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Calculation of Generalized Pauli Constraints
Murat Altunbulak
Department of Mathematics Dokuz Eylul University
April, 2016
Murat Altunbulak Oxford 2016
Notations
Murat Altunbulak Oxford 2016
Calculation of Generalized Pauli Constraints Murat Altunbulak - - PowerPoint PPT Presentation
Calculation of Generalized Pauli Constraints Murat Altunbulak - - PowerPoint PPT Presentation
Calculation of Generalized Pauli Constraints Murat Altunbulak Department of Mathematics Dokuz Eylul University April, 2016 Murat Altunbulak Oxford 2016 Notations Murat Altunbulak Oxford 2016 Notations Quantum States Quantum system A is
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Notations
Quantum States Quantum system A is described by a complex Hilbert space HA,
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Notations
Quantum States Quantum system A is described by a complex Hilbert space HA, called state space.
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Notations
Quantum States Quantum system A is described by a complex Hilbert space HA, called state space. Pure state = unit vector |ψ ∈ HA or projector operator |ψψ|
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Notations
Quantum States Quantum system A is described by a complex Hilbert space HA, called state space. Pure state = unit vector |ψ ∈ HA or projector operator |ψψ| Mixed state = classical mixture of pure states ρ =
- i
pi|ψiψi| ; pi ≥ 0 ;
- i
pi = 1
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Notations
Quantum States Quantum system A is described by a complex Hilbert space HA, called state space. Pure state = unit vector |ψ ∈ HA or projector operator |ψψ| Mixed state = classical mixture of pure states ρ =
- i
pi|ψiψi| ; pi ≥ 0 ;
- i
pi = 1 ρ is a non-negative (ρ ≥ 0) Hermitian operator with Trρ = 1, called Density matrix.
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Notations
Superposition Principle implies that the state space of composite system AB splits into tensor product of its components A and B HAB = HA ⊗ HB
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Notations
Superposition Principle implies that the state space of composite system AB splits into tensor product of its components A and B HAB = HA ⊗ HB Density Matrix of Composite Systems Density matrix of composite system can be written as linear combination ρAB =
- α
aαLα
A ⊗ Lα B
where Lα
A, Lα B are linear operators on HA, HB, respectively.
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Notations
Reduced state Its reduced matrices are defined by partial traces
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Notations
Reduced state Its reduced matrices are defined by partial traces ρA =
- α
aαTr(Lα
B)Lα A := TrB(ρAB)
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Notations
Reduced state Its reduced matrices are defined by partial traces ρA =
- α
aαTr(Lα
B)Lα A := TrB(ρAB)
ρB =
- α
aαTr(Lα
A)Lα B := TrA(ρAB)
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Notations
Reduced state Its reduced matrices are defined by partial traces ρA =
- α
aαTr(Lα
B)Lα A := TrB(ρAB)
ρB =
- α
aαTr(Lα
A)Lα B := TrA(ρAB)
they are called one-particle reduced density matrices.
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Pauli Exclusion Principle
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Pauli exclusion principle
Initial form (1925) state that no two identical fermions may occupy the same quantum state.
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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
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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 ρN : H⊗N, H⊗N = state space of N electrons.
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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 ρN : H⊗N, H⊗N = state space of N electrons. Define ρ = ρN
1 + ρN 2 + . . . + ρN N sum of all reduced states ρN i : H. ρ
is called electron density matrix.
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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 ρN : H⊗N, H⊗N = state space of N electrons. Define ρ = ρN
1 + ρN 2 + . . . + 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
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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 ρN : H⊗N, H⊗N = state space of N electrons. Define ρ = ρN
1 + ρN 2 + . . . + 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
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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 ρN : H⊗N, H⊗N = state space of N electrons. Define ρ = ρN
1 + ρN 2 + . . . + 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 ∧NH ⊂ H⊗N.
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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 ρN : H⊗N, H⊗N = state space of N electrons. Define ρ = ρN
1 + ρN 2 + . . . + 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 ∧NH ⊂ H⊗N. This implies the original Pauli principle, because ψ ∧ ψ = 0.
