Distribution of Bipartite Entanglement of a Random Pure State

Distribution of Bipartite Entanglement of a Random Pure State Satya N. Majumdar Laboratoire de Physique Théorique et Modèles Statistiques,CNRS, Université Paris-Sud, France Collaborators: C. Nadal (Oxford University, UK) M. Vergassola (Institut Pasteur, Paris, France) Refs: Phys. Rev. Lett. 104, 110501 (2010) J. Stat. Phys. 142, 403 (2011) (long version)

Plan Plan: Bipartite entanglement of a random pure state

Plan Plan: Bipartite entanglement of a random pure state Reduced density matrix random matrix theory

Plan Plan: Bipartite entanglement of a random pure state Reduced density matrix random matrix theory Distribution of the Renyi entropy Results Coulomb gas technique for large systems Phase transitions

Coupled Bipartite System A B N dim. M dim. i A α B Coupled Bipartite System N M Bipartite quantum system A B: Hilbert space H A H B subsystem A: dimension N (the system to study) subsystem B: dimension M N ( environment )

Pure State and Reduced Density Matrix A B N dim. M dim. i A α B Coupled Bipartite System N M

Pure State and Reduced Density Matrix A B N dim. M dim. i A α B Coupled Bipartite System N M Any pure state of the full system: ψ >= i,α x i,α i A > α B > X = [x i,α ] (N M) rectangular Coupling matrix

Pure State of Bipartite System ψ >= i,α x i,α i A > α B >

Pure State of Bipartite System ψ >= i,α x i,α i A > α B > If x i,α = a i b α then ψ >= i a i i A > α b α α B >= Φ A > Φ B >

Pure State of Bipartite System ψ >= i,α x i,α i A > α B > If x i,α = a i b α then ψ >= i a i i A > α b α α B >= Φ A > Φ B > Fully separable (factorised)

Pure State of Bipartite System ψ >= i,α x i,α i A > α B > If x i,α = a i b α then ψ >= i a i i A > α b α α B >= Φ A > Φ B > Fully separable (factorised) Otherwise Entangled (non-factorisable)

Pure State of Bipartite System ψ >= i,α x i,α i A > α B > If x i,α = a i b α then ψ >= i a i i A > α b α α B >= Φ A > Φ B > Fully separable (factorised) Otherwise Entangled (non-factorisable) Density matrix of the composite system ˆρ = ψ >< ψ with Tr[ˆρ] = 1

Reduced Density Matrix of subsystem A: A B N dim. M dim. i A α B Coupled Bipartite System N M

Reduced Density Matrix of subsystem A: A B N dim. M dim. i A α B Coupled Bipartite System N M Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [ ψ ψ ] with Tr[ˆρ A ] = 1

Reduced Density Matrix of subsystem A: A B N dim. M dim. i A α B Coupled Bipartite System N M Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [ ψ ψ ] with Tr[ˆρ A ] = 1 Measurement ÔA = Tr[Ô ˆρ A]

Reduced Density Matrix of subsystem A: A B N dim. M dim. i A α B Coupled Bipartite System N M Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [ ψ ψ ] with Tr[ˆρ A ] = 1 Measurement ÔA = Tr[Ô ˆρ A] ˆρ A = Tr B [ˆρ]

Reduced Density Matrix of subsystem A: A B N dim. M dim. i A α B Coupled Bipartite System N M Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [ ψ ψ ] with Tr[ˆρ A ] = 1 Measurement ÔA = Tr[Ô ˆρ A] M ˆρ A = Tr B [ˆρ] = < α B ˆρ α B > α=1

Reduced Density Matrix of subsystem A: A B N dim. M dim. i A α B Coupled Bipartite System N M Reduced Density Matrix: ˆρ A = Tr B [ˆρ] = Tr B [ ψ ψ ] with Tr[ˆρ A ] = 1 Measurement ÔA = Tr[Ô ˆρ A] [ M N M ] ˆρ A = Tr B [ˆρ] = < α B ˆρ α B > = x i,α xj,α i A >< j A α=1 i,j=1 α=1

Eigenvalues of W: In the diagonal representation N N ˆρ A = W i,j i A >< j A λ i λ A i >< λ A i i,j=1 i=1

Eigenvalues of W: In the diagonal representation N N ˆρ A = W i,j i A >< j A λ i λ A i >< λ A i i,j=1 i=1 {λ 1, λ 2,..., λ N } non-negative eigenvalues of W = XX

