Calculation of Generalized Pauli Constraints

Transcription

1 Calculation of Generalized Pauli Constraints Murat Altunbulak Department of Mathematics Dokuz Eylul University April, 2016

2 Notations

3 Notations Quantum States Quantum system A is described by a complex Hilbert space H A,

4 Notations Quantum States Quantum system A is described by a complex Hilbert space H A, called state space.

5 Notations Quantum States Quantum system A is described by a complex Hilbert space H A, called state space. Pure state = unit vector ψ H A or projector operator ψ ψ

6 Notations Quantum States Quantum system A is described by a complex Hilbert space H A, called state space. Pure state = unit vector ψ H A or projector operator ψ ψ Mixed state = classical mixture of pure states ρ = i p i ψ i ψ i ; p i 0 ; p i = 1 i

7 Notations Quantum States Quantum system A is described by a complex Hilbert space H A, called state space. Pure state = unit vector ψ H A or projector operator ψ ψ Mixed state = classical mixture of pure states ρ = i p i ψ i ψ i ; p i 0 ; p i = 1 i ρ is a non-negative (ρ 0) Hermitian operator with Trρ = 1, called Density matrix.

8 Notations Superposition Principle implies that the state space of composite system AB splits into tensor product of its components A and B H AB = H A H B

9 Notations Superposition Principle implies that the state space of composite system AB splits into tensor product of its components A and B H AB = H A H B Density Matrix of Composite Systems Density matrix of composite system can be written as linear combination ρ AB = α a α L α A L α B where L α A, Lα B are linear operators on H A, H B, respectively.

10 Notations Reduced state Its reduced matrices are defined by partial traces

11 Notations Reduced state Its reduced matrices are defined by partial traces ρ A = α a α Tr(L α B)L α A := Tr B (ρ AB )

12 Notations Reduced state Its reduced matrices are defined by partial traces ρ A = α a α Tr(L α B)L α A := Tr B (ρ AB ) ρ B = α a α Tr(L α A)L α B := Tr A (ρ AB )

13 Notations Reduced state Its reduced matrices are defined by partial traces ρ A = α a α Tr(L α B)L α A := Tr B (ρ AB ) ρ B = α a α Tr(L α A)L α B := Tr A (ρ AB ) they are called one-particle reduced density matrices.

14 Pauli Exclusion Principle

15 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state.

16 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows

17 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H N, H N = state space of N electrons.

18 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H N, H N = state space of N electrons. Define ρ = ρ N 1 + ρn ρn N sum of all reduced states ρn i : H. ρ is called electron density matrix.

19 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H N, H N = state space of N electrons. Define ρ = ρ N 1 + ρn ρn N sum of all reduced states ρn i : H. ρ is called electron density matrix. In terms of electron density matrix ρ, Pauli principle amounts to ψ ρ ψ 1 Specρ 1

20 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H N, H N = state space of N electrons. Define ρ = ρ N 1 + ρn ρn N sum of all reduced states ρn i : H. ρ is called electron density matrix. In terms of electron density matrix ρ, Pauli principle amounts to ψ ρ ψ 1 Specρ 1 Modern Version (1926) Heisenberg-Dirac replaced the Pauli exclusion principle by skew-symmetry of multi-electron wave function

21 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H N, H N = state space of N electrons. Define ρ = ρ N 1 + ρn ρn N sum of all reduced states ρn i : H. ρ is called electron density matrix. In terms of electron density matrix ρ, Pauli principle amounts to ψ ρ ψ 1 Specρ 1 Modern Version (1926) Heisenberg-Dirac replaced the Pauli exclusion principle by skew-symmetry of multi-electron wave function which implies that the state space of N electrons shrinks to N H H N.

22 Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H N, H N = state space of N electrons. Define ρ = ρ N 1 + ρn ρn N sum of all reduced states ρn i : H. ρ is called electron density matrix. In terms of electron density matrix ρ, Pauli principle amounts to ψ ρ ψ 1 Specρ 1 Modern Version (1926) Heisenberg-Dirac replaced the Pauli exclusion principle by skew-symmetry of multi-electron wave function which implies that the state space of N electrons shrinks to N H H N. This implies the original Pauli principle, because ψ ψ = 0.