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Statement of The Problem
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Statement of the problem
In the latter case, electron density matrix becomes ρ = NρN
1 ,
Trρ = N.
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Statement of the problem
In the latter case, electron density matrix becomes ρ = NρN
1 ,
Trρ = N. Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Specρ ≤ 1.
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Statement of the problem
In the latter case, electron density matrix becomes ρ = NρN
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).
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Statement of the problem
In the latter case, electron density matrix becomes ρ = NρN
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?”
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What was known before 2008?
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What was known before 2008?
Initial constraints Ordering inequalities λ1 ≥ λ2 ≥ . . . ≥ λr ≥ 0
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What was known before 2008?
Initial constraints Ordering inequalities λ1 ≥ λ2 ≥ . . . ≥ λr ≥ 0 Normalization condition Trρ =
i λi = N
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What was known before 2008?
Initial constraints Ordering inequalities λ1 ≥ λ2 ≥ . . . ≥ λr ≥ 0 Normalization condition Trρ =
i λi = N
Two-particle (∧2Hr) and Two-hole (∧r−2Hr) systems Constraints on electron density matrix ρ is given by even degeneracy
- f its eigenvalues, i.e. λ1 = λ2 ≥ λ3 = λ4 ≥ . . .
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What was known before 2008?
Initial constraints Ordering inequalities λ1 ≥ λ2 ≥ . . . ≥ λr ≥ 0 Normalization condition Trρ =
i λi = N
Two-particle (∧2Hr) and Two-hole (∧r−2Hr) systems Constraints on electron density matrix ρ is given by even degeneracy
- f its eigenvalues, i.e. λ1 = λ2 ≥ λ3 = λ4 ≥ . . .
Similar results hold for two-hole system.
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What was known before 2008?
Borland-Dennis system For the system ∧3H6 of three electrons of rank 6, the N-representability conditions are given by the following (in)equalities: λ1 + λ6 = λ2 + λ5 = λ3 + λ4 = 1, λ4 ≤ λ5 + λ6,
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What was known before 2008?
Borland-Dennis system For the system ∧3H6 of three electrons of rank 6, the N-representability conditions are given by the following (in)equalities: λ1 + λ6 = λ2 + λ5 = λ3 + λ4 = 1, λ4 ≤ λ5 + λ6, Sufficiency proved by Borland-Dennis (1972)
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What was known before 2008?
Borland-Dennis system For the system ∧3H6 of three electrons of rank 6, the N-representability conditions are given by the following (in)equalities: λ1 + λ6 = λ2 + λ5 = λ3 + λ4 = 1, λ4 ≤ λ5 + λ6, Sufficiency proved by Borland-Dennis (1972) Necessity proved by Ruskai (2007)
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Formal Solution of Mixed N-representability
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Formal Solution
Solution of Mixed N-representability Main Theorem. Let ρN : ∧NHr (TrρN = 1) be mixed state and ρ : Hr (Trρ = N) be its particle density matrix.
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Formal Solution
Solution of Mixed N-representability Main Theorem. Let ρN : ∧NHr (TrρN = 1) be mixed state and ρ : Hr (Trρ = N) be its particle density matrix. Then all constraints on Specρ = λ : λ1 ≥ λ2 ≥ . . . ≥ λr and SpecρN = µ : µ1 ≥ µ2 ≥ . . . ≥ µR (R = r
N
- = dim ∧NHr) are given by the following linear inequalities
(Generalized Pauli Constraints)
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Formal Solution
Solution of Mixed N-representability Main Theorem. Let ρN : ∧NHr (TrρN = 1) be mixed state and ρ : Hr (Trρ = N) be its particle density matrix. Then all constraints on Specρ = λ : λ1 ≥ λ2 ≥ . . . ≥ λr and SpecρN = µ : µ1 ≥ µ2 ≥ . . . ≥ µR (R = r
N
- = dim ∧NHr) are given by the following linear inequalities
(Generalized Pauli Constraints)
- i
aiλv(i) ≤
- k
(∧Na)kµw(k) (a, v, w)
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Formal Solution
Solution of Mixed N-representability Main Theorem. Let ρN : ∧NHr (TrρN = 1) be mixed state and ρ : Hr (Trρ = N) be its particle density matrix. Then all constraints on Specρ = λ : λ1 ≥ λ2 ≥ . . . ≥ λr and SpecρN = µ : µ1 ≥ µ2 ≥ . . . ≥ µR (R = r
N
- = dim ∧NHr) are given by the following linear inequalities
(Generalized Pauli Constraints)
- i
aiλv(i) ≤
- k
(∧Na)kµw(k) (a, v, w) Here, ∧Na consists of all sums ai1 + ai2 + · · · + aiN, 1 ≤ i1 < i2 < · · · < iN ≤ r arranged in decreasing order, v ∈ Sr and w ∈ SR subject to topological condition cw
v (a) = 0 explained soon.
