Random Matrix Theory in Quantum Information

Transcription

1 Random Matrix Theory in Quantum Information Some generic aspects of entanglement Université Claude Bernard Lyon 1 & Universitat Autònoma de Barcelona Cécilia Lancien BMC & BAMC Cambridge Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 1 / 25

2 Outline 1 Introduction 2 Interlude : reminder about GUE and Wishart matrices 3 Mean-width of the set of states which are separable vs satisfying a separability criterion 4 Entanglement vs Violating a separability criterion for random states 5 Summary and perspectives Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 2 / 25

3 Why Random Matrix Theory in Quantum Information? State of a quantum system : Positive and trace 1 operator ρ (density operator) on a Hilbert space H (state space). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 3 / 25

4 Why Random Matrix Theory in Quantum Information? State of a quantum system : Positive and trace 1 operator ρ (density operator) on a Hilbert space H (state space). Transformation on a quantum system : Completely Positive and Trace Preserving map φ : X B(H) Tr E (VXV ) B(H ) for some isometry V : H E H (quantum channel). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 3 / 25

5 Why Random Matrix Theory in Quantum Information? State of a quantum system : Positive and trace 1 operator ρ (density operator) on a Hilbert space H (state space). Transformation on a quantum system : Completely Positive and Trace Preserving map φ : X B(H) Tr E (VXV ) B(H ) for some isometry V : H E H (quantum channel). In finite dimension : Random states or transformations Random matrices. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 3 / 25

6 Why Random Matrix Theory in Quantum Information? State of a quantum system : Positive and trace 1 operator ρ (density operator) on a Hilbert space H (state space). Transformation on a quantum system : Completely Positive and Trace Preserving map φ : X B(H) Tr E (VXV ) B(H ) for some isometry V : H E H (quantum channel). In finite dimension : Random states or transformations Random matrices. Paradigm : Constructing explicitly an operator having certain properties may be harder than asserting that a suitably chosen random one should work with large probability. Usually, the curse of dimensionality then becomes an asset. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 3 / 25

7 Why Random Matrix Theory in Quantum Information? State of a quantum system : Positive and trace 1 operator ρ (density operator) on a Hilbert space H (state space). Transformation on a quantum system : Completely Positive and Trace Preserving map φ : X B(H) Tr E (VXV ) B(H ) for some isometry V : H E H (quantum channel). In finite dimension : Random states or transformations Random matrices. Paradigm : Constructing explicitly an operator having certain properties may be harder than asserting that a suitably chosen random one should work with large probability. Usually, the curse of dimensionality then becomes an asset. A few (partial) examples : Additivity conjectures in QI : Both counterexamples (Aubrun/Szarek/Werner...) and weak additivity results (Montanaro, Collins/Fukuda/Nechita...) were provided by RMT. Generic properties of states, measurements, evolutions etc. under certain constraints, such as noise, energy, locality etc. (Hayden/Leung/Winter, Linden/Popescu/Short/Winter, Aubrun/Lancien...) Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 3 / 25

8 Separability vs Entanglement for bipartite quantum systems Definition (Separable vs Entangled) A bipartite quantum state ρ AB on A B is separable if it may be written as a convex combination of product states, i.e. ρ AB = i p i σ (i) A τ(i) B. Otherwise, it is entangled. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 4 / 25

9 Separability vs Entanglement for bipartite quantum systems Definition (Separable vs Entangled) A bipartite quantum state ρ AB on A B is separable if it may be written as a convex combination of product states, i.e. ρ AB = i p i σ (i) A τ(i) B. Otherwise, it is entangled. If a compound system is in a separable global state, there are no intrinsically quantum correlations between its local constituents. Deciding whether a given bipartite state is entangled or (close to) separable is an important issue in quantum physics. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 4 / 25

10 Separability vs Entanglement for bipartite quantum systems Definition (Separable vs Entangled) A bipartite quantum state ρ AB on A B is separable if it may be written as a convex combination of product states, i.e. ρ AB = i p i σ (i) A τ(i) B. Otherwise, it is entangled. If a compound system is in a separable global state, there are no intrinsically quantum correlations between its local constituents. Deciding whether a given bipartite state is entangled or (close to) separable is an important issue in quantum physics. Problem : It is known to be a hard task, both from a mathematical and a computational point of view (Gurvits). Solution : Find set of states which are easier to characterize and which contain the set of separable states. Necessary conditions for separability that have a simple mathematical description and that may be checked efficiently on a computer (e.g. by a semi-definite programme). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 4 / 25

