quantum and tomographic metrics from relative entropies
Post on 23-Jan-2022
5 Views
Preview:
TRANSCRIPT
Quantum and tomographic metrics from relativeentropies
Marco Laudato
Dipartimento di Fisica Universita di Napoli “Federico II”
with F.M. Ciaglia, F. Di Cosmo, G. Marmo, F.M. Mele, F. Ventriglia and P. Vitale
Nottingham University
15.6.2017
Marco Laudato Quantum and tomographic metrics from relative entropies
Outline
1 Information Geometry: A SketchStatistical modelDivergence functionsDifferential calculus over the manifold of parameters
2 Quantum Metrics from Divergence FunctionsGeometric Quantum MechanicsQuantum metrics for N level systemsThe cases N = 2 and N = 3 and radial limit.Quantum metrics from q-z-relative entropy
3 Tomographic Reconstruction of Quantum MetricsTomographic approach to Quantum Mechanics (SpinTomography)Reconstruction formulaChanging the tomographic schemeThe inverse problem
4 Conclusions and Perspectives
Marco Laudato Quantum and tomographic metrics from relative entropies
Part I
Information Geometry: A Sketch
Marco Laudato Quantum and tomographic metrics from relative entropies
Information Geometry
Statistical Manifold, P(X ): is the manifold of probabilitydensities, p(x) on a measurable set X
Statistical Model: is a family of probability densities p(x ; ξ)parametrized by n variables ξ = [ξ1, ..., ξn] , s.t. the mappingξ → p(x ; ξ) is injective. ξ ∈ M with M, space of parameters,a finite dimensional differential manifold.
M can be equipped with two symmetric tensors:
g metric tensor
T 3-tensor, called the skewness tensor
(M, g ,T ) is the Statistical Model.
From g and T it is possible to construct affine connections
αΓjkl = gΓjkl +α
2Tjkl
Marco Laudato Quantum and tomographic metrics from relative entropies
Invariant divergence function
The metric and skewness tensor can be derived by a convexfunction D : M ×M → RDivergence function: D(ξ, ξ′) between p(x , ξ) and p(x , ξ′)
D(ξ, ξ′) > 0, ξ 6= ξ′
D(ξ, ξ′)|ξ=ξ′ = 0
gjk(ξ) := − ∂2D∂ξj∂ξ
′k|ξ=ξ′
Tljk(ξ) = ∂3D∂ξ′j∂ξ
′k∂ξl|ξ=ξ′ − ∂3D
∂ξj∂ξk∂ξ′l|ξ=ξ′
Invariance [Amari-Nagaoka]: Given a map x → y = φ(x) (e.g. coarsegraining) and p(y , ξ) =
∑x ,y=φ(x) p(x , ξ), D is said to be invariant
if it satisfies the monotonicity condition: D(ξ, ξ′) ≤ D(ξ, ξ′)Uniqueness [Chentsov-Amari-Nagaoka]: The metric deriving frominvariant divergence functions over M is unique:
Fisher-Rao metric tensor:
g(ξ) = k∑
p d ln p ⊗ d ln p
Marco Laudato Quantum and tomographic metrics from relative entropies
The classical case
Examples of divergence functions giving the same metric:for p ∈ P = (p1, ..., pn), pi ≡ p(xi ), x random variable assumingdiscrete values, P ≡ M
F (p, p) = 4(1−∑
j
√pj pj) = (q → 1/2)
H(p, p) =∑
j pj(ln pj − ln pj) (q → 1)
ST (p, p) = [q(1− q)]−1(1−∑
j pqj p
1−qj )
the latter: α-divergence function (with α = 2q − 1) related toTsallis relative entropy→ They all correspond to α-divergence functions for differentvalues of α
Marco Laudato Quantum and tomographic metrics from relative entropies
Differential calculus over the manifold of parameters
Given D : M ×M → R differentiable i : M → M ×M, diag.immersion i∗D = 0, i∗dD = 0.D a function on M ×M −→ we use bi-forms: γ of degree (p, q)∈ Ωp(M)⊗ Ωq(M) locally given by
γ(x , y) =∑
j1..jp ;k1,...,kq
f (x , y)j1..jp ;k1,...,kqdxj1∧...∧dx jp⊗dyk1∧...∧dykq
and exterior derivatived ⊗ 1 : Ωp(M)⊗ Ωq(M)→ Ωp+1(M)⊗ Ωq(M)1⊗ d : Ωp(M)⊗ Ωq(M)→ Ωp(M)⊗ Ωq+1(M)A metric tensor may be defined by setting:
g(X ,Y ) := −i∗(
(d ⊗ d D)(Xl ,Yr ))
= −i∗(LXlLYrD)
with X → Xl ⊕ 0 and Y → 0 ⊕ Yr .
