mixed states of a two level system: = Figure 1: Space of mixed state of a qubit: a) HS metric induces a flat geometry of a Bloch ball B 3 R 3 ; b) the Bures metric induces a curved space the Uhlmann hemisphere, 1 S3 R 4. One dimension suppressed. 5

6 . Measure in the set of pure quantum states A unique, unitary invariant measure exist (Fubini Study measure), P FS ( Ψ ) = P FS (U Ψ ), Ψ H ;. Algorithms for producing random pure states. a) Hurwitz parametrization. You do need: ( 1) polar angle ϕ k, uniformly distributed in [0, π) ( 1) azimuthal angle ϑ k, distributed in [0, π/] according to P (ϑ) = k sin (ϑ k ) (sin ϑ k ) k, k {1,,..., 1} (In practice: set ϑ k = arcsin ( ) ξ 1/k k, with ξk uniformly distributed in [0, 1]) b) Rescaling random gaussian variables generate Gaussian random C (or R) numbers {c i } i=1 P (c 1, c,..., c ) exp c i rescale such numbers {x i } i=1 dividing by their l norm i=1 y i = c i i=1 c i You got the coordinates {y i } i=1 H = C (or R )!!! of a random pure state Ψ 6

7 3. Measures in the set of mixed quantum states Unitary invariance P (ρ) = P ( UρU ) does not distinguish a single measure. This property is characteristic of all product measures µ = P ( λ ) ν H, where ν H is the Haar measure of U() (or O()), the vector λ represent the spectrum of ρ, while P ( λ ) is any distribution defined in the eigenvalues simplex. Examples a 1 ) Hilbert Schmidt measure :: P (β) (λ HS 1, λ,..., λ ) = C (β) δ 1 λ i i=1 j<k λ j λ k β (1) a ) Bures measure :: P B (λ 1, λ,..., λ ) = C δ ( 1 i=1 λ ) i λ1 λ λ ( ) λ j λ k λ j + λ k enjoy to following property: every ball (in sense of a given metric) of a fixed radius has the same measure. In formula (1), β is the repulsion exponent widely used in RMT. for GUE β = 1 for GOE j<k 7

8 b) Measure induced by random vectors :: we take for the vector of eigenvalues λ the squared moduli of the elements of a column of an auxiliary random unitary matrix V λ i = V i, j, i {1,..., }, j fixed Depending on V we can have: b 1 ) Unitary product measure when V is a random unitary drawn according to the Haar measure ν u of U (); b ) Orthogonal product measure when V is orthogonal drawn according to the Haar measure ν o of O (). equivalently b 1 ) {λ i } are squared moduli of components of a GUE matrix s eigenvector; b ) {λ i } are squared components of a GOE matrix s eigenvector. Both cases are described by the so called Dirichlet distribution P s (λ 1,..., λ ) = α s δ 1 λ i λi s 1, i=1 with the real parameter s 1 for V U() s = 1 for V O() ote that s = β, where β is the repulsion exponent. i s = 1 means {λi } uniformly distributed 8

9 c) Measure induced by partial tracing of composite systems :: In order to generate a random density matrix of size : Construct a composite Hilbert space H = H H K ; Generate generate a random pure state Ψ H according to the natural, FS measure on CP K 1 (or RP K 1 ); Obtain a mixed quantum state by partial tracing, ρ = Tr K ( Ψ Ψ ). The probability distribution induced in this way into the simplex of eigenvalues reads P (β),k ( λ) = C (β),k δ 1 i=1 λ i i λ [ β(k )+β ]/ i j<k λ j λ k β Observe that the induced distribution above can be recasted into the Hilbert Schmidt distribution, here rewritten, P (β) (λ HS 1, λ,..., λ ) = C (β) δ 1 λ i i=1 j<k λ j λ k β, provided that we choose K = 1 + /β, that is for complex ρ ( β = or abbr. C ) K = + 1 for real ρ ( β = 1 or abbr. R ) 9

10 3. Algorithms for producing density matrices HS distributed. The pure state of the bipartite system H = H H K can be represented in a product basis as ψ = i=1 K A i j i j. j=1 generate K Gaussian random C (or R) numbers { A i j } pick a K matrix A, with K suitably chosen, from the Ginibre ensemble of non Hermitian, complex (non symmetric, real) matrices; rescale such numbers { A i j } dividing by their l norm divide A for Tr AA ;... now ψ is FS distributed in H H K... obtain a mixed state by partial tracing, ρ = Tr K ρ = AA / Tr AA ; ( Ψ Ψ ) Random mixed states are produced by generating normalized Wishart matrices AA / Tr AA, with A belonging to the Ginibre ensemble. K = 1 for ρ = Ψ Ψ pure quantum state 1 + /β for ρ mixed quantum state HS distributed 10