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Remark
Solution of Pure N-representability Solution of pure N-representability problem can be deduced from the above theorem by specialization µi = 0 for i = 1, because pure states corresponds projector operators |ψψ|.
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Remark
Solution of Pure N-representability Solution of pure N-representability problem can be deduced from the above theorem by specialization µi = 0 for i = 1, because pure states corresponds projector operators |ψψ|. In this case all costraints together with ordering ineqaulities and normalization condition describe a convex polytope, called Moment Polytope.
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Nature of Topological Condition cv
w(a) = 0
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Nature of Topological Condition cv
w(a) = 0 Consider flag variety Fa(Hr) = {X : Hr → Hr | a = Spec(X)}
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Nature of Topological Condition cv
w(a) = 0 Consider flag variety Fa(Hr) = {X : Hr → Hr | a = Spec(X)} and morphism ϕa : Fa(Hr) → F∧Na(∧NHr) X → X (N)
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Nature of Topological Condition cv
w(a) = 0 Consider flag variety Fa(Hr) = {X : Hr → Hr | a = Spec(X)} and morphism ϕa : Fa(Hr) → F∧Na(∧NHr) X → X (N) Here, X (N) : ψ1 ∧ ψ2 ∧ . . . ∧ ψN →
i ψ1 ∧ ψ2 ∧ . . . ∧ Xψi ∧ . . . ∧ ψN.
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Nature of Topological Condition cv
w(a) = 0 The coefficients cv
w(a) are defined via induced morphism of cohomology
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Nature of Topological Condition cv
w(a) = 0 The coefficients cv
w(a) are defined via induced morphism of cohomology
ϕ∗
a : H∗(F∧Na(∧NHr))
→ H∗(Fa(Hr)), σw →
- v
cv
w(a)σv
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Nature of Topological Condition cv
w(a) = 0 The coefficients cv
w(a) are defined via induced morphism of cohomology
ϕ∗
a : H∗(F∧Na(∧NHr))
→ H∗(Fa(Hr)), σw →
- v
cv
w(a)σv
written in the basis of Schubert cocycles σw.
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Connection with Representation Theory
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Connection with Representation Theory
Plethysm Consider the m-th symmetric power of ∧NHr, called Plethysm
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Connection with Representation Theory
Plethysm Consider the m-th symmetric power of ∧NHr, called Plethysm It splits into irreducible components Hλ, parameterized by Young diagrams λ =
- f size N.m, with multiplicity mλ
Sm(∧NHr) =
- λ
mλHλ.
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Connection with Representation Theory
Plethysm Consider the m-th symmetric power of ∧NHr, called Plethysm It splits into irreducible components Hλ, parameterized by Young diagrams λ =
- f size N.m, with multiplicity mλ
Sm(∧NHr) =
- λ
mλHλ. Problem “Which irreducible representations Hλ of U(Hr) can appear in the decomposition of Sm(∧NHr)?”
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Connection with Representation Theory
Plethysm Consider the m-th symmetric power of ∧NHr, called Plethysm It splits into irreducible components Hλ, parameterized by Young diagrams λ =
- f size N.m, with multiplicity mλ
Sm(∧NHr) =
- λ
mλHλ. Problem “Which irreducible representations Hλ of U(Hr) can appear in the decomposition of Sm(∧NHr)?” Surprisingly solution of this problem coincides with that of pure N-representability problem.