23 Statement of The Problem

24 Statement of the problem In the latter case, electron density matrix becomes ρ = Nρ N 1, Trρ = N.

25 Statement of the problem In the latter case, electron density matrix becomes ρ = Nρ N 1, Trρ = N. Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Specρ 1.

26 Statement of the problem In the latter case, electron density matrix becomes ρ = Nρ N 1, Trρ = N. Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Specρ 1. Pure N-representability The above problem became known as (pure) N-representability problem after A.J. Coleman (1963).

27 Statement of the problem In the latter case, electron density matrix becomes ρ = Nρ N 1, Trρ = N. Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Specρ 1. Pure N-representability The above problem became known as (pure) N-representability problem after A.J. Coleman (1963). Mixed N-representability More generally we have mixed N-representability problem: What are the constraints on the spectra of a mixed state and its reduced matrix?

28 What was known before 2008?

29 What was known before 2008? Initial constraints Ordering inequalities λ 1 λ 2... λ r 0

30 What was known before 2008? Initial constraints Ordering inequalities λ 1 λ 2... λ r 0 Normalization condition Trρ = i λ i = N

31 What was known before 2008? Initial constraints Ordering inequalities λ 1 λ 2... λ r 0 Normalization condition Trρ = i λ i = N Two-particle ( 2 H r ) and Two-hole ( r 2 H r ) systems Constraints on electron density matrix ρ is given by even degeneracy of its eigenvalues, i.e. λ 1 = λ 2 λ 3 = λ 4...

32 What was known before 2008? Initial constraints Ordering inequalities λ 1 λ 2... λ r 0 Normalization condition Trρ = i λ i = N Two-particle ( 2 H r ) and Two-hole ( r 2 H r ) systems Constraints on electron density matrix ρ is given by even degeneracy of its eigenvalues, i.e. λ 1 = λ 2 λ 3 = λ 4... Similar results hold for two-hole system.

33 What was known before 2008? Borland-Dennis system For the system 3 H 6 of three electrons of rank 6, the N-representability conditions are given by the following (in)equalities: λ 1 + λ 6 = λ 2 + λ 5 = λ 3 + λ 4 = 1, λ 4 λ 5 + λ 6,

34 What was known before 2008? Borland-Dennis system For the system 3 H 6 of three electrons of rank 6, the N-representability conditions are given by the following (in)equalities: λ 1 + λ 6 = λ 2 + λ 5 = λ 3 + λ 4 = 1, λ 4 λ 5 + λ 6, Sufficiency proved by Borland-Dennis (1972)

35 What was known before 2008? Borland-Dennis system For the system 3 H 6 of three electrons of rank 6, the N-representability conditions are given by the following (in)equalities: λ 1 + λ 6 = λ 2 + λ 5 = λ 3 + λ 4 = 1, λ 4 λ 5 + λ 6, Sufficiency proved by Borland-Dennis (1972) Necessity proved by Ruskai (2007)

36 Formal Solution of Mixed N-representability

37 Formal Solution Solution of Mixed N-representability Main Theorem. Let ρ N : N H r (Trρ N = 1) be mixed state and ρ : H r (Trρ = N) be its particle density matrix.

38 Formal Solution Solution of Mixed N-representability Main Theorem. Let ρ N : N H r (Trρ N = 1) be mixed state and ρ : H r (Trρ = N) be its particle density matrix. Then all constraints on Specρ = λ : λ 1 λ 2... λ r and Specρ N = µ : µ 1 µ 2... µ R (R = ( r N) = dim N H r ) are given by the following linear inequalities (Generalized Pauli Constraints)

39 Formal Solution Solution of Mixed N-representability Main Theorem. Let ρ N : N H r (Trρ N = 1) be mixed state and ρ : H r (Trρ = N) be its particle density matrix. Then all constraints on Specρ = λ : λ 1 λ 2... λ r and Specρ N = µ : µ 1 µ 2... µ R (R = ( r N) = dim N H r ) are given by the following linear inequalities (Generalized Pauli Constraints) a i λ v(i) ( N a) k µ w(k) (a, v, w) i k

40 Formal Solution Solution of Mixed N-representability Main Theorem. Let ρ N : N H r (Trρ N = 1) be mixed state and ρ : H r (Trρ = N) be its particle density matrix. Then all constraints on Specρ = λ : λ 1 λ 2... λ r and Specρ N = µ : µ 1 µ 2... µ R (R = ( r N) = dim N H r ) are given by the following linear inequalities (Generalized Pauli Constraints) a i λ v(i) ( N a) k µ w(k) (a, v, w) i k Here, N a consists of all sums a i1 + a i2 + + a in, 1 i 1 < i 2 < < i N r arranged in decreasing order, v S r and w S R subject to topological condition c w v (a) 0 explained soon.

