On the orbit space of unitary actions for mixed quantum states Vladimir Gerdt, Arsen Khvedelidze and Yuri Palii Group of Algebraic and Quantum computation Laboratory of Information Technologies Joint Institute for Nuclear Research ACA 2015, Kalamata, Greece, July 20-23, 2015 Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 1 / 29
Contents Motivation 1 Basics of the bipartite entanglement 2 Orbit space and entanglement space in terms of local invariants 3 Example: 5-parameter subset of density matrices 4 Conclusions 5 Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 2 / 29
The problem statement Generic question: “C LASSICALITY OR QUANTUMNESS ” ? Mathematical problem: D ESCRIPTION OF THE ENTANGLEMENT SPACE Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 3 / 29
Space of states A complete information on a generic N -dimensional quantum system is accumulated in N × N density matrix ̺ . self-adjoint: ̺ = ̺ + , 1 positive semi-definite: ̺ ≥ 0 , 2 Unit trace: Tr ̺ = 1 , 3 The set P + , of all possible density matrices, is the space of ( mixed ) quantum states. Equivalence relation on P + , due to the adjoint action of SU ( N ) group ( Ad g ) ̺ = g ̺ g − 1 , g ∈ SU ( N ) , defines the orbit space P + | SU ( N ) that comprises a physically relevant knowledge. Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 4 / 29
Density matrix for binary composites Composition of two subsystems represented by the Hilbert spaces H A and H B defines tensor product space H A ∪ B = H A ⊗ H B . The density matrix of joint system ̺ acts on H A ⊗ H B For a binary system, N 1 ⊗ N 2 , the Local Unitary (LU) equivalence, ̺ ∼ ̺ ′ , means ̺ ′ = SU ( N 1 ) × SU ( N 2 ) ̺ ( SU ( N 1 ) × SU ( N 2 )) † . The LU equivalence decomposes P + into the local orbits. The union of these classes is customary to call as the “entanglement space” E n . Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 5 / 29
Entanglement A bipartite quantum system is separable if its density matrix can be written in the form M M � q j ρ A j ⊗ ρ B � ρ = j , q j ≥ 0 q j = 1 . j = 1 j = 1 where ρ A j and ρ B j are density matrices of the constituent systems. Otherwise the bipartite system is entangled. The property to be entangled (resp. separated) as well as the measure of entanglement is preserved by local unitary transformations. “The entanglement of a two-qubit system is a non-local property so that measures of entanglement should be independent of all local transformations of the two qubits separately. Since a mixed two-qubit system is described by its density matrix, its nonlocal entangling properties must be described by local invariants of the density matrix.” King & Welsh. Qubits and invariant theory. J. Phys: Conf. Series 30 , 1-8, 2006. Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 6 / 29
P + as semialgebraic variety The set of all N × N Hermitian matrices with unit trace is a manifold in hyperplane P ⊂ R N 2 The positive semi-definiteness ̺ ≥ 0 , restricts manifold further to a convex ( N 2 − 1 ) -dimensional body Since all roots of the characteristic equation det | λ I − ̺ | = λ N − S 1 λ N − 1 + · · · + ( − 1 ) N S N = 0 , are real, for their non-negativity it is necessary and sufficient that S k ≥ 0 , ∀ k . Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 7 / 29
Example: Pairs of 2-qubits The unit trace condition and semipositivity of ̺ define semialgebraic set 0 ≤ S k ≤ 1 , k = 1 , 2 , . . . , N . For 2 qubit case , S k are polynomials up to fourth order in 15 variables, e.g., in Fano parameters ̺ = 1 � � σ ⊗ I 2 + I 2 ⊗ � I 2 ⊗ I 2 + � a · � b · � σ + c ij σ i ⊗ σ j . 4 Parameters c ij determine the correlation matrix c ij = || C || ij Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 8 / 29
