Quantum Marginal Problems

Transcription

1 Quantum Marginal Problems David Gross (Colgone) Joint with: Christandl, Doran, Lopes, Schilling, Walter

2 Outline Overview: Marginal problems Overview: Entanglement Main Theme: Entanglement Polytopes Shortly: Beyond the Pauli principle.

3 Overview: Marginal Problems

4 Marginals A marginal is obtained by integrating out parts of high-dim object

5 Marginals A marginal is obtained by integrating out parts of high-dim object Not every set of marginals is compatible

6 Marginals A marginal is obtained by integrating out parts of high-dim object Not every set of marginals is compatible Deciding compatibility is the marginal problem

7 Marginals in classical probability Marginals are distributions of subsets of variables.

8 Marginals in classical probability Marginals are distributions of subsets of variables. One classical marginal prob well-known in quantum:

9 Marginals in classical probability Marginals are distributions of subsets of variables. One classical marginal prob well-known in quantum: Bell tests.

10 Bell tests as marginal problems There are four random variables: polarization along two axes, as seen by Alice/Bob

11 Bell tests as marginal problems There are four random variables: polarization along two axes, as seen by Alice/Bob Only certain pairs accessible Q: Are these marginals compatible with classical distribution?

12 Bell tests as marginal problems There are four random variables: polarization along two axes, as seen by Alice/Bob Only certain pairs accessible Q: Are these marginals compatible with classical distribution? Compatible marginals form convex polytope Facets are Bell inequalities.

13 Bell tests as marginal problems There are four random variables: polarization along two axes, as seen by Alice/Bob Only certain pairs accessible Q: Are these marginals compatible with classical distribution? Compatible marginals form convex polytope Facets are Bell inequalities. Testing locality NP-hard so is classical marginal problem

14 Marginals in quantum theory For subset S i specify state ρ i. Q: Are these compatible: for some global ρ? ρ i = tr \Si ρ

15 Marginals in quantum theory For subset S i specify state ρ i. Q: Are these compatible: for some global ρ? ρ i = tr \Si ρ Would solve all finite-dim. few-body ground-state probs!

16 Marginals in quantum theory For subset S i specify state ρ i. Q: Are these compatible: for some global ρ? ρ i = tr \Si ρ Would solve all finite-dim. few-body ground-state probs! E.g.: For two-body Hamiltonian compute min ρ trhρ = min ρ H = n h i,j, i,j=1 trh i,j ρ = i,j min {ρ i,j } comp. trh i,j ρ i,j. i,j

17 Marginals in quantum theory: Ground States Remarks: min ρ trhρ = min {ρ i,j } comp. Left-hand side optimizes over O(d n ) variables. R.h.s. over O(n 2 d 4 ). Exponential improvement! trh i,j ρ i,j. i,j

18 Marginals in quantum theory: Ground States Remarks: min ρ trhρ = min {ρ i,j } comp. Left-hand side optimizes over O(d n ) variables. R.h.s. over O(n 2 d 4 ). Exponential improvement! Optimization over convex set of compatible ρ i,j. trh i,j ρ i,j. i,j

19 Marginals in quantum theory: Ground States Remarks: min ρ trhρ = min {ρ i,j } comp. Left-hand side optimizes over O(d n ) variables. R.h.s. over O(n 2 d 4 ). Exponential improvement! Optimization over convex set of compatible ρ i,j. trh i,j ρ i,j. i,j General theory of convex optimization Computational complexity of 2-RDM method (r.h.s) dominated by deciding compatibility of ρ i,j s.

20 Marginals in quantum theory: Ground States Remarks: min ρ trhρ = min {ρ i,j } comp. Left-hand side optimizes over O(d n ) variables. R.h.s. over O(n 2 d 4 ). Exponential improvement! Optimization over convex set of compatible ρ i,j. trh i,j ρ i,j. i,j General theory of convex optimization Computational complexity of 2-RDM method (r.h.s) dominated by deciding compatibility of ρ i,j s. Two directions: Progress on q. marginal prob. info about ground states Hardness of ground-states hardness of q. marginals.