Eigenvalues of W: In the diagonal representation N N ˆρ A = W i,j i A >< j A λ i λ A i >< λ A i i,j=1 i=1 {λ 1, λ 2,..., λ N } non-negative eigenvalues of W = XX Tr[ˆρ A ] = 1 N λ i = 1 0 λ i 1 i=1

Eigenvalues of W: In the diagonal representation N N ˆρ A = W i,j i A >< j A λ i λ A i >< λ A i i,j=1 i=1 {λ 1, λ 2,..., λ N } non-negative eigenvalues of W = XX Tr[ˆρ A ] = 1 N λ i = 1 0 λ i 1 i=1 M Similarly ˆρ B = Tr A [ˆρ] = W α,β α B >< β B α,β=1

Eigenvalues of W: In the diagonal representation N N ˆρ A = W i,j i A >< j A λ i λ A i >< λ A i i,j=1 i=1 {λ 1, λ 2,..., λ N } non-negative eigenvalues of W = XX Tr[ˆρ A ] = 1 N λ i = 1 0 λ i 1 i=1 Similarly ˆρ B = Tr A [ˆρ] = M α,β=1 W α,β α B >< β B W = X X (M M) matrix with M eigenvalues (recall N M) {λ 1, λ 2,..., λ N, 0, 0,..., 0}

Entanglement N Schmidt decomposition: ψ >= λi λ A i > λ B i > N Reduced density matrix: ˆρ A = Tr B [ ψ >< ψ ] = λ i λ A i >< λ A i i=1 i=1

Entanglement N Schmidt decomposition: ψ >= λi λ A i > λ B i > N Reduced density matrix: ˆρ A = Tr B [ ψ >< ψ ] = λ i λ A i >< λ A i i=1 {λ 1, λ 2,..., λ N } eigenvalues of W = XX with N 0 λ i 1 and λ i = 1 i=1 i=1

Entanglement Schmidt decomposition: ψ >= N λi λ A i > λ B i > i=1 Reduced density matrix: ˆρ A = Tr B [ ψ >< ψ ] = {λ 1, λ 2,..., λ N } eigenvalues of W = XX with N 0 λ i 1 and λ i = 1 (i) Unentangled λ i0 = 1, λ j = 0 j i 0 ψ = λ A i 0 λ B i 0 is separable ˆρ A = λ A i 0 λ A i 0 is pure i=1 N λ i λ A i >< λ A i i=1 (ii) Maximally entangled λ j = 1/N for all j (all eigenvalues equal) ψ is a superposition of all product states ˆρ A = 1 N N i=1 λa i λ A i = 1 N 1 A is completely mixed

Entanglement entropy Subsystem A described by ˆρ A = N i=1 λ i λ A i λ A i Von Neuman entropy: S VN = Tr [ˆρ A ln ˆρ A ] = N i=1 λ i ln λ i

Entanglement entropy Subsystem A described by ˆρ A = N i=1 λ i λ A i λ A i Von Neuman entropy: S VN = Tr [ˆρ A ln ˆρ A ] = N i=1 λ i ln λ i [ Renyi entropy: S q = 1 1 q ln N ] i=1 λq i and Σ q = N i=1 λq i

Entanglement entropy Subsystem A described by ˆρ A = N i=1 λ i λ A i λ A i Von Neuman entropy: S VN = Tr [ˆρ A ln ˆρ A ] = N i=1 λ i ln λ i [ Renyi entropy: S q = 1 1 q ln N ] i=1 λq i and Σ q = N i=1 λq i S q 1 = S VN and S q = ln(λ max ) Σ 2 = exp[ S q=2 ] Purity (i) Unentangled λ i0 = 1, λ j = 0 j i 0 ˆρ A = λ A i 0 λ A i 0 is pure S VN = 0 = S q is minimal (ii) Maximally entangled λ j = 1/N for all j ˆρ A = 1 N N i=1 λa i λ A i mixed S VN = ln N = S q is maximal

A Simple Diagram for N=3 (0,0,1) λ 3 (1/3,1/3,1/3) MOST ENTANGLED S=log(N) (0,1,0) λ 2 λ 1 (1,0,0) LEAST ENTANGLED S=0

Random Pure State: Haar Measure ψ >= x i,α i A > α B > = i,α N λi λ A i > λ B i > i=1

Random Pure State: Haar Measure ψ >= i,α x i,α i A > α B > = N λi λ A i > λ B i > random pure state where X = [x i,α ] are uniformly distributed among the sets of {x i,α } satisfying i,α x i,α 2 = Trρ = 1 i=1 Haar measure Equivalently, for the N M matrix P(X ) δ ( Tr(XX ) 1 ) In the basis i A of H A : ˆρ A = W = XX Distribution of the eigenvalues λ i of ˆρ A? Distribution of the entanglement entropies S VN, S q?

