Entanglement Polytopes

Transcription

1

2 Entanglement Polytopes Detecting Genuine Multi-Particle Entanglement by Single-Particle Measurements Michael Walter joint work with Matthias Christandl, Brent Doran (ETH Zürich), and David Gross (Univ. Freiburg)

3 Multi-Particle Entanglement particle C d...n How entangled is a given multi-particle quantum state...n (prepared in the laboratory)?

4 Multi-Particle Entanglement via Single-Particle Measurements...N 2... N What can be said about the multi-particle entanglement of a pure state given its one-body density matrices only?

5 Multi-Particle Entanglement Operational approach: equivalent entanglement if related by operations which cannot create entanglement...n...n

6 Multi-Particle Entanglement Operational approach: equivalent entanglement if related by operations which cannot create entanglement...n SLOCC...N Stochastic Local Operations and Classical Comm.: non-zero success probability convenient mathematical characterization:...n pa b...b A N q...n Dür, Vidal & Cirac (2) invertible local operators

7 Multi-Particle Entanglement Entanglement class of a pure state :...N ( C SLOCC G yx Ñ PpHq G = SL(d)... SL(d)

8 Multi-Particle Entanglement Entanglement class of a pure state :...N ( C SLOCC G yx Ñ PpHq G = SL(d)... SL(d) Three Qubits: six classes (GHZ, W, bi-separable, product) Generically: infinitely many classes, labeled by exp(n) many continuous parameters

9 Single-Particle View of Multi- Particle Entanglement...N 2... N What can be said about the entanglement class of a pure state given its one-body density matrices only? What are the possible,..., N for states in a given entanglement class?

10 Single-Particle View of Multi- Particle Entanglement...N 2... N What can be said about the entanglement class of a pure state given its one-body density matrices only? What are the possible,..., N for states in a given entanglement class? only depends on local eigenvalues

11 Entanglement Polytopes ~ C ~ p ~,..., ~ N q for P C ( eigenvalues of,..., N

12 Entanglement Polytopes ~ C ~ p ~,..., ~ N q for P C ( entanglement polytope Our Main Result: convex polytope! finite hierarchy using results from Brion (987), Kempf & Ness (979) algebraic geometry / GIT algorithm to compute using computational invariant theory (difficult)

13 Entanglement Polytopes ~ C ~ p ~,..., ~ N q for P C ( entanglement polytope Our Main Result: convex polytope! finite hierarchy using results from Brion (987), Kempf & Ness (979) algebraic geometry / GIT algorithm to compute using computational invariant theory (difficult) cf. Quantum Marginal Problem Christandl-Mitchison (24) Klyachko (24) Daftuar-Hayden (24)

14 Entanglement Criterion ~ ~ R C ùñ R C efficient, requires only linearly many measurements robust against experimental noise ( «pure)

15 Entanglement Criterion Bell inequalities ~ ~ R C ùñ R C violation of Bell inequality efficient, requires only linearly many measurements robust against experimental noise ( «pure)

16 Three Qubits N =3 H = C 2 Entanglement classes: GHZy y` y (2) max W y y` y` y (3) max By y b p y ` yq, () max B2y, B3y y Han, Zhang & Guo (24) Botero & Mitchison (p.c.) Sawicki, W. & Kus (22) Dür, Vidal & Cirac (2)

17 Three Qubits N =3 2 H=C Entanglement classes: GHZy y ` y W y y ` y ` y (2) max (3) max By y b p y ` yq, B2y, B3y () max Han, Zhang & Guo (24) Botero & Mitchison (p.c.) Sawicki, W. & Kus (22) y Dür,Vidal & Cirac (2)

18 Three Qubits N =3 H = C 2 Entanglement classes: GHZy y` y (2) max W y y` y` y (3) max By y b p y ` yq, () max B2y, B3y y Han, Zhang & Guo (24) Botero & Mitchison (p.c.) Sawicki, W. & Kus (22) Dür, Vidal & Cirac (2)