21 Negative direction Computational complexity of 2-RDM method (r.h.s) dominated by deciding compatibility of ρ i,j s. But finding two-body ground-states is NP-hard (... and even QMA-hard)

22 Negative direction Computational complexity of 2-RDM method (r.h.s) dominated by deciding compatibility of ρ i,j s. But finding two-body ground-states is NP-hard (... and even QMA-hard) Thus: There is no efficient algorithm (quantum or classical) for the general two-body quantum marginal problem.

23 Negative direction Computational complexity of 2-RDM method (r.h.s) dominated by deciding compatibility of ρ i,j s. But finding two-body ground-states is NP-hard (... and even QMA-hard) Thus: There is no efficient algorithm (quantum or classical) for the general two-body quantum marginal problem. Remains hard for Fermions. Argument works for classical marginal prob. (hardness of Ising) Leaves room for outer approximations D. Mazziotti s talk.

24 Negative direction Computational complexity of 2-RDM method (r.h.s) dominated by deciding compatibility of ρ i,j s. But finding two-body ground-states is NP-hard (... and even QMA-hard) Thus: There is no efficient algorithm (quantum or classical) for the general two-body quantum marginal problem. Remains hard for Fermions. Argument works for classical marginal prob. (hardness of Ising) Leaves room for outer approximations D. Mazziotti s talk. Natural Question: Is there subproblem with enough structure to be tractable?

25 1-RDM marginal problem 1-RDM subproblem: marginals do not overlap, global state pure

26 1-RDM marginal problem 1-RDM subproblem: marginals do not overlap, global state pure Classical version: Globally pure no global randomness no local randomness.... trivial.

27 1-RDM marginal problem 1-RDM subproblem: marginals do not overlap, global state pure Classical version: Globally pure no global randomness no local randomness.... trivial. Quantum version: Globally pure no local randomness (in presence of entanglement).... seems non-trivial, but tractable!

28 1-RDM marginal problem Questions to be asked: Structure of set of 1-RDMs? What info about global ψ accessible from 1-RDM? Computational complexity of 1-RDM marginal prob.? Practical uses?

29 Structure of 1-RDMs.

30 Reduction to eigenvalues Local basis change does not affect compatibility can assume ρ i are diagonal described by eigenvalues λ (1),..., λ (n) R dn.

31 Reduction to eigenvalues Local basis change does not affect compatibility can assume ρ i are diagonal described by eigenvalues λ (1),..., λ (n) R dn. Question becomes: Which set of ordered local eigenvalues λ (i) can occur?

32 Reduction to eigenvalues Local basis change does not affect compatibility can assume ρ i are diagonal described by eigenvalues λ (1),..., λ (n) R dn. Question becomes: Which set of ordered local eigenvalues λ (i) can occur? Deep fact: Compatible spectra form convex polytope

33 Reduction to eigenvalues Local basis change does not affect compatibility can assume ρ i are diagonal described by eigenvalues λ (1),..., λ (n) R dn. Question becomes: Which set of ordered local eigenvalues λ (i) can occur? Deep fact: Compatible spectra form convex polytope Highly non-trivial! (Global set is not convex).

34 Reduction to eigenvalues Local basis change does not affect compatibility can assume ρ i are diagonal described by eigenvalues Question becomes: λ (1),..., λ (n) R dn. Which set of ordered local eigenvalues λ (i) can occur? Deep fact: Compatible spectra form convex polytope Highly non-trivial! (Global set is not convex). Several proofs, building on symplectic geometry & asymptotic rep theory [Klyachko, Kirwan, Christandl, Mitchison, Harrow, Daftuar, Hayden,... ]

35 Reduction to eigenvalues Local basis change does not affect compatibility can assume ρ i are diagonal described by eigenvalues Question becomes: λ (1),..., λ (n) R dn. Which set of ordered local eigenvalues λ (i) can occur? Deep fact: Compatible spectra form convex polytope Highly non-trivial! (Global set is not convex). Several proofs, building on symplectic geometry & asymptotic rep theory [Klyachko, Kirwan, Christandl, Mitchison, Harrow, Daftuar, Hayden,... ] No conceptually simple proof known to me!

