Thresholds for entanglement criteria in quantum information theory - PowerPoint PPT Presentation

Thresholds for entanglement criteria in quantum information theory Joint results with Nicolae Lupa, Ion Nechita and David Reeb Toulouse 2015

Content ◮ Entanglement via positive maps approach ◮ Reduction Criterion (RED) ◮ Absolute Reduction Criterion (ARED) ◮ Approximations of (A)SEP ◮ Thresholds for RED and ARED ◮ Comparing entanglement criteria via thresholds

Introduction: Entanglement versus separability Entanglement=inseparability Separable state : ρ = � i , p i ≥ 0 , � p i e i e ∗ i ⊗ f i f ∗ p i = 1 , e i ∈ C n , f i ∈ C k i i Goal : efficient methods to characterize entangled states PPT criterion : 1 if a state is separable, then the partial transpose respect to one of the subsystems is positive-semidef. SEP := { ρ AB : ρ AB − sep } ⊂ PPT = { ρ AB /ρ Γ ≥ 0 } Tool to detect entanglement: if the partial transpose is not positive-semidefinite, then the state is entangled Question : exists other postive maps ϕ such that ( id ⊗ ϕ )( ρ ) � 0, for some ρ entangled state 1 A. Peres, Separability criterion for density matrices, PRL 77,1996

Separability criteria based on positive maps approach Mathematical formulation 2 : ρ ∈ M n ⊗ M k separable iff ρ ϕ := [ id ⊗ ϕ ]( ρ ) ≥ 0 , ∀ ϕ ≥ 0 , ϕ : M k → M m , all positive integers m ∈ N SEP : ⊂ { ρ : ρ ϕ ≥ 0 } ◮ transposition map ϕ ⇒ Positive Partial Transposition (PPT) ◮ reduction map: ϕ ( X ) := I · Tr X − X ⇒ Reduction Criterion (RC) 3 : ρ AB − sep ⇒ ρ A ⊗ I B − ρ AB ≥ 0 , I A ⊗ ρ B − ρ AB ≥ 0 (1) ρ A = [ id ⊗ Tr ]( ρ AB ) partial trace over the second subsystem 2 Horodecki M, Separability of mixed states: necessary and sufficient conditions, Phys. Lett A, 1996 3 Horodecki and all.’99, Cerf and all. ’99

Separability via Reduction Criterion ◮ SEP ⊂ RED := { ρ : ρ red := ρ A ⊗ I B − ρ AB ≥ 0 } ◮ ρ AB rank one entangled state, then ρ red � 0 ◮ ρ Γ ≥ 0 ⇒ ρ red ≥ 0 , ( PPT ⊂ RED ) ◮ If dimB = 2, then PPT=RC ◮ RC connected to entanglement distillation: all states that violate RC are distillable. SEP ⊂ PPT ⊂ RED SEP PPT RLN RED D n,k

Absolutely Separable States Knill’s question 4 : given an self-adjoint positive semi-definite operator ρ , which are the conditions on the spectrum of ρ such that ρ is separable respect to any decomposition? ASEP : states that remain separable under any unitary transformation � U SEP n , k U ∗ ASEP n , k = U ∈U nk Goal = conditions on the spectrum such that to be separable! � APPT n , k := { ρ ∈ D n , k / ∀ U ∈ U nk : ( U ρ U ∗ ) Γ ≥ 0 } = U PPT n , k U ∗ U ∈U nk � ARED n , k := { ρ ∈ D n , k / ∀ U ∈ U nk : ( U ρ U ∗ ) red ≥ 0 } = U RED n , k U ∗ U ∈U nk 4 Open Problems in Quantum Information Theory, http://www. imaph.tu-bs.de/qi/problems