19 Three Qubits N =3 H = C 2 Entanglement classes: GHZy y` y (2) max W y y` y` y (3) max By y b p y ` yq, () max B2y, B3y y Han, Zhang & Guo (24) Botero & Mitchison (p.c.) Sawicki, W. & Kus (22) Dür, Vidal & Cirac (2)

20 Three Qubits N =3 H = C 2 Entanglement classes: GHZy y` y (2) max W y y` y` y (3) max By y b p y ` yq, () max B2y, B3y y Han, Zhang & Guo (24) Botero & Mitchison (p.c.) Sawicki, W. & Kus (22) Dür, Vidal & Cirac (2)

21 Three Qubits N =3 2 H=C Entanglement classes: GHZy y ` y W y y ` y ` y (2) max (3) max By y b p y ` yq, B2y, B3y () max Lower pyramid is witness for GHZ class! y

22 Further Examples () max Four Qubits: six nontrivial entanglement.75 polytopes (3d sections) Bosonic qubit systems, fermionic system,... Sym N C 2 ^3C 6 Genuine multi-particle entanglement: â bipartition S:S c S ˆ Sc

23 Invariant Theory H C d b...b C d G SLpdq ˆ...ˆ SLpdq Suppose there exists a G-invariant homogeneous polynomial P on H with P p yq.

24 Invariant Theory H C d b...b C d G SLpdq ˆ...ˆ SLpdq Suppose there exists a G-invariant homogeneous polynomial P on H with P p yq. G y ñ G y S

25 Invariant Theory H C d b...b C d G SLpdq ˆ...ˆ SLpdq Suppose there exists a G-invariant homogeneous polynomial P on H with P p yq. G y ñ G y S y Let y be a vector of minimal length.

26 Invariant Theory H C d b...b C d G SLpdq ˆ...ˆ SLpdq Suppose there exists a G-invariant homogeneous polynomial P on H with P p yq. G y ñ G y S y Let y be a vector of minimal length. Then d dt ˇ ˇt ke tx yk 2 2x X y for all traceless local observables X.

27 Invariant Theory H C d b...b C d G SLpdq ˆ...ˆ SLpdq Suppose there exists a G-invariant homogeneous polynomial P on H with P p yq. G y ñ G y S y Let y be a vector of minimal length. Then d dt ˇ ˇt ke tx yk 2 2x X y for all traceless local observables X. Thus is locally maximally mixed!

28 Invariant Theory H C d b...b C d G SLpdq ˆ...ˆ SLpdq Suppose there exists a G-invariant homogeneous polynomial P on H with P p yq. ñ G y S y G y Invariant Theory Let y be a vector of minimal length. Then d ˇ ˇt ke tx yk 2 2x X y dt The converse is also true! for all traceless local observables X. Kempf & Ness (979) Klyachko (24) Thus is locally maximally mixed! Local eigenvalues

29 Entanglement Polytopes and Covariants Covariant: G-equivariant homogeneous polynomial function : H Ñ V Irreducible G-representation with highest weight

30 Entanglement Polytopes and Covariants Covariant: G-equivariant homogeneous polynomial function : H Ñ V Choose a finite set of generators weight and degree. k d k k with highest

31 Entanglement Polytopes and Covariants Covariant: G-equivariant homogeneous polynomial function : H Ñ V Choose a finite set of generators weight and degree. k d k k with highest Theorem: The entanglement polytope of an entanglement class C is given by proof via Brion (987) convt k {d k : kp q u

32 Entanglement Polytopes and Covariants Covariant: G-equivariant homogeneous polynomial function : H Ñ V computational invariant theory Choose a finite set of generators weight and degree. k d k k with highest Theorem: The entanglement polytope of an entanglement class C is given by proof via Brion (987) convt k {d k : kp q u

33 arxiv: Summary convt k {d k : kp q u Detecting Multi-Particle Entanglement by Single-Particle Measurements

34 arxiv: Thank you! convt k {d k : kp q u See also Burak s talk on Recoupling Coefficients and Quantum Entropies! Thursday 9:55