36 Example: d = n = 2 Warm up: work out solution for two qubits.

37 Example: d = n = 2 Warm up: work out solution for two qubits. Schmidt-decomposition: ψ = λ (1) e 1 f 1 + λ (2) e 2 f 2 With ρ 1 = λ (1) e 1 e 1 +λ (2) e 2 e 2, ρ 2 = λ (1) f 1 f 1 +λ (2) f 2 f 2.

38 Example: d = n = 2 Warm up: work out solution for two qubits. Schmidt-decomposition: ψ = λ (1) e 1 f 1 + λ (2) e 2 f 2 With ρ 1 = λ (1) e 1 e 1 +λ (2) e 2 e 2, ρ 2 = λ (1) f 1 f 1 +λ (2) f 2 f 2. So eigenvalues must be equal: λ 1 = λ 2.

39 Example: d = n = 2 Warm up: work out solution for two qubits. Schmidt-decomposition: ψ = λ (1) e 1 f 1 + λ (2) e 2 f 2 With ρ 1 = λ (1) e 1 e 1 +λ (2) e 2 e 2, ρ 2 = λ (1) f 1 f 1 +λ (2) f 2 f 2. So eigenvalues must be equal: λ 1 = λ 2. In terms of largest eigenvalue, get simple polytope:

40 Further examples Three qubits: Three fermions on 6 modes ( Dennis-Borland ): (c.f. M. Christandl s talk)

41 Summary: Structure of 1-RDM s Compatible 1-RDMs described by convex polytopes of spectra.

42 Summary: Structure of 1-RDM s Compatible 1-RDMs described by convex polytopes of spectra. If this doesn t surprise you, I m terribly sad.

43 Computational aspects.

44 List inequalities? Polytopes characterized by finitely many linear ineqs D, x c. [Klyachko, Altunbulak]

45 List inequalities? Polytopes characterized by finitely many linear ineqs D, x c. [Klyachko, Altunbulak] Ansatz so far: Compute all ineqs Altunbulak s talk Doesn t seem to scale: too many ineqs as n, d go up.

46 List inequalities? There might be better algorithm than checking all ineqs. [Klyachko, Altunbulak]

47 List inequalities? There might be better algorithm than checking all ineqs. Ex.: l 1 -unit ball in R n has 2 n linear ineqs, but membership equivalent to [Klyachko, Altunbulak] x l1 = n x i 1. i=1

48 Computational complexity Thus, central open question: Q.: Is there a poly-time algorithm that decides the 1-RDM quantum marginal problem?

49 Computational complexity Thus, central open question: Q.: Is there a poly-time algorithm that decides the 1-RDM quantum marginal problem? Progress Nov [Burgisser, Christandl, Mulmuley, Walter]: Problem in NP conp Virtually guarantees that it can t be proven hard Suggests it might be in P.

50 Info about global state from 1-RDMs. Part 1: Selection rules.

51 Selection rules Selection rule, Generalized Hartree-Fock : If a state ψ maps to the boundary of the polytope, only few, special Slater determinants can appear in an expansion of ψ.

