Tomographic Reconstruction of Quantum Metrics Marco Laudato1,3 , Giuseppe Marmo1,2 , Fabio M. Mele1 , Franco Ventriglia1,2 , and Patrizia Vitale1,2
arXiv:1704.01334v1 [math-ph] 5 Apr 2017
1
Dipartimento di Fisica “E. Pancini” Universit` a di Napoli Federico II, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy. 2
3
INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy.
Turku Center for Quantum Physics, Department of Physics and Astronomy, University of Turku, FIN-20014, Turun Yliopisto, Finland. e-mail:
[email protected],
[email protected],
[email protected],
[email protected],
[email protected]
Abstract In classical information geometry one can use a potential function to generate a metric tensor and a dual pair of connections on the space of probability distributions. In a previous work, some of the authors have shown that, by using the quantum Tsallis q-entropy (which includes the von Neumann one in the limit q → 1) as a potential function and tomographic methods, it is possible to reconstruct quantum metrics from the “classical” one. Specifically, the unique Fisher-Rao metric, defined on the space of probability distributions associated with quantum states by means of a spin-tomography, is directly related to one particular metric on the space of quantum states in the sense of the Petz classification, i.e., to one operator monotone function. Our claim is that there exists a bijective relation between the choice of the tomographic scheme and the particular operator monotone function identifying a unique quantum metric tensor. As an explicit example, starting from the von Neumann entropy, we consider a non-linear tomogram and we show how such a choice selects a unique operator monotone function. Finally, we show in a general fashion that such a bijective relation exists for any tomographic scheme under the hypothesis of Lipschitzian continuity.
1
1
Introduction
In the last years, a considerable effort has been devoted to understand the role of Information Theory in Physics. In particular, due to the common probabilistic background, the study of Quantum Mechanics from the point of view of Information Theory has been very fruitful and it allowed the foundation of a brand-new field called Quantum Information Theory [1]. It is now a well-established subfield of physics, having relevance also to other areas of science. Starting from the pioneering work of Cramer, Rao and Fisher in the 40’s, Amari and others (see [2] and references therein) have shown that it is possible to approach Information Theory by using tools from Differential Geometry. Specifically, Geometric Information Theory deals with a Riemannian manifold of probability distributions on some sample space X endowed with a pair of dually related connections. The Riemannian structure can be defined in terms of the Hessian of a class of functions called potential functions, or alternatively contrast functions, divergence functions or distinguishability functions. They are defined on two copies of the statistical manifold 1 and, roughly speaking, give a measure of the distance between two probabilities. From the Chentsov uniqueness theorem [4] we know that, in the classical case at finite dimensions, there exists a unique monotone metric on statistical manifolds usually called Fisher-Rao metric tensor [5] (the generalization to the infinite-dimensional case was proven by Michor in [6]). On the other hand, in the quantum setting, the probabilistic interpretation of Quantum Mechanics deals with probability amplitudes instead of probabilities. It has been shown [7] that, by replacing probabilities with probability amplitudes, the Fisher-Rao metric tensor used in classical Information Theory may be obtained from the Fubini-Study metric on the space of pure states of a quantum system. Moreover, in order to include also mixed states, it is possible to“quantize” the simplex of classical probabilities by associating with every probability vector a coadjoint orbit of the unitary group acting on the dual space of its Lie algebra [8, 9]. The union of these orbits can be identified with the stratified manifold of the quantum state space [10]. It is possible to consider this space as a quantized statistical manifold and, by means of a potential function2 it is possible to define quantum Fisher metric tensors on it. However, unlike the classical case, these metric tensors are far from being unique. They were classified by Petz [11] in terms of operator monotone functions. In particular, as we will recall soon, the Petz classification theorem states that there exists a bijective correspondence between quantum metrics and operator monotone functions f : (0, ∞) → R which satisfy f (t) = tf (1/t). An equivalent approach to quantum mechanics is the quantum tomographic setting (for a recent review see [12, 13] and refs. therein). Since tomographic probability distributions are fair probabilities, Chenstov’s theorem ensures that there is a unique metric defined on them. It has been shown in [14] that, starting from the tomographic metric, it is possible to reconstruct the quantum metric obtained from the Tsallis entropy on the states space. Here the main point emerges. Since from the usual spin-tomography [15, 16, 17, 18], we can obtain one metric tensor which is in turn associated by Petz classification theorem to one operator monotone function, it is clear that a choice has been made somewhere to select a unique metric in the 1
A dynamical framework where two copies of the statistical manifold are connected with the tangent bundle by means of the Hamilton-Jacobi theory can be found in [3] . 2 To this aim, different quantum entropies have been suggested. The most celebrate is the von Neumann entropy, but also other candidates, like the Tsallis or the R´enyi’s entropy, have been considered.