g is symmetricf -linearpositive
Dual affine connections: g(∇XY ,Z ) = −i∗(LXlLYl
LZrD)g(∇∗XY ,Z ) = −i∗(LZl
LXrLYrD) and skewness tensor
T (X ,Y ,Z ) = g(∇XY −∇∗XY ,Z ), X ,Y ,Z ∈ X(M).
Marco Laudato Quantum and tomographic metrics from relative entropies
Partial Summary 1
Main tools we have used in the classical setting:[Amari, Information Geometry and its applications, Springer 2016]
(M, g ,T ) is a differential manifold
Geometric structures can be derived by a divergence function
Chentsov theorem on the unicity of the metric over M can bereformulated in terms of properties of the divergence function
Differential calculus over M which makes the formulationintrinsic
Marco Laudato Quantum and tomographic metrics from relative entropies
Part II
Quantum Metrics from Divergence Functions
Marco Laudato Quantum and tomographic metrics from relative entropies
Geometric Quantum Mechanics: Pure States
It is possible to formulate QM on the complex projective spaceassociated to the Hilbert space of a quantum system
Fiber bundle structure of QM:(We must take into account
normalization and global phases)
C0// H0
π
R(H) ∼= CPn−1
Complex projective space:
CPn−1 =
[|ψ〉] : |ψ〉 ∼ |ψ′〉 ⇔ |ψ′〉 = λ |ψ〉 , λ ∈ C0
Rays
realized as rank-1 projectors (embedded in u∗(H)):
ρ =|ψ〉 〈ψ|〈ψ|ψ〉
∈ D1(H) ⊂ u∗(H) s.t.
ρ† = ρ
Tr ρ = 1
ρ2 = ρ
Pure States
Marco Laudato Quantum and tomographic metrics from relative entropies
Geometric Quantum Mechanics: Mixed States
Mixed states are convex combinations of pure states. The resultingmanifold has a rich geometrical structure.
We can consider the action of the unitary group U(H) on thedual u∗(H) of its own Lie algebra (coadjoint action):Ad∗U(ρ) = UρU†. This action foliates the space of state intocoadjoint orbits, each one uniquely characterized by thespectrum of the state ρ.
If we want to describe processes which do not preserve thespectrum (e.g. decoherence [Ciaglia, Di Cosmo, L., Marmo]), wemay consider the GL(H). This action stratifies the space ofstate into orbits characterized by the rank of the state ρ.
Marco Laudato Quantum and tomographic metrics from relative entropies
From classical to quantum: N level systems
How to describe the parameter space for a quantum system tokeep contact with classical case?A simplex of classical probabilities may be “quantized”, byassociating with every probability vector a coadjoint orbit of theunitary group acting on the dual space of its Lie algebra: aprobability vector, (p1, p2, ..., pN) gives rise to a density matrix bysetting
ρ(U, ~p) = U
p1 ... 00 p2 ...... ... ...0 ... pN
U†
The parameter space is M ≈ SU(N)× Σ and it is (almost) thespace of states.
Marco Laudato Quantum and tomographic metrics from relative entropies
Quantum metrics from divergence function
We can define (quantum) metrics on the space of states by usingthe same methods of classical information geometry.
To define quantum metrics we use alpha-divergence functions ∼the quantum Tsallis relative entropy
S(ρ, ρ) = [q(1− q)]−11− Tr ρqρ1−q.
In the limit q → 1, 0 it is the relative von Neumann entropy (akaquantum relative entropy):
limq→1S(ρ, ρ) = Tr ρ(ln ρ− ln ρ).
Marco Laudato Quantum and tomographic metrics from relative entropies
The quantum metric for generic q
ρ, ρ parametrized in terms of diagonal matrices and unitarytransformations (we are working on the foliated space M)
ρ = Uρ0U−1, ρ = V ρ0V
−1
U,V ∈ SU(N)
given the diagonal immersion i : M → M ×M the metrictensor is :
g = −i∗(d ⊗ dST (ρ, ρ)
)= [q(1− q)]−1 Tr dρq ⊗ dρ1−q
The tensor product is related to forms → yields a bi-form, thetrace is over the U(N) generators.