11 The PPT criterion for separability Definition (Partial Transpose) The partial transpose of a state ρ AB on A B is defined as Γ AB (ρ AB ) = Id A T B (ρ AB ), where Id denotes the identity map and T denotes the transpose map. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 5 / 25

12 The PPT criterion for separability Definition (Partial Transpose) The partial transpose of a state ρ AB on A B is defined as Γ AB (ρ AB ) = Id A T B (ρ AB ), where Id denotes the identity map and T denotes the transpose map. NC for Separability (Peres) On a bipartite Hilbert space A B, if a state is separable, then it is positive under partial transpose (PPT). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 5 / 25

13 The PPT criterion for separability Definition (Partial Transpose) The partial transpose of a state ρ AB on A B is defined as Γ AB (ρ AB ) = Id A T B (ρ AB ), where Id denotes the identity map and T denotes the transpose map. NC for Separability (Peres) On a bipartite Hilbert space A B, if a state is separable, then it is positive under partial transpose (PPT). Remarks : This is obvious since Γ AB (σ A τ B ) = σ A T B (τ B ). NSC for separability on C 2 C 2 or C 2 C 3 (Horodecki). In higher dimensions, there exist PPT entangled states. Special instance in the class of separability relaxations built on : ρ AB is separable iff for any positive but not completely positive map Λ B, Id A Λ B (ρ AB ) is positive (Horodecki). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 5 / 25

14 The k-extendibility criterion for separability Definition (k-extendibility) Let k 2. A state ρ AB on A B is k-extendible with respect to B if there exists a state ρ AB k on A B k which is invariant under any permutation of the B subsystems and such that ρ AB = Tr B k 1 ρ AB k. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 6 / 25

15 The k-extendibility criterion for separability Definition (k-extendibility) Let k 2. A state ρ AB on A B is k-extendible with respect to B if there exists a state ρ AB k on A B k which is invariant under any permutation of the B subsystems and such that ρ AB = Tr B k 1 ρ AB k. NSC for Separability (Doherty/Parrilo/Spedalieri) On a bipartite Hilbert space A B, a state is separable if and only if it is k-extendible w.r.t. B for all k 2. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 6 / 25

16 The k-extendibility criterion for separability Definition (k-extendibility) Let k 2. A state ρ AB on A B is k-extendible with respect to B if there exists a state ρ AB k on A B k which is invariant under any permutation of the B subsystems and such that ρ AB = Tr B k 1 ρ AB k. NSC for Separability (Doherty/Parrilo/Spedalieri) On a bipartite Hilbert space A B, a state is separable if and only if it is k-extendible w.r.t. B for all k 2. Remarks : ρ AB separable [ ρ AB k-extendible ] w.r.t. B for all k 2 is obvious since σ A τ B = Tr B k 1 σa τ k B. ρ AB k-extendible w.r.t. B for all k 2 ρ AB separable relies on the quantum De Finetti theorem (Christandl/König/Mitchison/Renner). ρ AB k-extendible w.r.t. B ρ AB k -extendible w.r.t. B for k k. Hierarchy of NC for separability, which an entangled state is guaranteed to stop passing at some point (but one cannot tell when a priori). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 6 / 25

17 Quantifying the strength of separability relaxations Problem : When relaxing the separability constraint to one which is easier to check, how rough is the approximation? Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 7 / 25

18 Quantifying the strength of separability relaxations Problem : When relaxing the separability constraint to one which is easier to check, how rough is the approximation? Known : There exist states which are PPT or k-extendible, and nevertheless very entangled (i.e. far away from the set of separable states in some standard or operational distance measure). Instead of looking at worst case scenarios, can we say something stronger about average/typical behaviours? Hope : Implications regarding average case complexity of checking separability...? Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 7 / 25

19 Quantifying the strength of separability relaxations Problem : When relaxing the separability constraint to one which is easier to check, how rough is the approximation? Known : There exist states which are PPT or k-extendible, and nevertheless very entangled (i.e. far away from the set of separable states in some standard or operational distance measure). Instead of looking at worst case scenarios, can we say something stronger about average/typical behaviours? Hope : Implications regarding average case complexity of checking separability...? Two possible quantitative strategies Estimate the size of the set of states, either satisfying a given separability criterion or being indeed separable. Information on how much bigger than the separable set the relaxed set is. Characterize when certain random states are with high probability, either violating a given separability criterion or indeed entangled. Information on how powerful the separability test is to detect entanglement. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 7 / 25