11 Example: = 3 P( ) a) P( ) b) Figure : (a) unitary product measure (dotted line), orthogonal product measure (dashed line), HS measure (solid line) and Bures measure (dash-dotted line); (b) measures P,K induced by partial tracing; solid line represents HS measure i.e. K =, dashed line K = 3, dash-dotted line K = 4, dotted line K = 5. Example: = 3 a) (001) b) (001) c) (001) * * * (100) (010) (100) (010) (100) (010) d) (001) e) (001) f) (001) * * * (100) (010) (100) (010) (100) (010) Figure 3: (a) orthogonal product measure, (b) Bures measure, (c) HS measure equal to P () Other measures induced by partial tracing: (d) P () 3,4, (e) P() 3,5 and (f) P() 3,6. 3,3. 11

12 4. Pure state entanglement & entanglement monotones Consider two noninteracting systems A and B, with respective Hilbert spaces H A and H B. The Hilbert space of the composite system is the tensor product H A H B. If the first system is in state ψ A and the second in state φ B, the state of the composite system is ψ A φ B, or ψ A φ B, for short. States of the composite system which can be represented in this form are called separable states, or product states. ot all states are product states. Fix a basis { i A } for H A and a basis { j B } for H B. The most general state in the bipartite overall system H = H A H B = C C K is represented in a product basis as Ψ = i=1 K A i j i A j B, j=1 and admit a Schmidt decomposition Ψ = k=1 λk e (k) A f (k) B a separable state Ψ is characterized by a vector of Schmidt coefficients with only one non vanishing entry. Schmidt coefficients {λ k } coincide with the spectrum of the eigenvalues of the reduced density matrix ρ A = Tr B ( Ψ Ψ ). a state Ψ is separable iff its reduced density matrix is a pure state. 1

13 If a state is not separable, it is called an entangled state. a state is entangled iff its reduced density matrix is a mixed state. For example, given two basis vectors { 0 A, 1 A } of H A and two basis vectors { 0 B, 1 B } of H B, the following is an entangled state: Ψ = 1 ( ) 0 A 1 B 1 A 0 B If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure state. Instead, their states are superposed with one another. In this sense, the systems are entangled. OBS: Here, a local measurement causes a state reduction of the entire system state Ψ, and therefore changes the probabilities for potential future measurements on either subsystems. For example, the density matrix of A for the entangled state discussed above is ρ A = 1 ) ( 0 A 0 A + 1 A 1 A = 1 1 A This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of A for the pure product state ψ A φ B discussed above is ρ A = ψ A ψ A Entanglement Monotones: DEF: Quantity not increasing under the actions (even combined) of local unitaries, that is when the only admitted actions on the overall system are of the kind U A 1 B and/or 1 A U B. 13

14 5. Mean values: averaging entropy To characterize, to what extend a given state ρ is mixed, one may use the moments, Tr ρ k, with any k > 0, or the Von eumann entropy S (ρ) = Tr ρ ln ρ with S (ρ) = 0 for any pure state and S (ρ) = ln for the maximally mixed state. We have computed the following averages over a) Hilbert Smidth measure [ ( )] Tr ρ k = Γ (1 + k) 1 HS 1 k 1 + O, Γ (1 + k) Γ ( + k) S (ρ) HS = ln 1 ( ) ln + O () b) Bures measure Tr ρ k = k [ ( )] Γ [(3k + 1) /] 1 B 1 k 1 + O, Γ [(1 + k) /] Γ ( + k) ( ) ln S (ρ) B = ln ln + O (3) c) Induced measure [Page, 1993] S (ρ),k = K n=k+1 1 n 1 K = ψ (K + 1) ψ (K + 1) 1 (4) K + 1 K (5) Since S (ρ) B < S (ρ) HS, the Bures measure is more concentrated on the states of higher purity, than the Hilbert Schmidt measure. 14