41 Remark Solution of Pure N-representability Solution of pure N-representability problem can be deduced from the above theorem by specialization µ i = 0 for i 1, because pure states corresponds projector operators ψ ψ.

42 Remark Solution of Pure N-representability Solution of pure N-representability problem can be deduced from the above theorem by specialization µ i = 0 for i 1, because pure states corresponds projector operators ψ ψ. In this case all costraints together with ordering ineqaulities and normalization condition describe a convex polytope, called Moment Polytope.

43 Nature of Topological Condition c v w(a) 0

44 Nature of Topological Condition c v w(a) 0 Consider flag variety F a (H r ) = {X : H r H r a = Spec(X )}

45 Nature of Topological Condition c v w(a) 0 Consider flag variety F a (H r ) = {X : H r H r a = Spec(X )} and morphism ϕ a : F a (H r ) F a( N H N r ) X X (N)

46 Nature of Topological Condition c v w(a) 0 Consider flag variety F a (H r ) = {X : H r H r a = Spec(X )} and morphism ϕ a : F a (H r ) F a( N H N r ) X X (N) Here, X (N) : ψ 1 ψ 2... ψ N i ψ 1 ψ 2... X ψ i... ψ N.

47 Nature of Topological Condition c v w(a) 0 The coefficients c v w (a) are defined via induced morphism of cohomology

48 Nature of Topological Condition c v w(a) 0 The coefficients c v w (a) are defined via induced morphism of cohomology ϕ a : H (F N a( N H r )) H (F a (H r )), σ w v c v w (a)σ v

49 Nature of Topological Condition c v w(a) 0 The coefficients c v w (a) are defined via induced morphism of cohomology ϕ a : H (F N a( N H r )) H (F a (H r )), σ w v c v w (a)σ v written in the basis of Schubert cocycles σ w.

50 Connection with Representation Theory

51 Connection with Representation Theory Plethysm Consider the m-th symmetric power of N H r, called Plethysm

52 Connection with Representation Theory Plethysm Consider the m-th symmetric power of N H r, called Plethysm It splits into irreducible components H λ, parameterized by Young diagrams λ = of size N.m, with multiplicity m λ S m ( N H r ) = λ m λ H λ.

53 Connection with Representation Theory Plethysm Consider the m-th symmetric power of N H r, called Plethysm It splits into irreducible components H λ, parameterized by Young diagrams λ = of size N.m, with multiplicity m λ S m ( N H r ) = λ m λ H λ. Problem Which irreducible representations H λ of U(H r ) can appear in the decomposition of S m ( N H r )?

54 Connection with Representation Theory Plethysm Consider the m-th symmetric power of N H r, called Plethysm It splits into irreducible components H λ, parameterized by Young diagrams λ = of size N.m, with multiplicity m λ S m ( N H r ) = λ m λ H λ. Problem Which irreducible representations H λ of U(H r ) can appear in the decomposition of S m ( N H r )? Surprisingly solution of this problem coincides with that of pure N-representability problem.

55 Connection with Representation Theory Connection treat the diagrams λ as spectra. λ : λ 1 λ 2... λ r.

56 Connection with Representation Theory Connection treat the diagrams λ as spectra. λ : λ 1 λ 2... λ r. normalize them to a fixed size λ = λ/m s.t.

57 Connection with Representation Theory Connection treat the diagrams λ as spectra. λ : λ 1 λ 2... λ r. normalize them to a fixed size λ = λ/m s.t. Tr λ = N.

58 Connection with Representation Theory Connection treat the diagrams λ as spectra. λ : λ 1 λ 2... λ r. normalize them to a fixed size λ = λ/m s.t. Tr λ = N. Representation theoretical solution Theorem. Every λ obtained from irreducible component H λ S m ( N H r ) is a spectrum of one point reduced matrix ρ of a pure state ψ N H r.