20 Outline 1 Introduction 2 Interlude : reminder about GUE and Wishart matrices 3 Mean-width of the set of states which are separable vs satisfying a separability criterion 4 Entanglement vs Violating a separability criterion for random states 5 Summary and perspectives Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 8 / 25

21 Asymptotic spectrum of GUE and Wishart matrices Definitions (Gaussian Unitary Ensemble and Wishart matrices) G is a n n GUE matrix if G = (H + H )/ 2 with H a n n matrix having independent complex normal entries. W is a (n,s)-wishart matrix if W = HH with H a n s matrix having independent complex normal entries. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 9 / 25

22 Asymptotic spectrum of GUE and Wishart matrices Definitions (Gaussian Unitary Ensemble and Wishart matrices) G is a n n GUE matrix if G = (H + H )/ 2 with H a n n matrix having independent complex normal entries. W is a (n,s)-wishart matrix if W = HH with H a n s matrix having independent complex normal entries. Definitions (Semicircular and Marčenko-Pastur distributions) dµ SC(m,σ 2 )(x) = 1 2πσ 2 4σ2 (x m) 2 1 [m 2σ,m+2σ] (x)dx. { f dµ MP(λ) (x) = λ (x)dx if λ > 1, where f λ is defined by (1 λ)δ 0 + λf λ (x)dx if λ 1 f λ (x) = (λ+ x)(x λ ) 2πλx 1 [λ,λ + ](x), with λ ± = ( λ ± 1) 2. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 9 / 25

23 Asymptotic spectrum of GUE and Wishart matrices Definitions (Gaussian Unitary Ensemble and Wishart matrices) G is a n n GUE matrix if G = (H + H )/ 2 with H a n n matrix having independent complex normal entries. W is a (n,s)-wishart matrix if W = HH with H a n s matrix having independent complex normal entries. Definitions (Semicircular and Marčenko-Pastur distributions) dµ SC(m,σ 2 )(x) = 1 2πσ 2 4σ2 (x m) 2 1 [m 2σ,m+2σ] (x)dx. { f dµ MP(λ) (x) = λ (x)dx if λ > 1, where f λ is defined by (1 λ)δ 0 + λf λ (x)dx if λ 1 f λ (x) = (λ+ x)(x λ ) 2πλx 1 [λ,λ + ](x), with λ ± = ( λ ± 1) 2. For any Hermitian M on C n, denote by N M = 1 n n i=1 δ λi (M) its spectral distribution. (G n ) n N sequence of n n GUE matrices : (N Gn / n ) n N converges to µ SC(0,1). (W n ) n N sequence of (n,λn)-wishart matrices : (N Wn /λn) n N converges to µ MP(λ). (convergence in distribution, but also in probability, and even almost surely). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 9 / 25

24 Example : Density functions of a centered semicircular distribution of variance parameter 1 (on the left) and of a Marčenko-Pastur distribution of parameter 4 (on the right) Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 10 / 25

25 Moments of GUE and Wishart matrices via combinatorics of permutations Strategy : To identify the asymptotic spectrum of a sequence of random matrices (M n ) n N, one can compute the limit when n + of all p-order moments ETr(M p n ). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 11 / 25

26 Moments of GUE and Wishart matrices via combinatorics of permutations Strategy : To identify the asymptotic spectrum of a sequence of random matrices (M n ) n N, one can compute the limit when n + of all p-order moments ETr(M p n ). Tool : Gaussian Wick formula Let X 1,...,X q be centered Gaussian random variables (either real or complex). If q = 2p + 1, then E[X 1 X q ] = 0. ( p ) If q = 2p, then E[X 1 X q ] = E[X im X jm ]. {i 1,j 1 } {i p,j p }={1,...,2p} m=1 Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 11 / 25