Joint PDF of Eigenvalues the joint pdf P(λ 1, λ 2,..., λ N ) where {λ 1, λ 2,..., λ N } eigenvalues of the Wishart matrix W = XX (X N M Gaussian random matrix) with an additional constraint N λ i = 1 i=1 Joint distribution of Wishart eigenvalues (James 64): [ ( P({λ i }) exp β N λ i] λ β 2 (1+M N) 1 N ) i λ j λ k β δ λ i 1 2 i=1 i i=1 j<k (Llyod & Pagels 88, Zyczkowski & Sommers 2001)

Random Pure State: Well Studied Average von Neumann entropy: S VN ln N N 2M for M N 1 Near Maximal [Page ( 93), Foong and Kanno ( 94), Sánchez-Ruiz ( 95), Sen ( 96)]

Random Pure State: Well Studied Average von Neumann entropy: S VN ln N N 2M for M N 1 Near Maximal [Page ( 93), Foong and Kanno ( 94), Sánchez-Ruiz ( 95), Sen ( 96)] statistics of observables such as average density of eigenvalues, concurrence, purity, minimum eigenvalue λ min etc. [Lubkin ( 78), Zyczkowski & Sommers (2001, 2004), Cappellini, Sommers and Zyczkowski (2006), Giraud (2007), Znidaric (2007), S.M., Bohigas and Lakshminarayan (2008), Kubotini, Adachi and Toda (2008), Chen, Liu and Zhou (2010), Vivo (2010),...]

Results: Entropy Distribution Scaling for large N: λ typ 1 N as N i=1 λ i = 1

Results: Entropy Distribution Scaling for large N: λ typ 1 N as N i=1 λ i = 1 Renyi entropy: S q = 1 1 q ln [Σ q] with Σ q = N i=1 λq i, q > 1

Results: Entropy Distribution Scaling for large N: λ typ 1 N as N i=1 λ i = 1 Renyi entropy: S q = 1 1 q ln [Σ q] with Σ q = N i=1 λq i, q > 1 Σ q = N 1 q s for large N S q = ln N ln s q 1

Results: Entropy Distribution Scaling for large N: λ typ 1 N as N i=1 λ i = 1 Renyi entropy: S q = 1 1 q ln [Σ q] with Σ q = N i=1 λq i, q > 1 Σ q = N 1 q s for large N S q = ln N ln s q 1 (i) Unentangled S q = 0 is minimal Σ q = 1 (or s ) (ii) Maximally entangled S q = ln N is maximal Σ q = N 1 q (or s = 1) Average entropy for large N: Σ q N 1 q s(q) S q ln N ln s(q) q 1 where s(q) = Γ(q+1/2) πγ(q+2) 4 q

Results: pdf of Σ q = i λq i P ( Σ q = N 1 q s ) exp { βn 2 Φ I (s) } exp { βn 2 Φ II (s) } { exp βn 1+ 1 q } ΨIII (s) for 1 s < s 1 (q) for s 1 (q) < s < s 2 (q) for s > s 2 (q) P ( Σ q = N 1 q s ) s(q) (a) ln [ P ( Σ q = N 1 q s )] (b) I II III I II III 1 s 1 (q) s 2 (q) max. entangled unentangled s 1s 1 (q) s(q) s 2 (q) s

Computation of the pdf of Σ q : Coulomb gas method PDF of Σ q = i λq i ( ) ( ) P(Σ q, N) = P(λ 1,..., λ N ) δ λ q i Σ q dλ i. i i