52 Selection rules Selection rule, Generalized Hartree-Fock : If a state ψ maps to the boundary of the polytope, only few, special Slater determinants can appear in an expansion of ψ. Stated by Klyachko (2009). He didn t feel proof was necessary. True for for general scenarios stated here for Fermions. v v v (b) (d) [Schilling, DG, Christandl, PRL 13] (a) v (c)

53 Selection rules Consider n-fermion system with modes {φ 1,..., φ d }.

54 Selection rules Consider n-fermion system with modes {φ 1,..., φ d }. Expand ψ n C d in Slater dets: ψ = c i1,...,i n φ i1 φ in. (1) i 1 < <i n

55 Selection rules Consider n-fermion system with modes {φ 1,..., φ d }. Expand ψ n C d in Slater dets: ψ = c i1,...,i n φ i1 φ in. (1) i 1 < <i n Assume (wlog) ρ (1) (ψ) is diagonal with eigenvalues λ

56 Selection rules Consider n-fermion system with modes {φ 1,..., φ d }. Expand ψ n C d in Slater dets: ψ = c i1,...,i n φ i1 φ in. (1) i 1 < <i n Assume (wlog) ρ (1) (ψ) is diagonal with eigenvalues λ Let D, x c be a face of the 1-RDM polytope.

57 Selection rules Consider n-fermion system with modes {φ 1,..., φ d }. Expand ψ n C d in Slater dets: ψ = c i1,...,i n φ i1 φ in. (1) i 1 < <i n Assume (wlog) ρ (1) (ψ) is diagonal with eigenvalues λ Let D, x c be a face of the 1-RDM polytope. If the eigenvalues of ψ saturate the ineq. D, λ = c, then (1) only contains Slater dets whose eigenvalues do so as well.

58 Selection rules: Proof If the eigenvalues of ψ saturate the ineq. D, λ = c, then (1) only contains Slater dets whose eigenvalues do so as well. Elementary proof [Alex Lopes, PhD thesis; Lopes, Schilling, DG, in eternal prep.]

59 Selection rules: Proof If the eigenvalues of ψ saturate the ineq. D, λ = c, then (1) only contains Slater dets whose eigenvalues do so as well. Elementary proof [Alex Lopes, PhD thesis; Lopes, Schilling, DG, in eternal prep.] Trick: Introduce operator ˆD = i D i a i a i. Then D, λ = tr ˆDρ (1) = tr ˆD ψ ψ.

60 Selection rules: Proof If the eigenvalues of ψ saturate the ineq. D, λ = c, then (1) only contains Slater dets whose eigenvalues do so as well. Elementary proof [Alex Lopes, PhD thesis; Lopes, Schilling, DG, in eternal prep.] Trick: Introduce operator ˆD = i D i a i a i. Then Selection rule equivalent to: D, λ = tr ˆDρ (1) = tr ˆD ψ ψ. If tr ˆD ψ ψ = c, then ˆD ψ = c ψ. (Non-trivial, as c need not be extremal eigenvalue of ˆD).

61 Selection rules: Proof If the eigenvalues of ψ saturate the ineq. D, λ = c, then (1) only contains Slater dets whose eigenvalues do so as well. Elementary proof [Alex Lopes, PhD thesis; Lopes, Schilling, DG, in eternal prep.] Trick: Introduce operator ˆD = i D i a i a i. Then Selection rule equivalent to: D, λ = tr ˆDρ (1) = tr ˆD ψ ψ. If tr ˆD ψ ψ = c, then ˆD ψ = c ψ. (Non-trivial, as c need not be extremal eigenvalue of ˆD). Proof: Blackboard.

62 Info about global state from 1-RDMs. Part 2: Entanglement.

63 Entanglement Two pure states ψ, φ are in same entanglement class if they can be converted into each other with finite probability of success using local operations and classical communication.

64 Entanglement Two pure states ψ, φ are in same entanglement class if they can be converted into each other with finite probability of success using local operations and classical communication. Often referred to as SLOCC classes. But that sounds too unpleasant.