27 Moments of GUE and Wishart matrices via combinatorics of permutations Strategy : To identify the asymptotic spectrum of a sequence of random matrices (M n ) n N, one can compute the limit when n + of all p-order moments ETr(M p n ). Tool : Gaussian Wick formula Let X 1,...,X q be centered Gaussian random variables (either real or complex). If q = 2p + 1, then E[X 1 X q ] = 0. ( p ) If q = 2p, then E[X 1 X q ] = E[X im X jm ]. {i 1,j 1 } {i p,j p }={1,...,2p} m=1 Consequence : G n a n n GUE matrix, W n a (n,λn)-wishart matrix. For any p N, ( ETr Gn 2p ) = n (γα) NC 2 (2p) n p+1 = M (2p) α S 2 (2p) n + SC(0,1) np+1. ( ( ) ETr Wn p ) = λ (α) n (γα)+ (α) λ (α) n p+1 = M (p) α S(p) n + MP(λ) np+1. α NC(p) Sum over all permutations S / pairings S 2 : Exact moment formula. Sum over non-crossing permutations NC / pairings NC 2 : Dominant order in n. γ : canonical full cycle, ( ) : number of cycles in a permutation. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 11 / 25

28 Outline 1 Introduction 2 Interlude : reminder about GUE and Wishart matrices 3 Mean-width of the set of states which are separable vs satisfying a separability criterion 4 Entanglement vs Violating a separability criterion for random states 5 Summary and perspectives Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 12 / 25

29 Mean-width of a set of states Definitions Let K be a convex set of states on C n containing Id/n (maximally mixed state). For a n n Hermitian s.t. HS = 1, the width of K in the direction is w(k, ) = sup σ K Tr( (σ Id/n)). The mean-width of K is the average of w(k, ) over the Hilbert-Schmidt unit sphere of n n Hermitians, equipped with the uniform probability measure. It is equivalently defined as w(k ) = Ew(K,G)/γ n, where G is a n n GUE matrix and γ n = E G HS n + n. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 13 / 25

30 Mean-width of a set of states Definitions Let K be a convex set of states on C n containing Id/n (maximally mixed state). For a n n Hermitian s.t. HS = 1, the width of K in the direction is w(k, ) = sup σ K Tr( (σ Id/n)). The mean-width of K is the average of w(k, ) over the Hilbert-Schmidt unit sphere of n n Hermitians, equipped with the uniform probability measure. It is equivalently defined as w(k ) = Ew(K,G)/γ n, where G is a n n GUE matrix and γ n = E G HS n + n. The mean-width of a set of states is a certain measure of its size (for any reasonable K, w(k ) vrad(k ), where vrad(k ) is the volume-radius of K, i.e. the radius of the Euclidean ball with same volume as K ). Computing it amounts to estimating the supremum of some Gaussian process. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 13 / 25

31 Mean-width of the set of separable states Observation : Esup σ state Tr(G(σ Id/n)) = E G. So by Wigner s semicircle law, the mean-width of the set of all states on C n is asymptotically 2 n/γ n, i.e. 2/ n. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 14 / 25

32 Mean-width of the set of separable states Observation : Esup σ state Tr(G(σ Id/n)) = E G. So by Wigner s semicircle law, the mean-width of the set of all states on C n is asymptotically 2 n/γ n, i.e. 2/ n. Theorem (Aubrun/Szarek) Denote by S the set of separable states on C d C d. c There exist universal constants c,c > 0 such that w(s) C d 3/2 d. 3/2 Remark : The mean-width of the set of separable states is of order 1/d 3/2, hence much smaller than the mean-width of the set of all states (of order 1/d). On high dimensional bipartite systems, most states are entangled. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 14 / 25

33 Mean-width of the set of separable states Observation : Esup σ state Tr(G(σ Id/n)) = E G. So by Wigner s semicircle law, the mean-width of the set of all states on C n is asymptotically 2 n/γ n, i.e. 2/ n. Theorem (Aubrun/Szarek) Denote by S the set of separable states on C d C d. c There exist universal constants c,c > 0 such that w(s) C d 3/2 d. 3/2 Remark : The mean-width of the set of separable states is of order 1/d 3/2, hence much smaller than the mean-width of the set of all states (of order 1/d). On high dimensional bipartite systems, most states are entangled. Proof idea : Upper-bound : Approximate S by a polytope with few vertices, and use that Esup i I Z i C log I for (Z i ) i I a finite bounded Gaussian process (Pisier). Lower-bound : Estimate the volume-radius of S by convex geometry considerations, and use that vrad w (Urysohn). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 14 / 25

34 Mean-width of the set of k-extendible states Theorem Fix k 2 and denote by E k the set of k-extendible states on C d C d. 2 Then, w(e k ). d + kd Remark : The mean-width of the set of k-extendible states is of order 1/d, hence much bigger than the mean-width of the set of separable states. On high dimensional bipartite systems, the set of k-extendible states is a very rough approximation of the set of separable states. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 15 / 25