15 6. Determinants of reduced ρ A and their th roots Consider pure states Ψ on the bipartite system and their reduced density matrices H = H A H B = C C K ρ A = Tr B ( Ψ Ψ ), characterized by their spectrum of eigenvalues λ. We now focus on: a) the determinant D of the reduced density matrix ρ A D det ρ A, given by the product of all eigenvalues D = λ i, and i=1 b) its rescaled th root G D 1, called G concurrence, and given by the (rescaled) geometric mean of all λ G = λ i i=1 1 [Gour, 005] Both G and D are important entanglement monotones. We aim to compute mean values and to describe probability distributions for the determinant and G concurrence of random density matrices generated with respect to the flat, Euclidean, HS measure. 15

16 7. Average moments of G concurrence The moments of the G concurrence on the HS probability distribution P (β) (G) are given by G M C = M Γ ( ) Γ ( M + j ) Γ(M+ ) j=1 Γ( j) ( ) G M R = M Γ + ( Γ M + j+1 ) ; (6) ( ) ( ) j=1 j+1 Γ M+ + similar expression have been found for D M (β). Γ Figure 4: Average of G concurrence for complex and real random pure states of a ( + β) system distributed accordingly to the FS measure. The average is computed by means of equation (6); error bars represent the variance of P (β) (G). Dashed line represent the asymptote G = 1/e. 16

17 8. HS Probability distribution P (β) (G) When =, the HS Probability distribution P (β) (G) is given by P C (G) = 3 G 1 G P R (G) = G, G [0, 1] (7) For > we construct the HS distribution from all moments G(β) M. The distribution P (β) (G) is given by an inverse Laplace transformation, consisting in the following integral (see Figure 5): P (β) (G) = +i i dm πi G (1+M) G M (β) (8) The same relation holds between P (β) (D) and DM (β). Figure 5: G concurrence s distributions P (β) (G) are compared for different. The distributions are obtained by performing numerically the inverse Laplace transformation of equation (8). Dashed vertical line is centered in G = 1/e. 17

18 8. Asymptotic behavior of P C (G) for D 0. In this part of the spectrum, the probability P C (D) can be expanded in a power series with some logarithmic corrections, as follows: P C (D) ZC + XC D log D + X C D + O ( D (log D) ) (9) Figure 6: In the first panel, the exact formula for P (β) (G) is compared with a 100 bins histogram of 10 6 G concurrence of complex density matrices distributed accordingly to the HS measure. The other panels shows histograms (for different ) together with the distribution of G concurrence obtained by inverse Laplace transforming (plotted in solid lines). The left asymptote given by eq. (9) is also plotted in dashed line for comparison. Z C = Γ( ) Γ( ) Γ(), X C = Γ( ) Γ( ) Γ() Γ( 1) X C = XC [ + ψ( ) 4 ψ(1) ( )ψ( ) ] OTE P (β) (G) dg = P(β) (D) dd G P(β) (G) = D P(β) (D) From now on formulae and figures will be given indifferently for both G and D distribution, being clear their mutual relation. 18,

19 8. Asymptotic behavior of P R (G) for D 0. The expansion of probability P R (D), corresponding real ρ A of small determinant, is still a power series (plus logarithmic corrections) but the exponents are now semi integer: P R (D) ZR + YR D1 + X R D log D + X R D+ + W R D3 log D + W R D3 + O ( D (log D) ) (10) Figure 7: In this plot we shows a 100 bins histogram of 10 6 G concurrence of a 3 3 real density matrix distributed accordingly to the HS measure. The distribution of G concurrence obtained by inverse Laplace transforming is represented by red line. Blue lines represent the contribution to eq. (10), correspondent to Z R, YR, XR and X R, added one by one. ( Coefficients reads Z R = 1 Γ + ( and, for > 3, we have ( X R = 3 Γ + ) ( ) Γ 3 Γ() Γ( ) { ( X R = X R + ψ 3 Γ ) ) Γ(), YR = π ) 8 3 ψ ( 1 ( Γ ) ψ (1) + 3 ψ ( 3 ( 1 Γ + ) ) 4 ψ ( 4 ) Γ( +1 ) Γ( 1) For = 3 separate calculations yields X R 3 = 1 5! and XR 3 = 0. ) } 19

