Quantum Information Geometry: quantum metrics for finite ... · Patrizia Vitalea with V.I....
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Quantum Information Geometry: quantum metrics
for finite dimensional systems
Patrizia Vitalea
with V.I. Man’ko, G. Marmo and F. Ventriglia
a Dipartimento di Fisica Universita di Napoli “Federico II” and INFN
Policeta 13 july 2016
On the occasion of the 70th birthday of Beppe Marmo
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold,
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold, statistical model,
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold, statistical model, classical metrics fromdivergence function
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold, statistical model, classical metrics fromdivergence function
Quantum metrics from divergence functions
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold, statistical model, classical metrics fromdivergence function
Quantum metrics from divergence functions
Finite-dimensional quantum systems
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold, statistical model, classical metrics fromdivergence function
Quantum metrics from divergence functions
Finite-dimensional quantum systems
The cases N=2 and N=3
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Outline
Statistical manifold, statistical model, classical metrics fromdivergence function
Quantum metrics from divergence functions
Finite-dimensional quantum systems
The cases N=2 and N=3
The tomographic picture
Outlook
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Invariant divergence function
Notation
P(X )= statistical manifold of probability distributions p(x),x ∈ X
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Invariant divergence function
Notation
P(X )= statistical manifold of probability distributions p(x),x ∈ X
M = {p(x , ξ)} ⊂ P(X ) statistical model parametrized by ξ
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Invariant divergence function
Notation
P(X )= statistical manifold of probability distributions p(x),x ∈ X
M = {p(x , ξ)} ⊂ P(X ) statistical model parametrized by ξ
M is finite dimensional. ξ coordinates on MDivergence D(ξ, ξ′) between p(x , ξ) and p(x , ξ′)
D(ξ, ξ′) ≥ 0
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Invariant divergence function
Notation
P(X )= statistical manifold of probability distributions p(x),x ∈ X
M = {p(x , ξ)} ⊂ P(X ) statistical model parametrized by ξ
M is finite dimensional. ξ coordinates on MDivergence D(ξ, ξ′) between p(x , ξ) and p(x , ξ′)
D(ξ, ξ′) ≥ 0
D(ξ, ξ + dξ) = 12gjk(ξ)dξjdξk + O(dξ3)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Invariant divergence function
Notation
P(X )= statistical manifold of probability distributions p(x),x ∈ X
M = {p(x , ξ)} ⊂ P(X ) statistical model parametrized by ξ
M is finite dimensional. ξ coordinates on MDivergence D(ξ, ξ′) between p(x , ξ) and p(x , ξ′)
D(ξ, ξ′) ≥ 0
D(ξ, ξ + dξ) = 12gjk(ξ)dξjdξk + O(dξ3)
g = (gjk) positive-definite
D not symmetric, no triangular inequality .
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Invariant divergence function
Notation
P(X )= statistical manifold of probability distributions p(x),x ∈ X
M = {p(x , ξ)} ⊂ P(X ) statistical model parametrized by ξ
M is finite dimensional. ξ coordinates on MDivergence D(ξ, ξ′) between p(x , ξ) and p(x , ξ′)
D(ξ, ξ′) ≥ 0
D(ξ, ξ + dξ) = 12gjk(ξ)dξjdξk + O(dξ3)
g = (gjk) positive-definite
D not symmetric, no triangular inequality .Invariance : No information loss (information monotonicity) undercoarse graining x → y(x)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The classical case: Chentsov theorem
Classically: Given any invariant divergence, the invariant metric isunique → the Fisher Rao metric
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The classical case: Chentsov theorem
Classically: Given any invariant divergence, the invariant metric isunique → the Fisher Rao metric
g(ξ) = k p d ln p ⊗ d ln p
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The classical case: Chentsov theorem
Classically: Given any invariant divergence, the invariant metric isunique → the Fisher Rao metric
g(ξ) = k p d ln p ⊗ d ln p
α-divergence function or Tsallis relative entropy
ST (p, p) = (1− q)−1{1−∑
pqp1−q}
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The classical case: Chentsov theorem
Classically: Given any invariant divergence, the invariant metric isunique → the Fisher Rao metric
g(ξ) = k p d ln p ⊗ d ln p
α-divergence function or Tsallis relative entropy
ST (p, p) = (1− q)−1{1−∑
pqp1−q}with α = 2q − 1.