35 Mean-width of the set of k-extendible states Theorem Fix k 2 and denote by E k the set of k-extendible states on C d C d. 2 Then, w(e k ). d + kd Remark : The mean-width of the set of k-extendible states is of order 1/d, hence much bigger than the mean-width of the set of separable states. On high dimensional bipartite systems, the set of k-extendible states is a very rough approximation of the set of separable states. Proof strategy : sup σ k ext Tr(G(σ Id/d 2 )) may be expressed as G for some suitable G. So one has to estimate E G for the modified GUE matrix G. This is done by computing the p-order moments ETr Gp, and identifying the limiting spectral distribution (after rescaling by d/k) : a centered semicircular distribution µ SC(0,k). The latter has 2 k as upper-edge. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 15 / 25

36 Mean-width of the set of PPT states Theorem Denote by P the set of PPT states on C d C d. Then, there exist constants c d,c d e 1/2 2 1 such that c d w(p ) C d d + d d. Remark : The mean-width of the set of PPT states is of order 1/d, hence much bigger than the mean-width of the set of separable states. On high dimensional bipartite systems, the set of PPT states is a very rough approximation of the set of separable states. Proof strategy : D set of all states on C d C d. Upper-bound : w(p ) w(d) d + 2/d. Lower-bound : w(p ) vrad(d) 2 /2w(D) (Urysohn + Milman/Pajor) with vrad(d) d + e 1/4 /d (Sommers/Życzkowski) and w(d) d + 2/d. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 16 / 25

37 Outline 1 Introduction 2 Interlude : reminder about GUE and Wishart matrices 3 Mean-width of the set of states which are separable vs satisfying a separability criterion 4 Entanglement vs Violating a separability criterion for random states 5 Summary and perspectives Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 17 / 25

38 Random induced states Definition (Pure vs mixed) A quantum state ρ on H is pure if there exists a unit vector ψ in H such that ρ = ψ ψ. Otherwise, it is mixed. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 18 / 25

39 Random induced states Definition (Pure vs mixed) A quantum state ρ on H is pure if there exists a unit vector ψ in H such that ρ = ψ ψ. Otherwise, it is mixed. System space H C n. Ancilla space H C s. Random mixed state model on H : ρ = Tr H ψ ψ with ψ a uniformly distributed pure state on H H (quantum marginal). Equivalent description : ρ = W with W a (n,s)-wishart matrix. TrW Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 18 / 25

40 Random induced states Definition (Pure vs mixed) A quantum state ρ on H is pure if there exists a unit vector ψ in H such that ρ = ψ ψ. Otherwise, it is mixed. System space H C n. Ancilla space H C s. Random mixed state model on H : ρ = Tr H ψ ψ with ψ a uniformly distributed pure state on H H (quantum marginal). Equivalent description : ρ = W with W a (n,s)-wishart matrix. TrW Question : Fix d N and consider ρ a random state on C d C d induced by some environment C s. For which values of s is ρ typically separable? PPT? k-extendible? typically = with probability going to 1 as d grows. Hence 2 steps : Identify the range of s where ρ is, on average, separable/ppt/k-extendible. Show that the average behaviour is generic in high dimension (concentration of measure phenomenon : a sufficiently well-behaved function has an exponentially small probability of deviating from its average as the dimension grows). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 18 / 25

41 Separability of random induced states Theorem (Aubrun/Szarek/Ye) Let ρ be a random state on C d C d induced by C s. There exists a threshold s 0 satisfying cd 3 s 0 Cd 3 log 2 d for some universal constants c,c > 0 such that, if s < s 0 then ρ is typically entangled, and if s > s 0 then ρ is typically separable. Intuition : If s d 2 then ρ is uniformly distributed on the set of states of rank at most s, therefore generically entangled. If s d 2 then ρ is expected to be close to Id/d 2, therefore separable. Phase transition between these two regimes? Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 19 / 25

42 Separability of random induced states Theorem (Aubrun/Szarek/Ye) Let ρ be a random state on C d C d induced by C s. There exists a threshold s 0 satisfying cd 3 s 0 Cd 3 log 2 d for some universal constants c,c > 0 such that, if s < s 0 then ρ is typically entangled, and if s > s 0 then ρ is typically separable. Intuition : If s d 2 then ρ is uniformly distributed on the set of states of rank at most s, therefore generically entangled. If s d 2 then ρ is expected to be close to Id/d 2, therefore separable. Phase transition between these two regimes? Proof idea : Convex geometry + Comparison of random matrix ensembles (majorization) + Concentration of measure in high dimension. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 19 / 25