Absolutely PPT states ◮ necessary and sufficient conditions 5 on the spectrum under which the absolute PPT property holds ◮ the condition is to check the positivity of an exponential number of Hermitian matrices (the number of LMI is bounded above by e 2 plnp ( 1 + o ( 1 )) , p = min ( n , k ) ) ◮ if ρ ∈ H 2 n = H 2 ⊗ H n , then ρ ∈ APPT iff � λ 1 ≤ λ 2 n − 1 + 2 λ 2 n λ 2 n − 2 . ◮ if ρ ∈ H 3 n = H 3 ⊗ H n , then ρ ∈ APPT iff   2 λ 3 n λ 3 n − 1 − λ 1 λ 3 n − 2 − λ 2  ≥ 0 , λ 3 n − 1 − λ 1 2 λ 3 n − 3 λ 3 n − 4 − λ 3  λ 3 n − 2 − λ 2 λ 3 n − 4 − λ 3 2 λ 3 n − 5   2 λ 3 n λ 3 n − 1 − λ 1 λ 3 n − 3 − λ 2  ≥ 0 . λ 3 n − 1 − λ 1 2 λ 3 n − 2 λ 3 n − 4 − λ 3 (2)  λ 3 n − 3 − λ 2 λ 3 n − 4 − λ 3 2 λ 3 n − 5 5 Hildebrand, Positive partial transpose from spectra, PRA, 2007

Absolutely Reduced pure states ARED n , k := { ρ ∈ D n , k | ∀ U ∈ U nk : ( U ρ U ∗ ) red ≥ 0 } (3) Reduction of pure state 6 : Given a vector ψ ∈ C n ⊗ C k with Schmidt coefficients { x i } r i = 1 , the eigenvalues of the reduced matrix ( ψψ ∗ ) red are � ( ψψ ∗ ) red � = ( x 1 , . . . , x 1 , η 1 , x 2 , . . . , η r − 1 , x r , . . . , x r , 0 , . . . , 0 , η r ) spec � �� � � �� � � �� � k − 1 times k − 1 times ( n − r ) k times where x i ≥ η i ≥ x i + 1 for i ∈ [ r − 1 ] and η r = − � r − 1 i = 1 η i ≤ 0. The set { η i } r i = 1 \ { x i } r i = 1 is the set of solutions η ∈ R \ { x i } r i = 1 of the equation r x i � x i − η = 1 i = 1 . 6 M.A. Jivulescu, N. Lupa, I. Nechita, D. Reeb, Positive reduction from spectra, Linear Algebra and its Applications, 2014

Characterizing Absolutely Reduced states ARED n , k = { ρ ∈ D n , k : ∀ x ∈ ∆ min ( n , k ) , � λ ↓ x ↑ � ≥ 0 } , ρ , ˆ (4) x ↑ is the where λ ↓ ρ is the vect. of the eigenvalues of ρ and ˆ vector of Schmidt coefficients. ˆ x := ( x 1 , . . . , x 1 , η 1 , x 2 , . . . , x 2 , . . . , η r − 1 , x r , . . . , x r , 0 , . . . , 0 , η r ) , � �� � � �� � � �� � � �� � k − 1 times k − 1 times k − 1 times ( n − r ) k times η i are the solutions of the equation F x ( λ ) := � q m i x i x i − λ − 1 = 0 . i = 1 ◮ necessary and sufficient condition on the spectrum as family of linear inequalities in terms of the spectrum of reduced of a pure state ◮ given ρ ∈ M 2 ( C ) ⊗ M k ( C ) , then ρ ∈ ARED 2 , k if and only if � λ 1 ≤ λ k + 1 + 2 ( λ 2 + · · · + λ k )( λ k + 2 + · · · + λ 2 k ) .

Approximations of SEP � � ρ ∈ D n , k | Tr ( ρ 2 ) ≤ 1 SEPBALL n , k = nk − 1 ◮ largest Euclidian ball 7 inside D n , k , centered at I nk ◮ contains states on the boundary of D n , k ◮ all states within SEPBALL are separable ◮ depends only on the spectrum, i.e. SEPBALL n , k ⊂ ASEP ◮ it is smaller than other sets � � λ ∈ ∆ nk : � r − 1 i = 1 λ ↓ i ≤ 2 λ ↓ nk + � r − 1 i = 1 λ ↓ GER n , k = nk − i ◮ the defining equation 8 represents the sufficient condition provided by Gershgorin’s theorem for all Hildebrand APPT matrix inequalities to be satisfied ◮ lower approximation of APPT, since GER n , k ⊂ APPT n , k ◮ provides easily-checkable sufficient condition to be APPT, much simpler that Hildebrand’s conditions 7 Gurvitz L., PRA 2002 8 M.A. Jivulescu, N. Lupa, I. Nechita, D. Reeb, Positive reduction from spectra, Linear Algebra and its Applications, 2014