Marco Laudato Quantum and tomographic metrics from relative entropies
The quantum metric for generic q
since dρq = dUρq0U−1 + Udρq0U
−1 − Uρq0U−1dUU−1
we get the following general expression for the metric tensor[Man’ko, Marmo, Ventriglia, Vitale]
gq = g tanq + g trans
q
with
g tanq = [q(1− q)]−1 Tr
([U−1dU, ρq0 ]⊗ [U−1dU, ρ1−q
0 ])
andg transq = Tr ρ−1
0 dρ0 ⊗ dρ0
This metric is defined on the space of invertible (maximal rank)states.In the limit q → 1
g1 = Tr ρ−10 dρ0 ⊗ dρ0 + Tr [U−1dU, ln ρ0]⊗ [U−1dU, ρ0]
Marco Laudato Quantum and tomographic metrics from relative entropies
The case N = 2
U,V ∈ SU(2).
U−1dU = σjθj
ρq0 =
(( 1+w
2 )q 00 ( 1−w
2 )q
)=
aq + bq2
σ0 +aq − bq
2σ3
and similarly for ρ1−q0 .
The metric:
gq =2
q(1− q)(aq − bq)(a1−q − b1−q)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
+1
1− w2dw ⊗ dw
g1 =1
1− w2dw ⊗ dw + 2w ln
1 + w
1− w(θ1 ⊗ θ1 + θ2 ⊗ θ2)
Marco Laudato Quantum and tomographic metrics from relative entropies
Relation to Petz classification theorem
The theorem [Petz]: for N = 2, quantum metrics are of the form
g =1
1− w2dw ⊗ dw +
w2
(1 + w)f(
1−w1+w
)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
For q 6= 1, 0 we have
f (t) =(q(1− q)) (t − 1)2
(tq − 1)(t1−q − 1)
for q = 1, 0
f (t) =(t − 1)
ln t
where
t =1− w
1 + w
Marco Laudato Quantum and tomographic metrics from relative entropies
The case N = 3
U,V ∈ SU(3) ρ = Uρ0U−1, ρ = V ρ0V
−1,
ρq0 =
kq1 0 00 kq2 00 0 kq3
= (αqλ0 + βqλ3 + γqλ8)
with k1 + k2 + k3 = 1, and a similar expression for ρ1−q0 .
What is M: The strata are union of unitary orbits of SU(3)(four- and six-dimensional sub-manifolds in R8).
The metric:U−1dU = λjθ
j → we compute [U−1dU, ρq0 ]⊗ [U−1dU, ρ1−q0 ]
Marco Laudato Quantum and tomographic metrics from relative entropies
The case N = 3
gq = g tanq + g trans
q
g tanq =
2
q(1− q)
[(kq1 − kq2 )(k1−q
1 − k1−q2 )(θ1 ⊗ θ1 + θ2 ⊗ θ2)
+ (kq1 − kq3 )(k1−q1 − k1−q
3 )(θ4 ⊗ θ4 + θ5 ⊗ θ5)
+ (kq2 − kq3 )(k1−q2 − k1−q
3 )(θ6 ⊗ θ6 + θ7 ⊗ θ7)
]
g transq =
3∑j=1
1
kjdkj ⊗ dkj .
The limit to boundary states (pure states) and rank two states isperformed with same procedure as for N = 2.
Marco Laudato Quantum and tomographic metrics from relative entropies
Weak radial limit to pure states
The manifold of parameters is the three dimensional unit ballB2
Two strata: mixed states inside the ball (rank two matrices)and pure states on the boundary (w2 = 1)
gq only holds inside the ball
to get the metric on the boundary: weak radial limit [Petz-Sudar]
This amounts to evaluating the scalar product of two tangentialvectors, X ,Y , at a point ρ strictly inside the ball → only thetangential part of the metric contributes to gq(X ,Y )|ρ. Then, weperform the radial limit along the radius passing through ρ, up tothe pure state ρp
g0q = (q(1− q))−1 (θ1 ⊗ θ1 + θ2 ⊗ θ2)
Marco Laudato Quantum and tomographic metrics from relative entropies
Partial Summary 2
Main concepts:
We can define quantum metric tensors by using techniques ofinformation geometry
The resulting metrics split into a ”tangent part” (quantum)and a ”transversal part” (classical ∼ Fisher-Rao)
In a quantum framework there is no Chentsov uniquenesstheorem.