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Connection with Representation Theory
Connection treat the diagrams λ as spectra. λ : λ1 ≥ λ2 ≥ . . . ≥ λr.
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Connection with Representation Theory
Connection treat the diagrams λ as spectra. λ : λ1 ≥ λ2 ≥ . . . ≥ λr. normalize them to a fixed size λ = λ/m s.t.
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Connection with Representation Theory
Connection treat the diagrams λ as spectra. λ : λ1 ≥ λ2 ≥ . . . ≥ λr. normalize them to a fixed size λ = λ/m s.t. Tr λ = N.
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Connection with Representation Theory
Connection treat the diagrams λ as spectra. λ : λ1 ≥ λ2 ≥ . . . ≥ λr. normalize them to a fixed size λ = λ/m s.t. Tr λ = N. Representation theoretical solution
- Theorem. Every
λ obtained from irreducible component Hλ ⊂ Sm(∧NHr) is a spectrum of one point reduced matrix ρ of a pure state ψ ∈ ∧NHr.
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Connection with Representation Theory
Connection treat the diagrams λ as spectra. λ : λ1 ≥ λ2 ≥ . . . ≥ λr. normalize them to a fixed size λ = λ/m s.t. Tr λ = N. Representation theoretical solution
- Theorem. Every
λ obtained from irreducible component Hλ ⊂ Sm(∧NHr) is a spectrum of one point reduced matrix ρ of a pure state ψ ∈ ∧NHr. Moreover every one point reduced spectrum is a convex combination of such spectra λ with bounded m ≤ M.
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Practical Algorithm
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Practical Algorithm
For a fixed M the convex hull of the spectra λ from previous Theorem gives an inner approximation to the moment polytope,
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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.
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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.
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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 Hλ ⊂ Sm(∧NHr) for m ≤ M.
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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 Hλ ⊂ Sm(∧NHr) for m ≤ M. Calculate the convex hull of the corresponding spectra λ which gives an inner approximation Pin
M ⊂ P for the moment polytope P.
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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 Hλ ⊂ Sm(∧NHr) for m ≤ M. Calculate the convex hull of the corresponding spectra λ which gives an inner approximation Pin
M ⊂ P for the moment polytope P.
Identify the facets of Pin
M that are given by the inequalities of the
Main Theorem. They cut out an outer approximation Pout
M
⊃ P.
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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 Hλ ⊂ Sm(∧NHr) for m ≤ M. Calculate the convex hull of the corresponding spectra λ which gives an inner approximation Pin
M ⊂ P for the moment polytope P.
Identify the facets of Pin
M that are given by the inequalities of the
Main Theorem. They cut out an outer approximation Pout
M
⊃ P. Increase M and continue until Pin
M = Pout M .
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How it works?
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How it works?
The above algorithm works perfectly only for some small systems.
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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 ∧3H6 up to 4th degree. That is, M = 4.
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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 ∧3H6 up to 4th degree. That is, M = 4. Symmetric powers contains the following components
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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 ∧3H6 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]
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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]
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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.
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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.
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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.
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How it works?
N = 3, r = 8
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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.
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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, Pin
24 = Pout 24 .
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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, Pin
24 = Pout 24 .
There was one problematic facet.
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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, Pin
24 = 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.
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SLIDE 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, Pin
24 = 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.
Murat Altunbulak Oxford 2016
SLIDE 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, Pin
24 = 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.
Murat Altunbulak Oxford 2016
SLIDE 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, Pin
24 = 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 ∧3H8 which is given by 31 independent inequalities.
Murat Altunbulak Oxford 2016
SLIDE 85
Number of Constraints for systems of rank ≤ 10
r N Number of constraints 7 3 4 8 3 31 8 4 15 9 3 52 9 4 60 10 3 93 10 4 125 10 5 161
Murat Altunbulak Oxford 2016
SLIDE 86
Thank You for your attention!!!
Murat Altunbulak Oxford 2016