Mutltiparticle Entanglement. Andreas Osterloh Krakow,

Transcription

1 Mutltiparticle Entanglement Andreas Osterloh Krakow,

2 Outline Introduction to entanglement Genuine multipartite entanglement Balanced states Generalizations

3 single particle d ψ H = C d ; ψ = ψ i i i=1 d ρ H H = C d d ; ρ = ρ ij ij R ρ = r i u i u i ; R d i For R=1 the state is called pure. i,j=1

4 many constituents ψ H q =(C d ) q ; ψ = d i 1,...,i q =1 ψ i i 1,...,i q ρ H H = C dq dq ; ρ = ρ = R i r i u i u i ; R dq (d,...,d) i,j=(1,...,1) ρ ij i j Again, for R=1 the state is called pure. This will be the case also here.

5 what is Entanglement?

6 Pure states: 2 qubits ψ = ψ + ψ + ψ + ψ Question : ψ = Φ χ? Why is this an interesting question? Andernfalls > entangled product state: Acting on one spin can t affect the other no info transfer can be induced locally singlet states are entangled: polarized triplet states are disentangled ( )/ 2 J =1,m=1 = =

7 entanglement Indicators Reduced density matrix of pure product states is pure!!! local basis change doesn t affect entanglement local SU invariance of entanglement measures

8 Measures for 2 qubits τ 1 = 4 det ρ (1) (one)-tangle Coffman, Kundu, Wootters, PRA 51, 5236 (2) -Tr ρ (1) log 2 ρ (1) von Neumann entropy Negativity Concurrence Peres, PRL 77, 1413 (1996) Horodecki s, PLA 223, 1 (1996) Vidal, Werner, PRA 65, (22) entanglement measure C = ψ σ 2 σ 2 ψ E(ψ) Concept of entanglement is understood! Hill, Wootters, PRL s 1997 & 1998

9 Concurrence: 2 spins 1/2 Example Hill,Wootters PRL 1997; Wootters PRL 1998 ρ = ψ p BellBell = p +(1 + 1p) productproduct p Bell =( )/ 2 C = p Abouraddy et al. PRA 21 Ingredients R = ρσ 2 y ρ σy 2 ρ C = max{, 2λ max tr R} λ max : largest eigenvalue of R 9

10 Definition A q qudit state is called disentangled q : ρ = p i ψjψ i j i i j=1 normalized but not ON A q qudit state is called biseparable : ρ = i p i ψ i Aψ i A ψ i Bψ i B caution: being non-biseparable is sometimes called genuinely multipartite entangled

11 Genuine Multipartite Entanglement Osterloh, Siewert PRA 72, (25) Osterloh, Siewert IJQI 4, 531 (26) Lohmayer, Osterloh, Siewert, Uhlmann PRL 97, 2652 (26) Osterloh, Siewert, Uhlmann PRA 77, 3231 (28) Eltschka, Osterloh, Siewert, Uhlmann NJP 1, 4314 (28) Đjoković, Osterloh JMP 5, 3359 (29) Eltschka, Bastin, Osterloh, Siewert, Phys. Rev. A 85, 2231 (212) Osterloh, Siewert, Phys. Rev. A 86, 4232 (212)

12 tracing out NO entanglement left! Jedes Teilchen trägt bei Wissen über alle Teilchen nötig still entangled Wissen über 2 Teilchen ausreichend τ 1 & co. (somehow) detect all types C and τ 2 measure bipartite case τ 3 cares for tripartite case SL(2,C) Invarianten erlauben Differenzierung SL(2,C) invariants are the key Dür, Vidal, Cirac PRA 62, (2) Osterloh, J. Appl. Phys. B 98, 69 (21)

13 Vidal Kriterien PRA 2 Monotones local operations can destroy entanglement J =,m= = 1 2 ( ) projective measurement Kollaps der Wellenfunktion 5% or local operations can t entangle with other sites local operations must not enlarge entanglement! Unabhängige nicht WW Systeme bleiben unabhängig

14 local assistance local operations CAN induce entanglement... 1 ( ) 2... elsewhere projective measurement in x-direction 1 ( ± ) à 5% 2 Local unitaries cannot do this non-increasing is entanglement among/with these locally operated systems (e.g. qubits) General local operations needed!!!