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative Entropy
There is a plethora of divergence functions:
Quantum Relative Entropy
SvN(ρ|ρ) = Tr ρ(log ρ− log ρ).
Divergence function of Bures metric
SB(ρ|ρ) = 4[1− Tr
(ρρ) 1
2]
Divergence function of Wigner-Yanase metric
SWY (ρ|ρ) = 4[1− Tr
(ρ
12 ρ
12)]
Tsallis Relative Entropy, q-Renyi Relative Entropy andq-Quantum Renyi Divergence (more general):
SRRE (ρ|ρ) = [1− q]−1 log Tr(ρqρ1−q)
STs(ρ, ρ) = [q(1− q)]−1(1− Tr ρqρ1−q)
SQRD(ρ|ρ) =1
q − 1log Tr
(ρ
1−q2q ρρ
1−q2q)
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative Entropy
However, they do not contain all the possible divergence function(e.g. Bures) and DPI is not satisfied for all the range ofparameters.Most general [Audenaert, Datta] → q-z-Renyi Relative Entropy:
Sq,z(ρ|ρ) =1
q − 1log Tr
(ρq/z ρ(1−q)/z
)z, q ∈ (0, 1), z ∈ R+
Limits:
Sq,1(ρ|ρ) := limz→1
Sq,z(ρ|ρ) ≡ SRRE (ρ|ρ) =1
1− qlog Tr
(ρqρ1−q)
Sq,q(ρ|ρ) := limz→q
Sq,q(ρ|ρ) ≡ SQRD(ρ|ρ) =1
q − 1log Tr
(ρ
1−q2q ρρ
1−q2q)
S1,1(ρ|ρ) := limz=q→1
Sq,z(ρ|ρ) ≡ SvN(ρ|ρ) = Tr ρ(log ρ− log ρ)
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative Entropy
Since we want to compute a metric tensor, we consider theq-logarithm regularization:
logq ρ =1
1− q(ρ1−q − 1) with lim
q→1logq ρ = log ρ
We divide by q to make it symmetric under the exchangeq → (1− q):
Sq,z(ρ|ρ) =1
q(1− q)
[1− Tr
(ρq/z ρ
1−qz
)z ]Limits:
S1,1(ρ|ρ) := limz=q→1
Sq,z(ρ|ρ) ≡ SvN(ρ|ρ) = Tr ρ(log ρ− log ρ)
S 12,1(ρ|ρ) := lim
z=1,q→ 12
Sq,z(ρ|ρ) ≡ SWY (ρ|ρ) = 4[1− Tr
(ρ
12 ρ
12)]
S 12, 1
2(ρ|ρ) := lim
z=q→ 12
Sq,z(ρ|ρ) ≡ SB(ρ|ρ) = 4[1− Tr
(ρρ) 1
2]
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative EntropyN=2
Let compute the metric in the case N = 2. We indicate A = ρqz
and B = ρ1−qz . The metric tensor is:
gq,z = −i∗ddSq,z(ρ|ρ) =1
q(1− q)i∗ dA dBTr (AB)z
By consider the analytical expansion of the function (AB)z
(AB)z =∞∑n=0
cn(z)(AB − 1)n
and with a lot of patience:
gq,z =1
1− w2dw⊗dw+
2wz
q(1− q)
(a qz− b q
z)(a 1−q
z− b 1−q
z)
(a 1z− b 1
z)
δjk θj⊗θk
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative Entropy
Limits:
gq,1(ρ|ρ) := limz→1
gq,z(ρ|ρ) = gTs(ρ|ρ) (Tsallis)
=1
1− w2dw ⊗ dw + 2
(aq − bq)(a1−q − b1−q)
q(1− q)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
g 12,1(ρ|ρ) := lim
q→1/2z→1
gq,z(ρ|ρ) = gWY (ρ|ρ) (Wigner-Yanase)
=1
1− w2dw ⊗ dw + 8(1−
√1− w2)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
g 12, 1
2(ρ|ρ) := lim
z→1/2q→1/2
gq,z(ρ|ρ) = gB(ρ|ρ) (Bures)
=1
1− w2dw ⊗ dw + 4w2(θ1 ⊗ θ1 + θ2 ⊗ θ2)
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative EntropyGeneric N-level system
We use the standard (or natural) basis for the algebra u(N):
ταβα,β=1,...,N s.t. (ταβ)α′β′ = δαα′δββ′ .