59 Connection with Representation Theory Connection treat the diagrams λ as spectra. λ : λ 1 λ 2... λ r. normalize them to a fixed size λ = λ/m s.t. Tr λ = N. Representation theoretical solution Theorem. Every λ obtained from irreducible component H λ S m ( N H r ) is a spectrum of one point reduced matrix ρ of a pure state ψ N H r. Moreover every one point reduced spectrum is a convex combination of such spectra λ with bounded m M.

60 Practical Algorithm

61 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,

62 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,while any set of inequalities of Main Theorem amounts to its outer approximation.

63 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,while any set of inequalities of Main Theorem amounts to its outer approximation. This suggests the following approach to the pure N-representability problem, which combines both theorems.

64 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,while any set of inequalities of Main Theorem amounts to its outer approximation. This suggests the following approach to the pure N-representability problem, which combines both theorems. The Algorithm Find all irreducible components H λ S m ( N H r ) for m M.

65 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,while any set of inequalities of Main Theorem amounts to its outer approximation. This suggests the following approach to the pure N-representability problem, which combines both theorems. The Algorithm Find all irreducible components H λ S m ( N H r ) for m M. Calculate the convex hull of the corresponding spectra λ which gives an inner approximation PM in P for the moment polytope P.

66 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,while any set of inequalities of Main Theorem amounts to its outer approximation. This suggests the following approach to the pure N-representability problem, which combines both theorems. The Algorithm Find all irreducible components H λ S m ( N H r ) for m M. Calculate the convex hull of the corresponding spectra λ which gives an inner approximation PM in P for the moment polytope P. Identify the facets of PM in that are given by the inequalities of the Main Theorem. They cut out an outer approximation PM out P.

67 Practical Algorithm For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,while any set of inequalities of Main Theorem amounts to its outer approximation. This suggests the following approach to the pure N-representability problem, which combines both theorems. The Algorithm Find all irreducible components H λ S m ( N H r ) for m M. Calculate the convex hull of the corresponding spectra λ which gives an inner approximation PM in P for the moment polytope P. Identify the facets of PM in that are given by the inequalities of the Main Theorem. They cut out an outer approximation PM out P. Increase M and continue until PM in = Pout M.

68 How it works?

69 How it works? The above algorithm works perfectly only for some small systems.

70 How it works? The above algorithm works perfectly only for some small systems. Borland-Dennis System (N = 3, r = 6) We only need to calculate symmetric powers of 3 H 6 up to 4th degree. That is, M = 4.

71 How it works? The above algorithm works perfectly only for some small systems. Borland-Dennis System (N = 3, r = 6) We only need to calculate symmetric powers of 3 H 6 up to 4th degree. That is, M = 4. Symmetric powers contains the following components

72 How it works? The above algorithm works perfectly only for some small systems. Borland-Dennis System (N = 3, r = 6) We only need to calculate symmetric powers of 3 H 6 up to 4th degree. That is, M = 4. Symmetric powers contains the following components m λ 1 [1, 1, 1, 0, 0, 0] 2 [2, 2, 2, 0, 0, 0], [2, 1, 1, 1, 1, 0] 3 [2, 2, 2, 1, 1, 1], [3, 3, 3, 0, 0, 0], [3, 2, 2, 1, 1, 0] 4 [2, 2, 2, 2, 2, 2], [3, 3, 3, 1, 1, 1], [4, 4, 4, 0, 0, 0], [3, 3, 2, 2, 1, 1], [4, 3, 3, 1, 1, 0], [4, 2, 2, 2, 2, 0]

73 How it works? Borland-Dennis System (N = 3, r = 6) After the normalization we obtain the spectra λ m λ 1 [1, 1, 1, 0, 0, 0] 2 [1, 1, 1, 0, 0, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] 3 [ 2 3, 2 3, 2 3, 2 3, 1 3, 1 3 ], [1, 1, 1, 0, 0, 0], [1, 2 3, 2 3, 1 3, 1 3, 0] 4 [ 1 2, 1 2, 1 2, 1 2, 1 2, 1 2 ], [ 3 4, 3 4, 3 4, 1 4, 1 4, 1 4 ], [1, 1, 1, 0, 0, 0], [ 3 4, 3 4, 1 2, 1 2, 1 4, 1 4 ], [1, 3 4, 3 4, 1 4, 1 4, 0], [1, 1 2, 1 2, 1 2, 1 2, 0]