15 How to classify Entanglement? Basiswechsel local unitary invariance: SU q Verallgemeinerte Messungen e iασ ; α RC local linear invariance: SL q Interesting: Not all SU-invars are entangl. monotones... Caveat: SL-trafos but all do SL-invars not preserve ARE! normalization

16 How to classify Entanglement? SU-invariance does not discriminate global entanglement from entanglement of only parts SU SL-operations introduce bias and creation generalized GL trafo Ψ + b b 1 SL-invariance only covers Ψ genuine c multipartite entanglement; subtangles are elusive

17 Central idea: combs U SU(2) U ψ Hfor all ψ H ψ I ψ = for all ψ H ψ U IU ψ = Osterloh, Siewert PRA 25 IJQI 26 find anti-linear operators such that

18 ... and filter ideal Comb property zero expectation values for local factors; i.e. Filters are non-local anti-linear operators with zero expectation for all product states Crucial: for any odd number of σ y Filters generate an Ideal within the algebra of polynomial invariants

19 4 qubits τ (1) 4 = σ µ σ ν σ y σ y σ µ σ y σ λ σ y σ y σ ν σ λ σ y τ (2) 4 = σµ σ ν σ y σ y σ µ σ y σ λ σ y σ y σ ν σ y σ τ σ y σ y σ λ σ τ τ (3) 4 = σ µ σ ν σ y σ y σ µ σ ν σ y σ y (1, 3) (2, 3) Osterloh, Siewert PRA 72, (25) Three inequivalent states τ (1) 4 τ (2) 4 τ (3) 4 Φ 1 = GHZ/cat state = + 15 Φ 2 = X state = Φ 3 = cluster state = Complete generating set for the filter-ideal Đjoković, Osterloh JMP 5, 3359 (29)

20 4 Qubits

21 Back to the definition of a comb...

22 ψ P 12 ψ ψ ψ ψ 123 σ µ σ 2 σ 2 ψ ψ 123 σ µ σ 2 σ 2 ψ 123 P 23 = ψ I n ψ... ψ 123 σ µ σ 2 σ 2 ψ 123 P 3 12 ψ123 σ µ σ 2 σ 2 ψ 123

23 a third invariant have a glance at and at σ 2 σ 2 P = 1 2 (σ µ σ µ σ 2 σ 2 ) σ 2 σ 2 σ 2 P 12 P 23 = 1 4 [σ 2 σ 2 σ 2 (σ µ σ µ σ 2 +cyclic) + i klm τ k τ l τ m ] with τ 1 = σ, τ 2 = σ 1, τ 3 = σ 3. i klm τ k τ l τ m P 12 σ µ σ µ σ 2 +cyclic

24 The symmetric Group defining group relations PP=1 ; P 23 P 12 P 23 = P 12 P 23 P 12 = third SL-invariant operator together with and the operators are complete! σ 2 σ µ σ µ i klm τ k τ l τ m identities between different invariants... e.g. 8σ 2 σ 2 σ 2 σ 2 σ 2 σ 2 =(σ 2 σ 2 σ µ σ µ ) 3 = Đjoković, Osterloh JMP 5, 3359 (29) Đjoković, Osterloh JMP 5, 3359 (29) 3σ µ σ 2 σ 2 σ µ σ 2 σ 2 = σ µ σ ν σ λ σ µ σ ν σ λ (σ 2 σ 2 ) 3 3(µ µ)(σ 2 σ 2 ) 2 + 3(µ µ) 2 (σ 2 σ 2 ) (µ µ) 3