In this basis the usual σ basis can be expressed as C-linearcombinations of the τ :
σk = Mαβk ταβ , Mαβ
k ∈ C (k = 0, . . . ,N2 − 1)
The diagonal density matrix ρ0 can be decomposed in the standardbasis as
ρ0 =N∑α=1
pαταα
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative EntropyGeneric N-level system
With the same (more!) amount of patience, the metric tensor is:
gq,z = g transvq,z + g tang
q,z =N∑α=1
pαd ln pα ⊗ d ln pα+
+z
q(1− q)
N∑′
α,β=1
(pα − pβ)(pqzα − p
qzβ )(p
1−qz
α − p1−qz
β )
(p1zα − p
1zβ )
θαβ ⊗ θβα
In terms of the usual σ basis (by using the matrices M):
gq,z = g transvq,z + g tang
q,z =N∑α=1
pαd ln pα ⊗ d ln pα +N2−1∑j ,k=1
Cjk θj ⊗ θk
with
Cjk =N∑
α,β=1
Eαβ<[Mαβj Mβα
k ], Eαβ =(pα − pβ)(p
qzα − p
qzβ )(p
1−qz
α − p1−qz
β )
(p1zα − p
1zβ )
Marco Laudato Quantum and tomographic metrics from relative entropies
q-z-Relative EntropyN=3
Limit for N = 3 and z = 1. The only non-trivial part is the tangentpart:
8∑j ,k=1
Cjk θj ⊗ θk = 2[E12(θ1 ⊗ θ1 + θ2 ⊗ θ2) + E13(θ4 ⊗ θ4 + θ5 ⊗ θ5)
+ E23(θ6 ⊗ θ6 + θ7 ⊗ θ7)]
Then one recover the previous expression for N = 3:
g tanq =
2
q(1− q)
[(kq1 − kq2 )(k1−q
1 − k1−q2 )(θ1 ⊗ θ1 + θ2 ⊗ θ2)
+ (kq1 − kq3 )(k1−q1 − k1−q
3 )(θ4 ⊗ θ4 + θ5 ⊗ θ5)
+ (kq2 − kq3 )(k1−q2 − k1−q
3 )(θ6 ⊗ θ6 + θ7 ⊗ θ7)
]
Marco Laudato Quantum and tomographic metrics from relative entropies
Partial Summary 3
As far as we know, this is the most general expression forquantum metric tensors.
This form of the metric contains several (all?) quantummetric tensors −→ Chentsov’s uniqueness theorem?
Marco Laudato Quantum and tomographic metrics from relative entropies
Part III
Tomographic Reconstruction of Quantum Metrics
Marco Laudato Quantum and tomographic metrics from relative entropies
Tomographic Approach to QMQuantizer-Dequantizer [Ciaglia, Di Cosmo, Ibort, Marmo]
Quantization-dequantization maps:
A→ fA(ξ) = Tr (AD(ξ))
invertible A =∫dξfA(ξ)U(ξ)
s.t. Tr D(ξ)U(ξ′) = δ(ξ − ξ′)with a star product fA ? fB := Tr (ABD(ξ))
for states ρ. have to use dual maps (U and D exchanged)
example: Weyl-Wigner-Moyal scheme
classical like description of quantum mechanics
evolution equations for operators symbols fA and statessymbols fρ .....
Marco Laudato Quantum and tomographic metrics from relative entropies
Spin Tomography
What is quantum tomography?
ρ→Wρ(ξ) = Tr ρD(ξ) invertible under certain assumptionsfor the dequantizer.Since ρ is a quantum state Wρ(ξ) is a probability distribution.
For finite levels systems −→ Spin tomography
D = u−1|m〉〈m|u 1u =∑m
u |m〉 〈m| u† =∑m
|m, u〉 〈m, u|
Tomographic probability distribution (Tomograms):
Wρ(m; u) = 〈m|uρu−1|m〉 u ∈ SU(N)
Invertible if a sufficient set of reference frames is provided(quorum). For N generic the dimension of quorum is N + 1corresponding to the different ways one can embed the Cartansubalgebra in the Lie algebra.