2
family of quantum metrics. Our claim is that this choice is actually the choice of the particular tomographic scheme we have used. The scope of the present work is to show that there exists a bijective correspondence between the choice of tomographic schemes and operator monotone functions classifying quantum metrics. In order to avoid computational difficulties, we shall eventually restrict the analysis to qubits and shall derive such an invertible relation in terms of the solution of a non-linear first order, ordinary differential equation. The paper is organized as follows. In Section 2 we review the derivation of the tomographic metric [14] from a divergence function, in the spin-tomographic scheme. In Section 3 we analyze the meaning of a change of tomography from a geometrical point of view. We then discuss the relationship between the metric tensors defined on the space of tomographic probability distributions, before and after the change of the tomographic scheme. Then in Section 4 we discuss the corresponding transformation induced on the metric defined on the space of quantum states. Specifically, in Section 4.1 we consider the explicit example of a non-linear tomographic scheme for a two-level system and show that in this case our claim is correct. Finally, in Section 4.2 we show that such a relation always exists and that it is bijective. The main calculations for the qubit example are reported in Appendix A.
2
Spin-tomographic metric from relative Tsallis entropy
In the spin-tomographic setting [15, 16, 17, 18], a N -level quantum state represented by a density matrix, ρ, is associated with a tomographic probability distribution W(m|u), through a specific family of operators, D, according to the general dequantization procedure [19, 20] ρ 7→ W = Tr ρD. The tomography is realized by unitarily rotating the spin basis in the space of quantum states: by posing N = 2j + 1, with j the quantum number associated to the spin and −j ≤ m ≤ j, the operator D(m, u) = u† |mihm|u , is the spin-tomographic family of dequantizers, depending on m and u ∈ SU (N ). We have indeed a resolution of the identity for each u X X 1u = u|mihm|u† = |m, uihm, u| (2.1) m
m
with |m, ui providing a basis in the unitarily rotated reference frame in the Hilbert space of states and the unitary matrix u labelling the reference frame where the spin states are considered. The spin-tomographic probability distribution is then given by the diagonal matrix elements of the density operator ρ, calculated in the given reference frame associated to u, that is Wρ (u|m) = Tr ρu† |mi hm| u = hm| uρu† |mi .
(2.2)
Eq. (2.2) is invertible if a sufficient number of reference frames, called quorum, is provided. As discussed in [14], a minimal number is equal to N + 1. We can define a Fisher metric tensor on the space of (classical) tomographic probabilities starting from the associated relative Tsallis entropy ! X 1−q −1 q f (m|u) (2.3) 0 ≤ Sq (ρ, ρ˜, u) = [(q(1 − q)] 1− Wρ (m|u) W ρ˜ m
3
with ρ, ρ˜ corresponding to two different states, and u labelling the reference frame. The metric is then [14] X X ˜ q (Wρ , W fρ˜) = (q(1 − q))−1 Wρ d ln Wρ ⊗ d ln Wρ (2.4) dWρ q ⊗ dWρ 1−q = G = −i∗ d dS m
m
