Random quantum states Ion Nechita CNRS, Laboratoire de Physique Th´ eorique, Toulouse ement Pellegrini, Karol Penson and Karol ˙ joint work with Benoˆ ıt Collins, Cl´ Zyczkowski Open Quantum Systems meeting Grenoble, November 29th, 2010
Random quantum states — unstructured models —
Random pure quantum states • Pure states of a finite dimensional quantum system: | ψ � ∈ H ≃ C N . • Up to an unimportant phase, the set of pure states is the unit sphere of C N . • For N = 2, a pure qubit is a point on the Bloch sphere (unit sphere of R 3 ). • A random pure state in H = C N is a uniform point on the unit sphere of C N . • One can sample from this distribution by normalizing a vector of N i.i.d. complex Gaussian random variables, | ψ � = X / � X � 2 . • Equivalent definition: let U ∈ U N be a Haar-distributed random unitary matrix and let | ϕ 0 � be a fixed quantum state. Then, | ϕ � = U | ϕ 0 � has the same distribution as | ψ � . • If, instead of a uniform distribution, we wand a random state“concentrated around” | ϕ 0 � , use | ϕ t � = U t | ϕ 0 � , where U t is a random unitary Brownian motion. In the limit t → ∞ , one recovers the previous model. • The structure of H does not play any role here ❀ unstructured quantum states 3 / 40
Random pure states and the induced ensemble • Induced ensemble : partial trace a random pure state on a composite system H ⊗ K : ρ = Tr K | ψ �� ψ | , where | ψ � is a random pure state on C N ⊗ C K . • The random matrix ρ has the same distribution as a rescaled Wishart matrix W / Tr W , where W = XX ∗ with X a Ginibre (i.i.d. Gaussian entries) matrix from M N × K ( C ). • The eigenvalue density of ρ is given by N � λ K − N ∆( λ ) 2 , ( λ 1 , . . . , λ N ) �→ C N , K i i =1 where ∆( λ ) = � 1 � i < j � N ( λ i − λ j ). • Exact formula for the average von Neumann entropy [Page, ’95] E H ( ρ ) = Ψ( NK + 1) − Ψ( K + 1) − N − 1 ∼ ln( N ) − N / 2 K . 2 K 4 / 40
Random density matrices - asymptotics • In the limit N → ∞ , K ∼ cN , for a fixed constant c > 0, the empirical spectral distribution of the rescaled eigenvalues N L N = 1 � δ cN λ i , N i =1 converges almost surely to the Marchenko-Pastur distribution π (1) c . • The Marchenko-Pastur (or free Poisson) distribution is defined by � ( x − a )( b − x ) π (1) = max { 1 − c , 0 } δ 0 + 1 [ a , b ] ( x ) dx , c 2 π x where a = ( √ c − 1) 2 and b = ( √ c + 1) 2 . 5 / 40
Random density matrices - asymptotics 3.5 0.5 0.12 0.45 3 0.1 0.4 2.5 0.35 0.08 0.3 2 density density density 0.25 0.06 1.5 0.2 0.04 0.15 1 0.1 0.02 0.5 0.05 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 8 10 12 14 16 eigenvalues eigenvalues eigenvalues Figure: Empirical and limit measures for ( N = 1000 , K = 1000), ( N = 1000 , K = 2000) and ( N = 1000 , K = 10000). 6 / 40
Random quantum states associated to graphs — structured models —
Pure states associated to graphs • Total Hilbert space has a product structure H = H 1 ⊗ · · · ⊗ H k . • We want our randomness model to encode initial quantum correlation between different subsystems. • The structure of correlations will be encoded in a graph: • Vertices encode the different subsystems; • Edges encode the presence of entanglement. 8 / 40
Pure states associated to graphs - examples V 1 V 1 H 1 H 2 | Φ + 12 � (a) (b) V 1 V 2 V 1 V 2 | Φ + 12 � H 1 H 2 (c) (d) Figure: Graphs with one edge: a loop on one vertex, in simplified notation (a) and in the standard notation (b), and two vertices connected by one edge, in simplified notation (c) and in the standard notation (d). 9 / 40
Pure states associated to graphs - examples V 1 V 2 V 3 V 1 V 2 V 3 | Φ + | Φ + 12 � 34 � H 1 H 2 H 3 H 4 (a) (b) V 3 V 1 V 2 H 5 V 1 V 2 V 3 | Φ + | Φ + 12 � 34 � | Φ + 56 � H 4 H 1 H 2 H 3 H 6 (c) (d) Figure: A linear 2-edge graph, in the simplified notation (a) and in the standard notation (b). Graph consisting of 3 vertices and 3 bonds (c), one of which is connected to the same vertex so it forms a loop; (d) the corresponding ensemble of random pure states defined in a Hilbert space composed of 6 subspaces represented by dark dots. 10 / 40