25 Genuine multipartite entangled states Balanced states Osterloh, Siewert NJP 12, 7525 (21)

26 σ 2 = σ µ σ µ = i i (+, +) (+, ) (, +) (, ) only balanced part is filtered out Problem: every state has a balanced form

27 Irreducibly balanced states Definition (binary and alternating matrix): ψ = J j=1 Definition (balanced state): J j=1 c j bin{m j } ; c j = B ψ = (bin{m 1 }... bin{m J }) A ψ = 2B ψ Z q J (A ψ ) k,j n j = k {1,...,q} ; n j IN +

28 irreducible balanced: balanced & no balanced part completely unbalanced: without balanced part all modulo local SU-transformations

29 Theorem: Every (irreducibly) balanced state is SL-semistable sketch of the proof: T (i) LF O = t z i t z i ; z i R, t > 1 t q i=1 z ia ij = t w j j =1,...,L q >w j = z i A ij i=1 L 1 Balancedness: A il = w L = and assume that w j < for all j=1,...,l. q i=1 j=1 A ij z i A il = i =1,...,q L 1 j=1 w j! >

30 Results Product states are not irreducibly balanced full products are completely unbalanced (also W) SL(2) irreducibly balanced states stochastic states (irreducibly) balanced states are SL-semistable irreducibly balanced states of normal form are minimal in their length

31 Generalization J j=1 (A ψ ) k,j z j = k {1,...,q} ; z j Z \{} comes from topological phases distinguishes SL-equivalent states as no (additional) top. phase (add.) topological phase W = W = GHZ = GHZ = α α α α 3 1 Johansson, Ericsson, Singh, Sjöqvist, Williamson, PRA 85, 3232 (212)

32 Top. Phases and Entanglement SL-invariants of polynomial degree 2n lead to χ = πm n if several invars are non-zero, it is πm χ = gcd(n 1,...) Example: for GHZ Cl X = d=2n=6 d=2n=4 d=2n=2

33 Question: What (invariants) is highlighting those states? One hint given: if SU-invariant with bidegree (d 1,d 2 ) exists, then the topological phase is χ = 2πm d 1 d 2 Johansson, Ericsson, Singh, Sjöqvist, Williamson, PRA 85, 3232 (212) For three qubits, there is only a single invariant of bidegree (3,1) [and one of bidegree (1,3)] Luque, Thibon, Toumazet, Math. Struct. Comp. Sci. 17, 1133 (27) τ (3,1) =( ψ ij 2 ψ 1ij 2 )(ψ ψ ψ 1 ψ 11 ψ 11 ψ 1 ψ 1 ψ 11 ) +2 ψ ij (ψ 1 ψ 111 ψ 11 ψ 11 )ψ1ij ψ 1ij (ψ ψ 11 ψ 1 ψ 1 )ψij

34 (3,1) evaluated τ (3,1) top. phase GHZ W GHZ W π π π/2 W = W = GHZ = α α α α 3 1 GHZ = χ = 2πm = 2πm d 1 d = πm Generalizations of this concept to more qubits?

35 The partial spin flip c-balanced states as measured by (2n,)-invariant idea: correspondence through partial spin flip Remark: invariant with resp. to local SU(2) transforms α α α α α 4 1 α α α α α 4 1 value is SU(2)-invariant σ q y C σ q y C

36 The derived series SL SU U (2n,) (2n-1,1)... (n+1,n-1) (n,n) last element of series is U-invariant (n,n) is either a product (see example) or a completely unbalanced entangled state W-like (2n-p,p) : states SU-invariant τ (2n-p,p) SU-invars divide zero-class of SL down to U

37 Summary Entanglement classifications needs SL (SU too fine) local SL invariants genuinely N-tangled states irreducibly balanced states are SL-semistable for every irreducible balanced state exists a series of affinely balanced states

38 Thanks to the audience Thierry Bastin Dragomir Đoković Christopher Eltschka Markus Johansson Jens Siewert Eric Sjöqvist Armin Uhlmann