43 PPT of random induced states Theorem (Aubrun) Let ρ be a random state on C d C d induced by C s. If s < 4d 2 then ρ is typically not PPT, and if s > 4d 2 then ρ is typically PPT. Remark : The threshold environment dimension at which random induced states are generically either PPT or NPT is of order d 2, hence much smaller than the one at which they are generically either separable or entangled (of order d 3 ). In the range d 2 s d 3, typical entanglement of random induced states is typically not detected by the PPT test. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 20 / 25

44 PPT of random induced states Theorem (Aubrun) Let ρ be a random state on C d C d induced by C s. If s < 4d 2 then ρ is typically not PPT, and if s > 4d 2 then ρ is typically PPT. Remark : The threshold environment dimension at which random induced states are generically either PPT or NPT is of order d 2, hence much smaller than the one at which they are generically either separable or entangled (of order d 3 ). In the range d 2 s d 3, typical entanglement of random induced states is typically not detected by the PPT test. Proof strategy : Everything boils down to characterizing when a partially transposed (d 2,s)-Wishart matrix W is positive. This is done by computing the p-order moments ETrΓ(W) p, and identifying the limiting spectral distribution (after rescaling by s) : a non-centered semicircular distribution µ SC(1,d 2 /s). The latter has positive support iff 1 2 d 2 /s 0 i.e. iff s 4d 2. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 20 / 25

45 k-extendibility of random induced states Theorem Let ρ be a random state on C d C d induced by C s. If s < (k 1)2 d 2 then ρ is typically 4k not k-extendible. Proof strategy : If sup σ k ext Tr(ρσ) < Tr(ρ 2 ), then ρ is not k-extendible. To identify when such is the case, one should characterize when Esup σ k ext Tr(Wσ) < ETr(W 2 )/ETrW for W a (d 2,s)-Wishart matrix. (+ Concentration around their average of the considered functions) RHS : In the limit d,s +, ETr(W 2 ) = d 4 s + d 2 s 2 and ETrW = d 2 s. LHS : Write sup σ k ext Tr(Wσ) = W, and estimate E W for the modified Wishart matrix W. This may be done by computing the p-order moments ETr W p, and identifying the limiting spectral distribution (after rescaling by s/k) : a Marčenko-Pastur distribution µ MP(ks/d 2 ). The latter s support has ( ks/d 2 + 1) 2 as upper-edge. For s < (k 1) 2 d 2 /4k, ( ks/d 2 + 1) 2 < (d 2 + s)k/s. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 21 / 25

46 Outline 1 Introduction 2 Interlude : reminder about GUE and Wishart matrices 3 Mean-width of the set of states which are separable vs satisfying a separability criterion 4 Entanglement vs Violating a separability criterion for random states 5 Summary and perspectives Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 22 / 25

47 Summary, generalizations, open questions Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 23 / 25

48 Summary, generalizations, open questions On high dimensional bipartite systems, the volume of PPT or k-extendible states is more like the one of all states than like the one of separable states. Asymptotic weakness of these NC for separability : most PPT or k-extendible states are entangled in high dimension. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 23 / 25

49 Summary, generalizations, open questions On high dimensional bipartite systems, the volume of PPT or k-extendible states is more like the one of all states than like the one of separable states. Asymptotic weakness of these NC for separability : most PPT or k-extendible states are entangled in high dimension. ρ a random state on C d C d induced by C s. When d +, ρ is w.h.p. entangled if s < cd 3, and this entanglement is w.h.p. detected by the PPT or k-extendibility test if s < Cd 2. But in the range d 2 s d 3, PPT or k-extendible entanglement is generic. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 23 / 25

50 Summary, generalizations, open questions On high dimensional bipartite systems, the volume of PPT or k-extendible states is more like the one of all states than like the one of separable states. Asymptotic weakness of these NC for separability : most PPT or k-extendible states are entangled in high dimension. ρ a random state on C d C d induced by C s. When d +, ρ is w.h.p. entangled if s < cd 3, and this entanglement is w.h.p. detected by the PPT or k-extendibility test if s < Cd 2. But in the range d 2 s d 3, PPT or k-extendible entanglement is generic. Similar features are exhibited by most other known separability criteria, e.g. realignment (Aubrun/Nechita) or reduction (Jivulescu/Lupa/Nechita). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 23 / 25