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The classical case: Chentsov theorem
Classically: Given any invariant divergence, the invariant metric isunique → the Fisher Rao metric
g(ξ) = k p d ln p ⊗ d ln p
α-divergence function or Tsallis relative entropy
ST (p, p) = (1− q)−1{1−∑
pqp1−q}with α = 2q − 1.
in the limit q → 1 the relative Shannon entropy
limq→1
ST (p, p) =∑
p(ln p − ln p)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The classical case: Chentsov theorem
Classically: Given any invariant divergence, the invariant metric isunique → the Fisher Rao metric
g(ξ) = k p d ln p ⊗ d ln p
α-divergence function or Tsallis relative entropy
ST (p, p) = (1− q)−1{1−∑
pqp1−q}with α = 2q − 1.
in the limit q → 1 the relative Shannon entropy
limq→1
ST (p, p) =∑
p(ln p − ln p)
the metric
gjk(ξ) := − ∂2S
∂ξ∂ξ′|ξ=ξ′
is the Fisher Rao metric for any qPatrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum case: quantum Fisher-Rao metric
The statistical model is replaced by the manifold of (N-dim)quantum states
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The quantum case: quantum Fisher-Rao metric
The statistical model is replaced by the manifold of (N-dim)quantum states
the potential function is now the quantum Tsallis relativeentropy
ST (ρ, ρ) = (1− q)−1{1− Tr ρqρ1−q}
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum case: quantum Fisher-Rao metric
The statistical model is replaced by the manifold of (N-dim)quantum states
the potential function is now the quantum Tsallis relativeentropy
ST (ρ, ρ) = (1− q)−1{1− Tr ρqρ1−q}
in the limitq → 1 we get the relative von Neumann entropy
limq→1
ST (ρ, ρ) = Tr ρ(ln ρ− ln ρ)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum case: quantum Fisher-Rao metric
The statistical model is replaced by the manifold of (N-dim)quantum states
the potential function is now the quantum Tsallis relativeentropy
ST (ρ, ρ) = (1− q)−1{1− Tr ρqρ1−q}
in the limitq → 1 we get the relative von Neumann entropy
limq→1
ST (ρ, ρ) = Tr ρ(ln ρ− ln ρ)
the metric
gjk(ξ) = − ∂2S
∂ξ∂ξ′|ξ=ξ′
is the quantum Fisher Rao metric only for q = 1Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum metric for generic q
ρ, ρ parametrized in terms of diagonal matrices and unitarytransformations
ρ = Uρ0U−1, ρ = V ρ0V
−1
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The quantum metric for generic q
ρ, ρ parametrized in terms of diagonal matrices and unitarytransformations
ρ = Uρ0U−1, ρ = V ρ0V
−1
U,V ∈ SU(N)
given the diagonal immersion i : M → M ×M the metrictensor is :
g(X ,Y ) := −i∗(LXLYS)
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The quantum metric for generic q
ρ, ρ parametrized in terms of diagonal matrices and unitarytransformations
ρ = Uρ0U−1, ρ = V ρ0V
−1
U,V ∈ SU(N)
given the diagonal immersion i : M → M ×M the metrictensor is :
g(X ,Y ) := −i∗(LXLYS)
=⇒
g = −i∗d dST (ρ, ρ) = (1− q)−1Tr dρq ⊗ dρ1−q
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The quantum metric for generic q
ρ, ρ parametrized in terms of diagonal matrices and unitarytransformations
ρ = Uρ0U−1, ρ = V ρ0V
−1
U,V ∈ SU(N)
given the diagonal immersion i : M → M ×M the metrictensor is :
