quantum and tomographic metrics from relative entropies

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


Part I

Information Geometry: A Sketch


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


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


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


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).


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


Part II

Quantum Metrics from Divergence Functions


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


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.


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.


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).


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.


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]


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)


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


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 ]


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.


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)


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.


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)



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)



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]


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


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


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 )


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)

]


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?


Part III

Tomographic Reconstruction of Quantum Metrics


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 .....


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.


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


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)


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]


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 =?



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)


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


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


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


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).


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).


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


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].


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)


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


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


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


quantum and tomographic metrics from relative entropies

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