Computation of the pdf of Σ q : Coulomb gas method PDF of Σ q = i λq i ( ) ( ) P(Σ q, N) = P(λ 1,..., λ N ) δ λ q i Σ q dλ i. i i The joint pdf of the eigenvalues can be interpreted as a Boltzmann weight at inverse temperature β: P(λ 1,..., λ N ) δ ( λ i 1 ) i N i=1 exp { βe [{λ i }]} λ β 2 (M N+1) 1 i λ i λ j β i<j where E [{λ i }] = γ N i=1 ln λ i i<j ln λ i λ j (with i λ i = 1) effective energy of a 2D Coulomb gas of charges

Charge Density and Effective Energy E{λ i } = γ N ln λ i ln λ i λ j with λ i = 1 i<j i i=1

Charge Density and Effective Energy E{λ i } = γ N ln λ i ln λ i λ j with λ i = 1 i<j i i=1 Continuous charge density in the large N limit (regimes I, II): {λ i } ρ(x) = 1 δ (x λ i N) N i P ( Σ q = N 1 q s, N ) D [ρ] exp { βn 2 E s [ρ] }

Charge Density and Effective Energy E{λ i } = γ N ln λ i ln λ i λ j with i<j i i=1 λ i = 1 Continuous charge density in the large N limit (regimes I, II): {λ i } ρ(x) = 1 δ (x λ i N) N i P ( Σ q = N 1 q s, N ) D [ρ] exp { βn 2 E s [ρ] } with effective energy E s [ρ] = 1 dx dx ρ(x)ρ(x ) ln x x 2 0 0 ( ) +µ 0 dx ρ(x) 1 0 ( ) ( +µ 1 dx x ρ(x) 1 + µ 2 where µ 0, µ 1 and µ 2 are Lagrange multipliers 0 0 ) dx x q ρ(x) s

Saddle Point Solution E s [ρ]= 1 ( dxdx ρ(x)ρ(x ) ln x x + µ 0 2 0 0 ( ) ( +µ 1 dx x ρ(x) 1 + µ 2 0 0 ) dx ρ(x) 1 0 ) dx x q ρ(x) s

Saddle Point Solution E s [ρ]= 1 ( dxdx ρ(x)ρ(x ) ln x x + µ 0 2 0 0 ( ) ( +µ 1 dx x ρ(x) 1 + µ 2 0 0 ) dx ρ(x) 1 0 ) dx x q ρ(x) s Saddle point method for large N: P ( Σ q = N 1 q s, N ) D [ρ] exp { βn 2 E s [ρ] } exp { βn 2 E s [ρ c ] } where ρ c minimizes the energy: 0 δe s δρ = 0 giving ρ=ρc dx ρ c (x ) ln x x = µ 0 + µ 1 x + µ 2 x q V (x) V (x) effective potential for the charges.

Steps for Computing the PDF of Σ q (i) find the solution ρ c (x) of the integral equation P 0 dx ρ c(x ) x x = µ 1 + q µ 2 x q 1 = V (x) Tricomi s solution: density ρ c (x) with finite support [L 1, L 2 ]: [ 1 L2 ] ρ c (x) = π dy y L1 L2 y C P V (y), x L 1 L2 x π x y where C = L 2 L 1 dx ρ c (x) is a constant (ii) for a given s, the unknown Lagrange multipliers µ 0, µ 1 and µ 2 are fixed by the three conditions: 0 ρ c (x) dx = 1, 0 L 1 x ρ c (x) dx = 1 and 0 x q ρ c (x) dx = s

Phase Transitions Regime I: 1 < s < s 1 (s 1 (q = 2) = 5/4 ) Regime II: s 1 < s < s 2 (s 2 (q = 2) = 2 + 24/3 N 1/3 ) V (x) V (x) V (x) Regime III: s > s 2 (weakly entangled) I II III 0 x x 0 x 0 x (a) (b) (c) ρ c (λ, N) ρ c (λ, N) ρ c (λ, N) I II III L 1 /N L 2 /N λ λ λ 0 L/N 0 ζ t

q = 2 : Purity Regime I: 1 < s < 5/4 ρ c (x) = L2 x x L 1 2π (s 1) P (Σ 2 = s ) N, N e βn2 Φ I (s), Φ I (s) = 1 4 ln (s 1) 1 8,

q = 2 : Purity Regime I: 1 < s < 5/4 ρ c (x) = L2 x x L 1 2π (s 1) P (Σ 2 = s ) N, N e βn2 Φ I (s), Φ I (s) = 1 4 ln (s 1) 1 8 Regime II: 5/4 < s < 2 + 24/3 N 1/3 ρ c (x) = 1 L x ( (A + B x) with L = 2 3 ) 9 4s π x P (Σ 2 = s ) N, N e βn2 Φ II (s), Φ II (s) = 1 ( ) L 2 ln + 6 4 L 2 5 L + 7 8,