g(X ,Y ) := −i∗(LXLYS)
=⇒
g = −i∗d dST (ρ, ρ) = (1− q)−1Tr dρq ⊗ dρ1−q
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum metric for generic q
givinggq = g tan
q + g transq
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum metric for generic q
givinggq = g tan
q + g transq
with
g tanq = (1− q)−1
Tr
(
[U−1dU, ρq0]⊗ [U−1dU, ρ1−q0 ]
)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The quantum metric for generic q
givinggq = g tan
q + g transq
with
g tanq = (1− q)−1
Tr
(
[U−1dU, ρq0]⊗ [U−1dU, ρ1−q0 ]
)
andg transq = qρ−1
0 dρ0 ⊗ dρ0
In the limit q → 1
g1 = Tr ρ−10 dρ0 ⊗ dρ0 + Tr [U−1dU, ln ρ0]⊗ [U−1dU, ρ0]
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The quantum metric for generic q
givinggq = g tan
q + g transq
with
g tanq = (1− q)−1
Tr
(
[U−1dU, ρq0]⊗ [U−1dU, ρ1−q0 ]
)
andg transq = qρ−1
0 dρ0 ⊗ dρ0
In the limit q → 1
g1 = Tr ρ−10 dρ0 ⊗ dρ0 + Tr [U−1dU, ln ρ0]⊗ [U−1dU, ρ0]
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The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
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The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
ρq0 =
(
(1+r2 )q 00 (1−r
2 )q
)
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The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
ρq0 =
(
(1+r2 )q 00 (1−r
2 )q
)
=aq + bq
2σ0 +
aq − bq
2σ3
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The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
ρq0 =
(
(1+r2 )q 00 (1−r
2 )q
)
=aq + bq
2σ0 +
aq − bq
2σ3
and similarly for ρ1−q0 .
The metric:
gq = q1
1− r2dr ⊗ dr
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The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
ρq0 =
(
(1+r2 )q 00 (1−r
2 )q
)
=aq + bq
2σ0 +
aq − bq
2σ3
and similarly for ρ1−q0 .
The metric:
gq = q1
1− r2dr ⊗ dr
− 2
1− q(aq − bq)(a1−q − b1−q)(θ
1 ⊗ θ1 + θ2 ⊗ θ2)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
ρq0 =
(
(1+r2 )q 00 (1−r
2 )q
)
=aq + bq
2σ0 +
aq − bq
2σ3
and similarly for ρ1−q0 .
The metric:
gq = q1
1− r2dr ⊗ dr
− 2
1− q(aq − bq)(a1−q − b1−q)(θ
1 ⊗ θ1 + θ2 ⊗ θ2)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
U−1dU = λjθj
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
U−1dU = λjθj
ρq0 =
rq1 0 00 r
q2 0
0 0 rq3
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
U−1dU = λjθj
ρq0 =
rq1 0 00 r
q2 0
0 0 rq3
= (αqλ0 + βqλ3 + γqλ8)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
U−1dU = λjθj
ρq0 =
rq1 0 00 r
q2 0
0 0 rq3
= (αqλ0 + βqλ3 + γqλ8) (1)
with r3 = 1− r1 − r2, and a similar expression for ρ1−q0 .
The metric:
g tranq = q
1
rjdrj ⊗ drj
g tanq = −8
{
βqβ1−q
∑
j 6=3,8
θj ⊗ θj
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
U−1dU = λjθj
ρq0 =
rq1 0 00 r
q2 0
0 0 rq3
= (αqλ0 + βqλ3 + γqλ8) (1)
with r3 = 1− r1 − r2, and a similar expression for ρ1−q0 .
The metric:
g tranq = q
1
rjdrj ⊗ drj
g tanq = −8
{
βqβ1−q
∑
j 6=3,8
θj ⊗ θj
+√3(√
3γqγ1−q + βqγ1−q + β1−qγq
)
∑
j 6=1,2,3,8
θj ⊗ θj}
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The case N = 3
U,V ∈ SU(3)
U−1dU = λjθj
ρq0 =
rq1 0 00 r
q2 0
0 0 rq3
= (αqλ0 + βqλ3 + γqλ8) (1)
with r3 = 1− r1 − r2, and a similar expression for ρ1−q0 .