65 Entanglement Two pure states ψ, φ are in same entanglement class if they can be converted into each other with finite probability of success using local operations and classical communication. Often referred to as SLOCC classes. But that sounds too unpleasant. Formally: ψ φ ψ = (g 1 g n )φ with g i local invertible matrices (filtering operations). Mathematically: We re looking at SL(C d ) n -orbits in ( C d) n.

66 SLOCC, SLOCC! Who s There? For three qubits (d = 2, n = 3), equivalence classes known since mid-1800s. Re-discovered in 2000 to great effect:

67 Examples Classes: Products ψ = φ 1 φ 2 φ 3. Three classes of bi-separable states: ψ = φ 1 φ 2,3. The W-class: W = The GHZ-class: GHZ =

68 Further examples 4 qubits: Classification apparently first obtained in QI community [Verstraete et al. (2002)]. Nine families of four complex parameters each.

69 Further examples 4 qubits: Classification apparently first obtained in QI community [Verstraete et al. (2002)]. Nine families of four complex parameters each. Beyond: Number of parameters required to label orbits increases exponentially. Only sporadic facts known.

70 Desiderata Can we come up with theory that is systematic (any number of particles, local dimensions, symmetry constraints), is efficient (only polynomial number of parameters have to be learned), experimentally feasible (parameters easily accessible, robust to noise)?

71 Desiderata Can we come up with theory that is systematic (any number of particles, local dimensions, symmetry constraints), is efficient (only polynomial number of parameters have to be learned), experimentally feasible (parameters easily accessible, robust to noise)? Claim: The single-site quantum marginal problem lives up to these standards.

72 Entanglement Polytopes

73 Central observation, entanglement polytopes Set of allowed eigenvalues may depend on entanglement class of global state.

74 Central observation, entanglement polytopes Set of allowed eigenvalues may depend on entanglement class of global state. Thus: To every class C, associated set C of local eigenvalues of states in (closure of) C.

75 Central observation, entanglement polytopes Set of allowed eigenvalues may depend on entanglement class of global state. Thus: To every class C, associated set C of local eigenvalues of states in (closure of) C. Turns out: C is again polytope: the entanglement polytope associated with C. [Walter, Doran, Gross, Christandl, Science 2013]

76 Central observation, entanglement polytopes Set of allowed eigenvalues may depend on entanglement class of global state. Thus: To every class C, associated set C of local eigenvalues of states in (closure of) C. Turns out: C is again polytope: the entanglement polytope associated with C. [Walter, Doran, Gross, Christandl, Science 2013] Clearly: the position of λ(ψ) w.r.t. the entanglement polytopes contains all local information about global entanglement class.

77 Examples re-visited: 3 qubit entanglement polytopes For three qubits, polytopes resolve all 6 entanglement classes: [Hang et al. (2004), Sawicki et al. (2012), our paper]

78 Examples re-visited: 3 qubit entanglement polytopes For three qubits, polytopes resolve all 6 entanglement classes: [Hang et al. (2004), Sawicki et al. (2012), our paper] W-class corresponds to upper pyramid : λ (1) max + λ (2) max + λ (3) max 2. Any violation of that witnesses GHZ-type entanglement.

79 Examples re-visited: 4 qubit entanglement polytopes 4 qubits: Entanglement classes: 9 families with up to four complex parameters each [Verstraete et al. (2002)].

80 Examples re-visited: 4 qubit entanglement polytopes 4 qubits: Entanglement classes: 9 families with up to four complex parameters each [Verstraete et al. (2002)]. Entanglement Polytopes: 13 polytopes, 7 of which are genuinely 4-party entangled.

81 Examples re-visited: 4 qubit entanglement polytopes 4 qubits: Entanglement classes: 9 families with up to four complex parameters each [Verstraete et al. (2002)]. Entanglement Polytopes: 13 polytopes, 7 of which are genuinely 4-party entangled. We feel: attractive balance between coarse-graining and preserving structure. Example: 4-qubit W-class C W again an upper pyramid : λ (1) max + λ (2) max + λ (3) max + λ (4) max 3.