Marco Laudato Quantum and tomographic metrics from relative entropies
Spin Tomography for N=2
Qubit case: N = 2
ρ =1
2
(σ0 +
∑j
yjσj
),
∑k
y2k = w2 ≤ 1
Quorum:
u1 = exp(iπ
4σ2
), u2 = exp
(−i π
4σ1
), u3 = 1
Tomographic probability distributions (j = 1, 2, 3):
W (±|uj) = 〈±| ujρu†j |±〉 =1± yj
2, W (+|uj) + W (−|uj) = 1.
It is possible to write the parameter of the state in terms of thetomograms:
yj = 2W (+|uj)− 1
Marco Laudato Quantum and tomographic metrics from relative entropies
Spin Tomography for N=2
Wρ is a fair probability distribution−→ we can define a divergencefunction (Tsallis)
S(Wρ,Wρ) = (1− q)−1(
1−∑m
Wqρ (m; u)W1−q
ρ (m; u))
and a metric
G = −i∗d dSq(Wρ, Wρ) = (q(1− q))−1∑m
dWρq ⊗ dWρ
1−q
which is the unique metric tensor defined for classical probabilitydistributions (Fisher-Rao):
GW(y , uj) =∑m
Wρ(uj)d lnWρ(uj)⊗ d lnWρ(uj)
for N=2:
Gq(y , uj) =1
4
1
Wρ(uj)(1−Wρ(uj))dWρ(uj)⊗Wρ(uj)
= Gjjdyj ⊗ dyj =1
1− y2j
dyj ⊗ dyj (no sum over j)
Marco Laudato Quantum and tomographic metrics from relative entropies
Reconstruction Formula
Quantum metrics are written in terms of the parameter of thestates:
gPetz =1
1− w2dw ⊗ dw +
w2
(1 + w)f(
1−w1+w
)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
The parameter of the states can be written in terms of thetomographic symmetric tensors (Reconstruction formula):
yj = ±√
1− G−1jj
→ It is possible to reconstruct quantum metrics starting fromtomographic symmetric tensors. [Man’ko, Marmo, Ventriglia, Vitale]
Marco Laudato Quantum and tomographic metrics from relative entropies
Changing the Tomographic Scheme
Consider a function F which is invertible, analytic and s.t F (ρ) is aquantum state. We change the tomographic scheme:
Wρ = 〈m| uρu† |m〉 −→ WF = 〈m| uF (ρ)u† |m〉
Question: if we change the tomographic scheme, which metric dowe obtain by using the reconstruction formula?
G = G(Fisher)W
yj=±√
1−G−1jj
F (ρ) //______ G = G(Fisher)WF
yj=±√
1−G−1jj
g = g
(Petz)S ,f F
//________ g =?
Marco Laudato Quantum and tomographic metrics from relative entropies
Changing the Tomographic Scheme
Usual (linear) tomography:
Wρ(u|m) = Tr(uρu† |m〉 〈m|
)= Tr
((uU)ρ0(uU)† |m〉 〈m|
)= Tr
(vρ0v
† |m〉 〈m|)≡ Wρ0(v |m) ,
New tomography:
WF (u|m) = Tr(uF (ρ)u† |m〉 〈m|
)= Tr
(uF (Uρ0U
†)u† |m〉 〈m|)
= Tr(uUF (ρ0)U†u† |m〉 〈m|
)= Tr
(vF (ρ0)v † |m〉 〈m|
)ρ = Uρ0U
† M ≈ SU(N)× Σ
M iP //p
iD
""DDD
DDDD
DDDD
DDDD
DDD P(X )
D
W
OO F(ρ0(~p)
)= ρ0(~p)
Diff(M) 3 F =
~p 7−→ ~p = ~p(~p)
1U(N)
Marco Laudato Quantum and tomographic metrics from relative entropies
Changing the Tomographic SchemeN=2 case
M ≈ SU(2)×Σ1 ~p = (p1, p2) =(
1+w2 , 1−w
2
)− 1 ≤ w ≤ 1
A smart way to realize the embedding → consider the SU(2) basis:
σw =
(1 00 −1
)σθ =
(0 e iθ−iφ
e−iθ+iφ 0
)σφ =
(0 −i e iθ−iφ
i e−iθ+iφ 0
)written in the basis the eigenstates of σw .The state becomes:
ρ =1
2
(σ0 + wσw
)Quorum:
uθ = exp(iπ
4σφ
), uφ = exp
(−i π
4σθ
), uw = 1
Tomograms:
Wρ(±|uw ) =1± w
2Wρ(±|uθ) =
1
2Wρ(±|uφ) =
1
2