20 8. Asymptotic behavior of P (β) (G) for D (1/). Stirling approximation P C (D) AC ( log D [ log )( ] 3)/ D ( 3)/! P R (D) AR ( log D [ log )( + 6)/4 ] D ( + 6)/4! Figure 8: In the first panel a 100 bins histogram of 10 8 determinants of 3 3 complex density matrices distributed accordingly to the HS measure is compared with the right asymptote given by equation (11) (plotted in solid line). Same analysis is depicted in the second panel, but for 3 3 real density matrices. log D log 1 D = 1 G P C (G) ÃC (1 G ) ( 3)/ G P R (G) ÃR (1 G ) ( + 6)/4 G 0

21 9. Asymptotic behavior for P (β) (G) at large When the system becomes eventually large, we found the general result G(M) := lim G M = (β) e M (11) that holds for both real and complex density matrices HS distributed. Example (β = ) G M C = M 1 k=0 + k {[ M ( M )] } Γ ( M ) k k k=1 Equation (11) display µ = G (β) = 1/e = σ = G (β) G (β) = 0 The limiting distribution, can be earned by performing the Laplace inverse transform, and reads again for both β = 1,. P (β) (G) lim P (β) (G) = δ(g e 1 ) (1) A similar concentration of reduced density matrices around the maximally mixed state has been recently quantified for bipartite K systems [Hayden et al., 006]. Concerning the G concurrence, we have concentration of the distribution around its mean, although in this case the mean is not coinciding with the maximally mixed state (on which G = 1). 1

22 10. Induced distributions P (β) l 1,l (G) at large The moments of the G concurrence on the induced probability distribution P (β) K 1,K (G) are given by G M C K 1,K = K1 M Γ (K 1 K ) K 1 Γ (K K 1 + j + M/K 1 ) Γ (K 1 K + M) Γ (K K 1 + j) G M R K 1,K = K1 M Γ ( ) K 1 K Γ ( K 1 K + M ) j=1 K 1 j=1 Γ ( ) K K 1 + j + M K 1 Γ ( ) K K 1 + j ; We now consider the following high dimensional joint limit: K 1 = l 1 and K = l, with l 1, l +, l > l 1 and eventually large. In terms of the parameter q l /l 1 Q, we find G(M) lim G M (β) = X l 1,l q M, β {1, } with X q 1 e ( q q 1 ) q 1 The limiting distribution P (l1 ),(l )(G), can be earned as before and reads P (l1 ),(l )(G) lim P (β) l 1,l (G) = δ(g X q ) again for both β = 1,. OBS: q 1 (HS distribution) X q 1/e (previous example) q (large environment) X q 1 (compl. mixed state)

23 11. Concentration of measure and Levy s Lemma Levy s Lemma. Let f : S k R be a function with Lipschitz constant η (with respect to the Euclidean norm) and a point X S k be chosen uniformly at random. Then 1. Pr { f (X) E f ±α} exp ( C 1 (k + 1) α /η ) and. Pr { f (X) m ( f ) ±α} exp ( C (k 1) α /η ) for absolute constant C i > 0 that may be chosen as C 1 = (9π 3 ) 1 and C = (π ) 1. (E f is the mean value of f, m ( f ) a median for f). Trivial example Take f (x 1, x,..., x k+1 ) = x 1. Then Levy s Lemma says that the probability of finding a random point on S k outside a band of the equator of size α approach to 0 as exp ( C 1 (k + 1) α ). (k ) LEVY S LEMMA Every equator is Fat Less trivial example The function f could be a measure, an entangle monotones... In particular, for f = S (ρ A ), the Von eumann Entropy, we have Theorem [Hayden et al., 006]. Let ψ ψ be a random pure state FS distributed on H = H A H B = C C K, with K 3. Then { [ ] } Pr S Tr ( ψ ψ ) < ln α β exp ( (K 1) C 3α ) B (ln ) where β = /K and C 3 = (8π ) 1., 3

24 Outlook & Perspective Main aim of the project proposed consist in: deriving the distribution of eigenvalues induced by differen measures and finding a link to the RMT; analyzing ensemble of random matrices related to different measures in the set M. Concerning the G concurrence: finding its mean value over the set of random pure states together with all higher moments; describing its complete probability distribution over the set of random pure states: showing analytically the effect of concentration of the measure. (G concurrence is likely to be the first measure of pure state entanglement for which such a complete analysis is performed). Moreover, our work may also be considered as a contribution to the Random Matrix Theory: we have found the distribution of the determinants of random Wishart matrices AA, normalized by fixing their trace. 4