82 Example: 4 qubit entanglement polytopes

83 Example: Bosonic qubits Consider n bosonic qubits: ψ Sym n ( C 2).

84 Example: Bosonic qubits Consider n bosonic qubits: ψ Sym n ( C 2). Symmetry all local reductions are equal: ρ (1) i,j = ψ a i a j ψ. single number captures all: λ max [0.5, 1].

85 Example: Bosonic qubits Consider n bosonic qubits: ψ Sym n ( C 2). Symmetry all local reductions are equal: ρ (1) i,j = ψ a i a j ψ. single number captures all: λ max [0.5, 1]. Analyze polytopes: 0,..., 0 in all C s C = [γ C, 1].

86 Example: Bosonic qubits Consider n bosonic qubits: ψ Sym n ( C 2). Symmetry all local reductions are equal: ρ (1) i,j = ψ a i a j ψ. single number captures all: λ max [0.5, 1]. Analyze polytopes: 0,..., 0 in all C s C = [γ C, 1]. Turns out: Possible choices are { } { } 1 N k γ C : k = 0, 1,..., N/ N... with innermost point γ the image of W -type states.

87 Example: No Solipsism A vector is genuinely n-partite entangled if it does not factorize w.r.t. any bi-partition: ψ ψ 1 ψ 2.

88 Example: No Solipsism A vector is genuinely n-partite entangled if it does not factorize w.r.t. any bi-partition: ψ ψ 1 ψ 2. Observation: sometimes detectable from local spectra alone.

89 Example: No Solipsism A vector is genuinely n-partite entangled if it does not factorize w.r.t. any bi-partition: ψ ψ 1 ψ 2. Observation: sometimes detectable from local spectra alone. spectra ( λ (1),..., λ (n) ) compatible, but no bi-partition is. Example: (λ (1) max,..., λ (n) max) = ( n 1, 1 1 n 1,..., 1 1 ). n 1

90 Example: No Solipsism A vector is genuinely n-partite entangled if it does not factorize w.r.t. any bi-partition: ψ ψ 1 ψ 2. Observation: sometimes detectable from local spectra alone. spectra ( λ (1),..., λ (n) ) compatible, but no bi-partition is. Example: (λ (1) max,..., λ (n) max) = Interpretation: ( n 1, 1 1 n 1,..., 1 1 ). n 1 no solipsism: love needs a partner! (And entangled qubits need their counter-parts).

91 Example: Distillation Entanglement measures from local information: (Linear) entropy of entanglement E(ψ) = 1 1 trρ 2 i N simple function of Euclidean distance of eigenvalue point to origin. Closer to origin more entanglement. i

92 Example: Distillation Entanglement measures from local information: (Linear) entropy of entanglement E(ψ) = 1 1 trρ 2 i N simple function of Euclidean distance of eigenvalue point to origin. Closer to origin more entanglement. can bound distillable entanglement from local information! i

93 Example: Distillation Entanglement measures from local information: (Linear) entropy of entanglement E(ψ) = 1 1 trρ 2 i N simple function of Euclidean distance of eigenvalue point to origin. Closer to origin more entanglement. can bound distillable entanglement from local information! Can even give distillation procedure without need to know state beyond local densities (generalizing [Verstraete et al. 2002]). i

94 Pure??? Yeah, but no pure state exists in Nature.

95 Pure??? Yeah, but no pure state exists in Nature. Results are epsilonifiable: if distance d of spectrum to a polytope exceeds 4N 1 p, then ρ conv( ). p = tr ρ 2 is purity, which an be lower-bounded from local information alone.

96 Summary of Entanglement Polytopes Locally accessible info about global entanglement encoded in entanglement polytopes subpolytopes of the set of admissible local spectra. Provides a systematic and efficient way of obtaining information about entanglement classes.

97 Thank you for your attention! David Gross (Uni Cologne) Oxford, April 2016