Approximations of (A)SEP SEPBALL ASEP GER APPT ARED ∆ n,k LS p := { λ ∈ ∆ nk : λ ↓ 1 ≤ λ ↓ nk − p + 1 + λ ↓ nk − p + 2 + · · · + λ ↓ nk } ◮ LS P -the sets of eigenvalue vectors for which the largest eigenvalue is less or equal than the sum of the p smallest (arbitrary p ∈ [ nk ] ) ◮ For n , k ≥ 3, APPT ⊆ LS 3 ⊆ LS k ⊆ ARED n , k ⊆ LS 2 k − 1 .

Threshold concept Threshold =the value c of the parameter, giving an scaling of the environment, value at which a sharp phase transition of the system occurs! Mathematical characterisation 9 : Consider a random bipartite quantum state ρ AB ∈ M n ( C ) ⊗ M k ( C ) , obtained by partial tracing over C s a uniformly distributed, pure state x ∈ C n ⊗ C k ⊗ C s . When one (or both)of the system dimensions n and k are large, a threshold phenomenon occurs: if s ∼ cnk , then there is a threshold value c 0 such that 1. for all c < c 0 , as dimension nk grows, P ( ρ AB satisfies the entangled criterion) = 0; 2. for all c > c 0 , as dimension nk grows, P ( ρ AB satisfies the entangled criterion) = 1; 9 Aubrun, G. Partial transposition of random states and non-centered semicircular distributions. Random Matrices: Theory Appl. (2012)

Reduction Criterion: RMT approach W = XX ∗ ∈ M d ( C ) -Wishart matrix of parmeters d and s . Wishart matrices – physically reasonable models for random density matrices on a tensor product space. The spectral properties of ρ red → reduced matrix R = W red := W A ⊗ I k − W AB , where W AB is a Wishart matrix of parameters nk and s , W A is its partial trace with respect to the second subsystem B . Issues: ◮ study the distribution of the eigenvalues of the random matrix R = W red ◮ evaluating the probability that R is positive semidefinite

Moment formula for R Theorem The moments of the random matrix R = W red = W A ⊗ I k − W AB ∈ M nk ( C ) are given by 10 � ( − 1 ) | f − 1 ( 2 ) | s # α n #( γ − 1 α ) k 1 f ≡ 1 +#( P − 1 E Tr ( R p ) = α ) , ∀ p ≥ 1 , f α ∈S p , f ∈F p (5) ( # -number of cycles of α , f : { 1 , . . . , p } → { 1 , 2 } ) . Examples: E Tr ( R ) = nk ( k − 1 ) s � R 2 � � ( ks ) 2 n + ksn 2 � + nks 2 + ( nk ) 2 s . E Tr = ( k − 2 ) 10 M.A. Jivulescu, N. Lupa, I. Nechita, On the reduction criterion for random quantum states, Journal of Mathematical Physics, Volume: 55, Issue: 11, 2014

Moment formula The proof is based on ◮ the development of R p using the non-commutative binomial formula R p = � ( − 1 ) | f − 1 ( 2 ) | R f f ∈F p ◮ f : { 1 , . . . , p } → { 1 , 2 } encodes the choice of the term (choose the f ( i ) -th term) in each factor in the product R p = ( W A ⊗ I k − W AB )( W A ⊗ I k − W AB ) · · · ( W A ⊗ I k − W AB ) . ◮ R f denotes the ordered product − − − → � R f = R f ( 1 ) R f ( 2 ) · · · R f ( p ) = R f ( i ) , 1 ≤ i ≤ p for the two possible values of the factors R 1 = W A ⊗ I k , R 2 = W AB . ◮ Wick graphical calculus to compute E Tr R f ;