Coefficients S k for two qubits 1 − 1 a 2 + b 2 + c 2 � � S 2 = 3 � 3 � � a 2 + b 2 + c 2 � � S 3 = 1 − − 2 c 1 c 2 c 3 − a i b i c i , i = 1 � 3 � a 2 + b 2 + c 2 �� 2 � � � S 4 = 1 − + 8 c 1 c 2 c 3 − a i b i c i i = 1 a 2 b 2 + ( a 2 + ( c 2 ) 2 − c 4 i + b 2 i ) c 2 � . − 2 2 i − a i b i c j c k i cyclic c 1 , c 2 , c 3 - singular numbers of correlation matrix C Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 9 / 29
Peres–Horodecki separability criterion Peres–Horodecki separability criterion: The system is in a separable state iff partially transposed density matrix ̺ T B = I ⊗ T ̺ , T − transposition operator satisfies the conditions for a density operator. Coefficients of the characteristic equation for ̺ T B : S T B = S 2 , 2 = S 3 − 1 S T B 4det ( C ) , 3 = S 4 + 1 S T B 16det ( M ) , 4 M = ̺ − ̺ A ⊗ ̺ B − Schlienz & Mahler matrix, ̺ A = tr B ̺ and ̺ B = tr A ̺ - density matrices of subsystems A , and B . Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 10 / 29
3-parameter family of 2-qubits states A sample density matrix (GKP , Phys. Atom. Nucl. 74(6),893-900,2011) 1 + α 0 0 0 ρ = 1 0 1 − β i γ 0 0 − i γ 1 + β 0 4 0 0 0 1 − α Its partially transposed 1 + α 0 0 i γ ρ T B = 1 0 1 − β 0 0 0 0 1 + β 0 4 − i γ 0 0 1 − α Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 11 / 29
Semipositivity domains ρ ≥ 0 : α 2 ≤ 1 β 2 + γ 2 ≤ 1 ρ T B ≥ 0 : β 2 ≤ 1 α 2 + γ 2 ≤ 1 Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 12 / 29
Domains of Separability vs. Entanglement Separability domain Entanglement domain Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 13 / 29
Bipartite ( r × s − dimensional) quantum system r 2 − 1 s 2 − 1 r 2 − 1 s 2 − 1 1 � � � � ρ = I r · s + a i λ i ⊗ I s + b i I r ⊗ µ i + c ij λ i ⊗ µ j r · s i = 1 i = 1 i = 1 j = 1 ρ is an element in the universal enveloping algebra of su ( r · s ) . Matrix C := || c ij || c ij = Tr ( ρ · λ i ⊗ µ j ) accounts for correlations of parts. Local unitary transformations: ( U 1 × U 2 ) · ρ · ( U 1 × U 2 ) † , ρ �→ U 1 ∈ SU ( r ) , U 2 ∈ SU ( s ) It is natural to describe the orbit space in terms of elements in the invariant ring K [ X ] G X := { a i , b j , c ij | 1 ≤ i ≤ r 2 − 1 , 1 ≤ j ≤ s 2 − 1 } ⊂ R ( r 2 − 1 )( s 2 − 1 ) Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 14 / 29
Elements of Invariant Theory I Let G be a compact Lie group. Then, The invariant ring R [ X ] G := { p ∈ R [ X ] | p ( v ) = p ( g ◦ v ) ∀ v ∈ V , g ∈ G } is finitely generated (Hilbert’s finiteness theorem). There exist algorithms to construct generators of R [ X ] G . There exist a set of algebraically independent homogeneous primary invariants P := { p 1 , . . . , p q } ⊂ R [ X ] G such that R [ X ] G is integral over R [ P ] (Noether normalization lemma). Criterion: the variety in C q given by P is { 0 } . There exist a set S := { s 1 , . . . , s m } of secondary invariants, homogeneous generators of R [ X ] G as a module over R [ P ] . Together, primary and secondary invariants (integrity basis) generate R [ X ] G . Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 15 / 29
Elements of Invariant Theory II R [ X ] G is Cohen-Macaulay and there is a Hironaka decomposition R [ X ] G = ⊕ m k = 0 s k R [ P ] . Orbit separation: (Onishchik & Vinberg. Lie Groups and Algebraic Groups. Springer, 1990; Th.3, Chap.3, §4) ∀ u , v ∈ V s.t. G ◦ u � = G ◦ v : ∃ p ∈ R [ X ] G s.t. p ( u ) � = p ( v ) . Syzygy ideal: I P := { h ∈ R [ y 1 , . . . , y q ] | h ( p 1 , p 2 , . . . , p q ) = 0 in R [ x 1 , . . . , x d ] } , R [ y 1 , . . . , y q ] / I P ≃ R [ X ] G . Gerdt, Khvedelidze, Palii (LIT, JINR) Orbit space of composite quantum systems ACA 2015, July 20, 2015 16 / 29
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