Marco Laudato Quantum and tomographic metrics from relative entropies
Changing the Tomographic Scheme
We can define the new tomographic scheme:
WF (u|mw ) = Tr(uF (ρ)u† |mw 〉 〈mw |
)= Tr
(uρu† |mw 〉 〈mw |
)=Wρ(u|mw )
where ρ = 12 (σ0 + wσw ). Then, by comparing
F(ρ(w)
)= ρ(w)
We obtain the diffeomorphism on the parameter space induced bychanging the tomographic scheme:
F =
w 7−→ w = w(w)
1
Marco Laudato Quantum and tomographic metrics from relative entropies
Consequences on Tomograms
Let start by analyzing what happens to tomographic symmetrictensors when we change tomographic scheme.We can compute the tomographic symmetric tensor (Fisher-Rao)associated to the tomographic scheme F (ρ) = ρ:
GW =∑mw
1
WρdWρ ⊗ dWρ =
1
1− w2dw ⊗ dw
And now, we perform the pull-back of this tensor w.r.t. the diffeoinduced by the change of tomographic scheme:
GWF= F ∗G
W=
1
1− w2(w)
(dw(w)
dw
)2
dw ⊗ dw
=1− w2
1− w2(w)
(dw(w)
dw
)2
GW
Marco Laudato Quantum and tomographic metrics from relative entropies
Consequences on Tomograms
The pulled-back symmetric tensor can be factorized as:
F ∗G = GWF= AFGW
Where the conformal factor is:
AF ≡ A(w(w)) =1− w2
1− w2(w)
(dw(w)
dw
)2
and GW is the Fisher-Rao symmetric tensor associated to the lineartomography.
G = G(Fisher)
W
w=±√
1−G−1w w
F∗ //______ F ∗G = AF G(Fisher)W
w=±√
1−G−1ww
g = g(Petz)S ,f F∗
//_______ F ∗g = AF h(Petz)
S ,f
Marco Laudato Quantum and tomographic metrics from relative entropies
Consequences on Quantum States
N = 2 WF (u|mw ) = Tr(uF (ρ)u† |mw 〉 〈mw |
)F (ρ) = e−βρ
Tr(e−βρ)
The diffeomorphism is:
w(w) = − tanh
(βw
2
)We have to choose a starting quantum metric (von Neumann):
g(w) =1
1− w2dw ⊗ dw + 2w ln
(1 + w
1− w
)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
We let the diffeo act:
g(w(w)
)=
1
1− tanh2(βw2
) −β
2 cosh2(βw2
)2
dw ⊗ dw+
− 2 tanh
(βw
2
)ln
1− tanh(βw2
)1 + tanh
(βw2
) (θ1 ⊗ θ1 + θ2 ⊗ θ2).
Marco Laudato Quantum and tomographic metrics from relative entropies
Consequences on Quantum States
The metric factorizes again as g(w(w)
)= A(w)h
(w(w)
)and the
conformal factor is
A(w) =β2(1− w2)
4 cosh2(βw2
) ,We have to prove that this tensor is a metric tensor satisfying thePetz’s classification theorem:
h(w(w)
)=
1
1− w2dw ⊗ dw+
− 2
β2tanh
(βw
2
)ln
1− tanh(βw2
)1 + tanh
(βw2
) 1− tanh2
(βw2
)1− w2
×
×(
4 cosh4
(βw
2
))(θ1 ⊗ θ1 + θ2 ⊗ θ2).
Marco Laudato Quantum and tomographic metrics from relative entropies
Quantum States Space
By comparing it with the expression of quantum metrics in termsof OMF:
gf =1
1− w2dw ⊗ dw +
w2
(1 + w)f ( 1−w1+w )
(θ1 ⊗ θ1 + θ2 ⊗ θ2)
We obtain a different OMF:
f (t) =β
2
t(1− t)
(1 + t)2 sinh(β 1−t
1+t
)
f (t) = tf
(1
t
)0 ≤ β ≤ 1
Marco Laudato Quantum and tomographic metrics from relative entropies
The Inverse Problem
We want to state the inverse problem: If we choose the startingmetric tensor and the target metric tensor, are we able to find adiffeo induced by a change of tomographic scheme which relates
them?