74 How it works? Borland-Dennis System (N = 3, r = 6) After the normalization we obtain the spectra λ m λ 1 [1, 1, 1, 0, 0, 0] 2 [1, 1, 1, 0, 0, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] 3 [ 2 3, 2 3, 2 3, 2 3, 1 3, 1 3 ], [1, 1, 1, 0, 0, 0], [1, 2 3, 2 3, 1 3, 1 3, 0] 4 [ 1 2, 1 2, 1 2, 1 2, 1 2, 1 2 ], [ 3 4, 3 4, 3 4, 1 4, 1 4, 1 4 ], [1, 1, 1, 0, 0, 0], [ 3 4, 3 4, 1 2, 1 2, 1 4, 1 4 ], [1, 3 4, 3 4, 1 4, 1 4, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] Taking convex hull of these spectra we obtain the full moment polytope which is described by 3 equalities and 1 inequality.

75 How it works? Borland-Dennis System (N = 3, r = 6) After the normalization we obtain the spectra λ m λ 1 [1, 1, 1, 0, 0, 0] 2 [1, 1, 1, 0, 0, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] 3 [ 2 3, 2 3, 2 3, 2 3, 1 3, 1 3 ], [1, 1, 1, 0, 0, 0], [1, 2 3, 2 3, 1 3, 1 3, 0] 4 [ 1 2, 1 2, 1 2, 1 2, 1 2, 1 2 ], [ 3 4, 3 4, 3 4, 1 4, 1 4, 1 4 ], [1, 1, 1, 0, 0, 0], [ 3 4, 3 4, 1 2, 1 2, 1 4, 1 4 ], [1, 3 4, 3 4, 1 4, 1 4, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] Taking convex hull of these spectra we obtain the full moment polytope which is described by 3 equalities and 1 inequality. N = 3, r = 7 M = 8.

76 How it works? Borland-Dennis System (N = 3, r = 6) After the normalization we obtain the spectra λ m λ 1 [1, 1, 1, 0, 0, 0] 2 [1, 1, 1, 0, 0, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] 3 [ 2 3, 2 3, 2 3, 2 3, 1 3, 1 3 ], [1, 1, 1, 0, 0, 0], [1, 2 3, 2 3, 1 3, 1 3, 0] 4 [ 1 2, 1 2, 1 2, 1 2, 1 2, 1 2 ], [ 3 4, 3 4, 3 4, 1 4, 1 4, 1 4 ], [1, 1, 1, 0, 0, 0], [ 3 4, 3 4, 1 2, 1 2, 1 4, 1 4 ], [1, 3 4, 3 4, 1 4, 1 4, 0], [1, 1 2, 1 2, 1 2, 1 2, 0] Taking convex hull of these spectra we obtain the full moment polytope which is described by 3 equalities and 1 inequality. N = 3, r = 7 M = 8. N = 4, r = 8 M = 10.

77 How it works? N = 3, r = 8

78 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope.

79 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope. That is, P24 in Pout 24.

80 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope. That is, P24 in Pout 24. There was one problematic facet.

81 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope. That is, P24 in Pout 24. There was one problematic facet. To get rid of this, we resort to use a numerical minimization of the linear form obtained from the problematic facet over all particle density matrices.

82 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope. That is, P24 in Pout 24. There was one problematic facet. To get rid of this, we resort to use a numerical minimization of the linear form obtained from the problematic facet over all particle density matrices. It turns out that the form attains its minimum at a single point.

83 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope. That is, P24 in Pout 24. There was one problematic facet. To get rid of this, we resort to use a numerical minimization of the linear form obtained from the problematic facet over all particle density matrices. It turns out that the form attains its minimum at a single point. Adding this new point gives a polytope P whose all facets are covered by the Main Theorem.

84 How it works? N = 3, r = 8 Using the above algorithm we manage to calculate symmetric powers up to M = 24. Resulting polytope was not the full moment polytope. That is, P24 in Pout 24. There was one problematic facet. To get rid of this, we resort to use a numerical minimization of the linear form obtained from the problematic facet over all particle density matrices. It turns out that the form attains its minimum at a single point. Adding this new point gives a polytope P whose all facets are covered by the Main Theorem. Thus the polytope is the genuine moment polytope for 3 H 8 which is given by 31 independent inequalities.

85 Number of Constraints for systems of rank 10 r N Number of constraints

86 Thank You for your attention!!!