Pure states associated to graphs - formal definition • Consider an undirected graph Γ consisting of m edges (or bonds ) B 1 , . . . , B m and k vertices V 1 , . . . V k . • We associate to Γ a pure state | ˜ Ψ �� ˜ Ψ | ∈ H = H 1 ⊗ · · · ⊗ H 2 m : | ˜ � | Φ + Ψ � = i , j � , { i , j } edge where | Φ + i , j � denotes a maximally entangled state: d i N 1 | Φ + � ij � = √ d i N | e x � ⊗ | f x � , x =1 • dim H i = d i N , with d i fixed parameters and N → ∞ . For each edge { i , j } , we have d i = d j . • At each vertex, a Haar unitary matrix acts on the subsystems n =2 m � � � � | ˜ H i ∋ | Ψ Γ � = U C Ψ � i =1 C vertex • The random unitary matrices U 1 , . . . , U k are independent. 11 / 40
Marginals of graph states — moments and entropy —
Partial tracing random pure graph states • Non-local properties of the random graph state | Ψ � ❀ partition of the set of all 2 m subsystems into two groups, { S , T } . • Total Hilbert space can be decomposed as a tensor product, H = H T ⊗ H S . • Reduced density operator ρ S = Tr T | Ψ �� Ψ | . • Graphically, partial traces are denoted at the graph by“crossing”the spaces H i which are being traced out. H 1 H 2 H 3 V 1 V 2 V 3 | Φ + | Φ + | Φ + 14 � 25 � 36 � V 4 H 4 H 5 H 6 Figure: The random pure state supported on n = 6 subspaces is partial traced over the subspace H T defined by the set T = { 2 , 4 , 6 } , represented by crosses.The reduced state ρ S supported on subspaces corresponding to the set S = { 1 , 3 , 5 } . 13 / 40
Moments • Use the method of moments: compute lim N →∞ E Tr( X p ) for a random matrix X . • Using matrix coordinates, we can reduce our problem to computing integrals over the unitary group. Theorem (Weingarten formula) Let d be a positive integer and i = ( i 1 , . . . , i p ) , i ′ = ( i ′ 1 , . . . , i ′ p ) , j = ( j 1 , . . . , j p ) , j ′ = ( j ′ 1 , . . . , j ′ p ) be p-tuples of positive integers from { 1 , 2 , . . . , d } . Then � U i 1 j 1 · · · U i p j p U i ′ 1 · · · U i ′ p dU = 1 j ′ p j ′ U ( d ) � β ( p ) Wg( d , αβ − 1 ) . δ i 1 i ′ α (1) . . . δ i p i ′ α ( p ) δ j 1 j ′ β (1) . . . δ j p j ′ α,β ∈S p If p � = p ′ then � U i 1 j 1 · · · U i p j p U i ′ 1 · · · U i ′ p ′ dU = 0 . 1 j ′ p ′ j ′ U ( d ) 14 / 40
Network associated to a marginal • Using the Weingarten formula, one has to find the dominating term in a sum indexed by permutations of p objects. • This optimization problem is equivalent to finding the maximum flow in a network. V 3 V 1 V 2 H 5 | Φ + | Φ + 12 � 34 � | Φ + 56 � H 4 H 1 H 2 H 3 H 6 γ 1 2 β 1 β 2 β 3 1 1 1 2 id 15 / 40
Network associated to a marginal • Network ( V , E , w ) with vertex set V , edge set E and edge capacities w . • The vertex set V = { id , γ, β 1 , . . . , β k } , with two distinguished vertices: the source id and the sink γ . • The edges in E are oriented and they are of three types: E = { (id , β i ) ; | T i | > 0 } ⊔ { ( β i , γ ) ; | S i | > 0 } ⊔ { ( β i , β j ) , ( β j , β i ) ; | E ij | > 0 } , where S i , T i is are the surviving and traced out subsystems at vertex i and E ij are the edges from vertex i to vertex j . • The capacities of the edges are given by: w (id , β i ) = | T i | > 0 w ( β i , γ ) = | S i | > 0 w ( β i , β j ) = w ( β j , β i ) = | E ij | > 0 . 16 / 40
Main result Theorem (Collins, N., ˙ Zyczkowski ’10) Asymptotically, as N → ∞ , the p-th moment of the reduced density matrix behaves as S ) ∼ N − X ( p − 1) · [ combinatorial term + o (1)] , E Tr( ρ p where X is the maximum flow in the network associated to the marginal. The combinatorial part can be expressed in terms of the residual network obtained after removing the capacities of the edges that appear in the maximum flow solution. 17 / 40
Fuss-Catalan limit distributions
Definition • Matrix model: π ( s ) is the limit eigenvalue distribution of the random matrix X s = G s · · · G 2 G 1 G ∗ 1 G ∗ 2 · · · G ∗ s , with i.i.d. Gaussian matrices G i . • Combinatorics: moments given by � 1 � sp + p � x p d π ( s ) ( x ) = sp + 1 p = |{ ˆ 0 p � σ 1 � σ 2 � · · · � σ s � ˆ 1 p ∈ NC ( p ) }| . π (1) � ⊠ s , where π (1) is the free Poisson (or • Free probability: π ( s ) = � Marchenko-Pastur) distribution (of parameter c = 1). 19 / 40
Graph marginals with limit Fuss-Catalan distribution, s = 1 V 1 γ β 1 id V 1 (a) (b) (c) Figure: A vertex with one loop (a) and a marginal (b) having as a limit eigenvalue distribution the Marchenko-Pastur law π (1) . In the network (c), both edges have capacity one. • This is the simplest graph state having the Marchenko-Pastur asymptotic distribution. • The reduced matrix is obtained by partial tracing an uniformly distributed pure state, hence it is an element of the induced ensemble . 20 / 40
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