If this diffeo always exists and it is unique, we have shown aone-to-one correspondence between quantum metric tensors andtomographic schemes [L., Mele, Marmo, Ventriglia, Vitale].
Marco Laudato Quantum and tomographic metrics from relative entropies
The Inverse Problem
Now w(w) is unknown and we fix the OMF f (t) of the targetmetric. Let start again with the metric obtained from VN relativeentropy and formally perform the pull-back:
g(w(w)
)=
1
1− w(w)2
(dw(w)
dw
)2
dw ⊗ dw
+ 2w(w) ln
(1 + w(w)
1− w(w)
)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
The tensor factorizes again as: g(w(w)
)= A(w)h
(w(w)
), where
A(w(w)
)=
(1− w2)
1− w(w)2
(dw(w)
dw
)2
h(w(w)
)=
1
1− w2dw ⊗ dw + 2w(w) ln
(1 + w(w)
1− w(w)
)1− w(w)2
1− w2×
×(dw(w)
dw
)−2
(θ1 ⊗ θ1 + θ2 ⊗ θ2)
Marco Laudato Quantum and tomographic metrics from relative entropies
The Inverse Problem
The target metric is:
gf =1
1− w2dw ⊗ dw + Cf (w)(θ1 ⊗ θ1 + θ2 ⊗ θ2)
where
Cf (w) =w2
(1 + w)f ( 1−w1+w )
We have the following non-linear, first order, ordinary differentialequation:
dw(w)
dw=
√1
Cf (w)2w(w) ln
(1 + w(w)
1− w(w)
)1− w(w)2
1− w2
or also: dw(w)dw =
√2
q(1−q)1
Cf (w) (aq − bq)(a1−q − b1−q) 1−w(w)2
1−w2
Lipschitz continuity → uniqueness and existence
Marco Laudato Quantum and tomographic metrics from relative entropies
Conclusions and Perspectives
Conclusions:
We have developed an intrinsic differential calculus on the space ofparameters in information geometry
We have used techniques of information geometry to define ageneral form of quantum metric tensor on the space of (invertible)states, which depends on the parameters of the state and of theparameter of the q-z-divergence function
We have shown that there exists (for N = 2) a one-to-onecorrespondence between quantum metrics and tomographicschemes.
Perspectives:
We want to further investigate the relation between tomographicschemes and divergence functions
We want to investigate the Chentsov theorem by unfolding thesemetrics to the general linear group
Quantum Fisher-Rao metric in Lie Group Thermodynamics
Marco Laudato Quantum and tomographic metrics from relative entropies
References and Work in Progress
Quantum Information Geometry and Tomography:
V.I. Man’ko, G. Marmo, F. Ventriglia, P. Vitale, Metric on the Space ofQuantum States from Relative Entropy. Tomographic Reconstruction, acceptedby Journal of Physics A, ArXiv: 612.07986v2 [quant-ph], 2017
L.M, G. Marmo, F.M. Mele, F. Ventriglia, P. Vitale, TomographicReconstruction of Quantum Metrics, submitted to Journal of Physics A, ArXiv:1704.01334 [math-ph], 2017
F.M. Ciaglia, F. Di Cosmo, M.L., G. Marmo, F.M. Mele, F. Ventriglia, P. Vitale,Quantum Metrics from q, z-relative Entropies, prepared for submission toInformation Geometry, Springer
Open Quantum Systems:
F. Ciaglia, F. Di Cosmo, M.L., G. Marmo, Differential Calculus on Manifoldswith a Boundary. Applications., Int. J. Geom. Methods Mod. Phys.,DOI:10.1142/S0219887817400035, February 2017.
Quantum Gravity:
G. Chirco, F.M. Mele, D. Oriti, P. Vitale, Fisher Metric, GeometricEntanglement and Spin Networks, submitted to Physical Review D, ArXiv:1703.05231 [gr-qg], 2017
G. Chirco, I. Kotecha, M.L., F.M. Mele, D. Oriti, Statistical Aspects in GroupField Theories: Constraints from Equilibrium in Reparametrization-InvariantSystems, in preparation
Marco Laudato Quantum and tomographic metrics from relative entropies
Marco Laudato Quantum and tomographic metrics from relative entropies
top related