Regime III Regime III: s > 2 + 24/3 Continuous density with support [0, ζ] with N 1/3 ζ 4 N and separated λ max ζ: λ max = t s 2 N P (Σ 2 = s ) N, N e βn 3 2 Ψ III (s), Ψ III (s) = s 2 2

Numerical simulations Monte Carlo Simulations (non-standard Metropolis algorithm): pdf of Σ 2 (purity) for N = 1000 (second phase transition) ln P (Σ 2 = s N ) 0.008 βn 2 0.006 0.004 Φ III = Ψ III N 0.002 Φ II 0.000 2.0 2.1 2.2 2.3 2.4 2.5 s(2) s 2 (2) s

Numerical Simulations (2) Jump of the maximal eigenvalue (rightmost charge) at s = s 2 for q = 2 and different values of N N t = N λ max for Σ 2 = s N 20 15 N = 1000 N = 500 10 5 N = 50 0 2.0 2.1 2.2 2.3 2.4 2.5 s s 2 (N = 1000) s2 (N = 500) s

The limit q 1 von Neumann Entropy S VN = S q 1 = N i=1 λ i ln(λ i ) = ln(n) z

The limit q 1 von Neumann Entropy 0 z 01 01 01 01 S VN = S q 1 = N i=1 λ i ln(λ i ) = ln(n) z exp { βn 2 φ I (z) } for 0 z < z 1 = 2 3 ln ( 3 2) = 0.26. (S VN = ln(n) z) exp { βn 2 φ II (z) } for 0.26.. < z < z 2 = 1/2 { } exp β N2 ln(n) φ III (z) for z > z 2 = 1/2 P(S VN = ln(n) z) II III z=0 00 11 01 1 I maximally entangled z 1 =0.26.. 2 = 0.5 z

The limit q Maximal eigenvalue S q = ln (λ max )

The limit q Maximal eigenvalue S q = ln (λ max ) ( P λ max = t ) N exp { βn 2 χ I (t) } for 1 t < t 1 = 4 3 exp { βn 2 χ II (t) } for 4 3 < t < t 2 = 4 exp { βn χ III (t)} for t > t 2 = 4

The limit q Maximal eigenvalue S q = ln (λ max ) ( P λ max = t ) N exp { βn 2 χ I (t) } for 1 t < t 1 = 4 3 exp { βn 2 χ II (t) } for 4 3 < t < t 2 = 4 exp { βn χ III (t)} for t > t 2 = 4 Around its mean, λ max = 4/N, the typical fluctuations ( O(N 5/3 )) are described by the Tracy-Widom distribution λ max = 4 N + 24/3 N 5/3 Y β Y β random variable distributed via Tracy-Widom law

Summary and Conclusions Exact distribution of the Renyi entanglement entropy S q for a random pure state in a large bipartite quantum system (M N 1)

Summary and Conclusions Exact distribution of the Renyi entanglement entropy S q for a random pure state in a large bipartite quantum system (M N 1) Coulomb gas method saddle point solution

Summary and Conclusions Exact distribution of the Renyi entanglement entropy S q for a random pure state in a large bipartite quantum system (M N 1) Coulomb gas method saddle point solution 2 critical points direct consequence of 2 phase transitions in the associated Coulomb gas at the second critical point: λ max suddenly jumps to a value N (1 1/q) much larger than the other λ i 1/N While the average entropy of a random pure state is indeed close to its maximal value ln N, the probability of occurrence of the maximally entangled state (all λ i = 1/N) is actually very small.

References On the large N distribution of the Renyi entropy: C. Nadal, S.M. and M. Vergassola, Phys. Rev. Lett. 104, 110501 (2010) long version: J. Stat. Phys. 142, 403 (2011) On the distribution of the minimum Schmidt number λ min : S.M., O. Bohigas and A. Lakshminarayan, J. Stat. Phys. 131, 33 (2008) see also S.M. in Handbook of Random Matrix Theory (ed. by G. Akemann, J. Baik and P. Di Francesco and forwarded by F.J. Dyson) (Oxford Univ. Press, 2011) (arxiv:1005.4515)