51 Summary, generalizations, open questions On high dimensional bipartite systems, the volume of PPT or k-extendible states is more like the one of all states than like the one of separable states. Asymptotic weakness of these NC for separability : most PPT or k-extendible states are entangled in high dimension. ρ a random state on C d C d induced by C s. When d +, ρ is w.h.p. entangled if s < cd 3, and this entanglement is w.h.p. detected by the PPT or k-extendibility test if s < Cd 2. But in the range d 2 s d 3, PPT or k-extendible entanglement is generic. Similar features are exhibited by most other known separability criteria, e.g. realignment (Aubrun/Nechita) or reduction (Jivulescu/Lupa/Nechita). Possible generalization to the unbalanced case A C d A, B C d B, d A d B : straightforward if d A,d B +, more subtle if d A or d B is fixed (free probability approach usually more relevant and powerful in this latter setting). Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 23 / 25

52 Summary, generalizations, open questions On high dimensional bipartite systems, the volume of PPT or k-extendible states is more like the one of all states than like the one of separable states. Asymptotic weakness of these NC for separability : most PPT or k-extendible states are entangled in high dimension. ρ a random state on C d C d induced by C s. When d +, ρ is w.h.p. entangled if s < cd 3, and this entanglement is w.h.p. detected by the PPT or k-extendibility test if s < Cd 2. But in the range d 2 s d 3, PPT or k-extendible entanglement is generic. Similar features are exhibited by most other known separability criteria, e.g. realignment (Aubrun/Nechita) or reduction (Jivulescu/Lupa/Nechita). Possible generalization to the unbalanced case A C d A, B C d B, d A d B : straightforward if d A,d B +, more subtle if d A or d B is fixed (free probability approach usually more relevant and powerful in this latter setting). For k-extendibility : what happens if k is not fixed, but instead grows with d? For instance : to have w(e k ) w(s), do we need k log α d, k d β...? Work in progress...! Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 23 / 25

53 Broader perspectives Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 24 / 25

54 Broader perspectives Generalizations to the multi-partite case? The problem becomes much richer because checking separability across every bi-partite cut is not even enough to assert full separability... Small number of big subsystems can generally be dealt with by the same techniques, but the picture changes completely in the opposite high-dimensional setting, i.e. a big number of small subsystems. Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 24 / 25

55 Broader perspectives Generalizations to the multi-partite case? The problem becomes much richer because checking separability across every bi-partite cut is not even enough to assert full separability... Small number of big subsystems can generally be dealt with by the same techniques, but the picture changes completely in the opposite high-dimensional setting, i.e. a big number of small subsystems. Other random state models (perhaps physically more relevant)? Random mixtures are asymptotically equivalent to random inductions. But what about more specific ensembles, such as random tensor product states (Ambainis/Harrow/Hastings), random low-rank states...? Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 24 / 25

56 A few references A. Ambainis, A.W. Harrow, M.B. Hastings, Random tensor theory : extending random matrix theory to mixtures of random product states, arxiv : G. Aubrun, Partial transposition of random states and non-centered semicircular distributions, arxiv : G. Aubrun, I. Nechita, Realigning random states, arxiv : G. Aubrun, S. Szarek, Tensor products of convex sets and the volume of separable states on N qudits, arxiv :quant-ph/ G. Aubrun, S. Szarek, D. Ye, Entanglement threshold for random induced states, arxiv : M. Christandl, R. König, G. Mitchison, R. Renner, One-and-a-half quantum De Finetti theorems, arxiv :quant-ph/ A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, A complete family of separability criteria, arxiv :quant-ph/ P. Hayden, D. Leung, A. Winter, Aspects of generic entanglement, arxiv :quant-ph/ M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states : necessary and sufficient conditions, arxiv :quant-ph/ M.A Jivulescu, N. Lupa, I. Nechita, On the reduction criterion for random quantum states, arxiv : C. Lancien, k-extendibility of high-dimensional bipartite quantum states, in preparation A. Peres, Separability criterion for density matrices, arxiv :quant-ph/ Cécilia Lancien Random Matrix Theory in Quantum Information BMC & BAMC Cambridge 25 / 25