The metric:
g tranq = q
1
rjdrj ⊗ drj
g tanq = −8
{
βqβ1−q
∑
j 6=3,8
θj ⊗ θj
+√3(√
3γqγ1−q + βqγ1−q + β1−qγq
)
∑
j 6=1,2,3,8
θj ⊗ θj}
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉 with u ∈ SU(2) labeling thetomographic reference frame
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉 with u ∈ SU(2) labeling thetomographic reference frame
it is a probability distribution −→S(Wρ,Wρ) = (1− q)−1
(
1−∑
m Wqρ (m; u)W1−q
ρ (m; u))
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉 with u ∈ SU(2) labeling thetomographic reference frame
it is a probability distribution −→S(Wρ,Wρ) = (1− q)−1
(
1−∑
m Wqρ (m; u)W1−q
ρ (m; u))
Gq = q( 1Wρ(+;u) + ( 1
Wρ(−;u))(
dWρ(+; u)⊗ dWρ(+; u))
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉 with u ∈ SU(2) labeling thetomographic reference frame
it is a probability distribution −→S(Wρ,Wρ) = (1− q)−1
(
1−∑
m Wqρ (m; u)W1−q
ρ (m; u))
Gq = q( 1Wρ(+;u) + ( 1
Wρ(−;u))(
dWρ(+; u)⊗ dWρ(+; u))
varying u we reconstruct the state ρ ≡ (x1, x2, x3) (inverseformula), but for this we only need a sufficiency set
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉 with u ∈ SU(2) labeling thetomographic reference frame
it is a probability distribution −→S(Wρ,Wρ) = (1− q)−1
(
1−∑
m Wqρ (m; u)W1−q
ρ (m; u))
Gq = q( 1Wρ(+;u) + ( 1
Wρ(−;u))(
dWρ(+; u)⊗ dWρ(+; u))
varying u we reconstruct the state ρ ≡ (x1, x2, x3) (inverseformula), but for this we only need a sufficiency set(
u13 = exp(iπ/4σ1, u23 = exp(−iπ/4σ2), Id)
yield
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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The tomographic metric for N=2Spin tomography
ρ = 12(σ0 + ~x · ~σ)
Wρ(m; u) = 〈m|uρu−1|m〉 with u ∈ SU(2) labeling thetomographic reference frame
it is a probability distribution −→S(Wρ,Wρ) = (1− q)−1
(
1−∑
m Wqρ (m; u)W1−q
ρ (m; u))
Gq = q( 1Wρ(+;u) + ( 1
Wρ(−;u))(
dWρ(+; u)⊗ dWρ(+; u))
varying u we reconstruct the state ρ ≡ (x1, x2, x3) (inverseformula), but for this we only need a sufficiency set(
u13 = exp(iπ/4σ1, u23 = exp(−iπ/4σ2), Id)
yield
W(+, u13) =1 + x1
2W(+, u23) =
1 + x2
2W(+, Id) =
1 + x3
2
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Once the sufficiency set chosen, we can replace
dxk = 2dWk(+; u)
and get the quantum metric in terms of tomograms
gq = gjk(q, x)dxj ⊗ dxk −→ gq(W)
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Summary
We have computed the quantum metric for finite dimensionalsystems
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Summary
We have computed the quantum metric for finite dimensionalsystems
We have computed the tomographic metric
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Summary
We have computed the quantum metric for finite dimensionalsystems
We have computed the tomographic metric
We are looking for a reconstruction formula g(G)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional
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Summary
We have computed the quantum metric for finite dimensionalsystems
We have computed the tomographic metric
We are looking for a reconstruction formula g(G)
Patrizia Vitalea with V.I. Man’ko, G. Marmo and F. Ventriglia Quantum Information Geometry: quantum metrics for finite dimensional