with i : M ,→ M × M the diagonal embedding of the parameters manifold M into the Cartesian ˜ exterior derivatives on M × M respectively acting on the left product, i∗ its pull-back and d, d, and right components of the Cartesian product. As expected, this has the form of the Fisher-Rao metric on the space of tomographic probabilities. Let us restrict to the qubit case, where X 1 ρ = (σ0 + yj σ j ) 2
(2.5)
j
σ0 , σj , j = 1, .., 3 being the generators of the u(2) algebra and quorum is equal to N + 1 = 3 and we can choose for example π u1 = exp(i σ2 ) 4 so to have
π u2 = exp(−i σ1 ) 4
,
1 ± yj 1 1 1 W(± |uj ) ≡ h± |uj ρu†j | ± i = 2 2 2 2
2 k yk
P
,
= w2 ≤ 1. In this case the
u3 = 1
(2.6)
j = 1, ..., 3
(2.7)
with W( 21 |uj ) + W(− 12 |uj ) = 1 for each j. On inverting this relation for the parameters yj we thus get an expression of the qubit state (2.5) in terms of tomograms Wj ≡ W( 12 |uj ) yj = 2Wj − 1
(2.8)
For each of the reference frames of the tomographic setting, labelled by uj (cfr. (2.6)), we get from (2.4) a symmetric tomographic tensor, which can be expressed in terms of the parameters yj GW (y, uj ) = Grs (y, uj )dyr ⊗ dys (2.9) with Grs (y, uj ) =
1 1 1 δrs δsj δrs δsj = 4 Wj (1 − Wj ) 1 − yj2
(2.10)
from which we get 1±
p
1 − 1/Gjj (2.11) 2 Notice that the tensor (2.9) is degenerate. Finally, on using Eq. (2.8) , we arrive at a direct expression of the parameters yk labelling the quantum state, in terms of the tomographic tensor q yj = ± 1 − G−1 (2.12) jj . Wj =
It is now possible to assemble together a sufficient number of these symmetric tomographic tensors so as to build a metric tensor on the space of states. This will be the subject of next section.
4
2.1
Quantum metrics and reconstruction formula
A family of metric tensors on the manifold of quantum states, M , has been obtained in [14] starting from the associated quantum relative Tsallis entropy ST s (ρ, ρ˜) = (1 − Tr ρq ρ˜1−q )[q(1 − q)]−1
(2.13)
The density matrices ρ, ρ˜ can be parametrized in terms of diagonal matrices and unitary transformations ρ = U ρ0 U −1 , ρ˜ = V ρ˜0 V −1 (2.14) where U, V ∈ SU (N ) are special unitary N × N matrices, with N labeling the levels. According to [14] we have ˜ T s (ρ, ρ˜) = (q(1 − q))−1 i∗ Tr dρq ⊗ d˜ ˜ρ1−q g = −i∗ d dS (2.15) ˜ exterior derivatives with i : M ,→ M ×M the diagonal immersion, i∗ the pull-back map, and d, d, respectively acting on ρ, ρ˜. The tensor product is to be understood in the space of one-forms, whereas the trace is defined on the space of N × N matrices. On using dρq = d(U ρq0 U −1 ) = dU ρq0 U −1 + U dρq0 U −1 − U ρq0 U −1 dU U −1 ˜ρ1−q = d(V d˜
−1 ρ˜1−q ) 0 V
= dV
ρ˜01−q V −1
+V
−1 d˜ ρ1−q 0 V
−V
ρ˜01−q V −1 dV
(2.16) V −1
(2.17)
and performing the pullback to the manifold M (which amounts to put ρ˜0 = ρ0 and U = V, U −1 = V −1 ), we get [14] (2.18) gq = (q(1 − q))−1 Tr [U −1 dU, ρq0 ] ⊗ [U −1 dU, ρ01−q ] + q(1 − q)ρ−1 0 dρ0 ⊗ dρ0 . For N = 2 the relation between the parametrization in (2.5) and that in (2.14) acquires the simple form 1 (2.19) yj = w Tr (U σ3 U −1 σj ) 2 whereas the metric (2.18) becomes gq =
2 1 dw ⊗ dw + (aq − bq )(a1−q − b1−q )(θ1 ⊗ θ1 + θ2 ⊗ θ2 ), 1 − w2 q(1 − q)
with
aα =
1+w 2
α
,
bα =
1−w 2
(2.20)
α ,
α = q, 1 − q.
(2.21)
In polar coordinates y1 = w sin θ cos φ, y2 = w sin θ sin φ, y3 = w cos θ, the metric (2.20) reads gq (w, θ, φ) =
1 1 dw ⊗ dw + (aq − bq )(a1−q − b1−q ) dθ ⊗ dθ + sin2 θdφ ⊗ dφ (2.22) 2 1−w 2q(1 − q)
Thus, thanks to Eq. (2.1), it is possible [14] to reconstruct the metric (2.22) in terms of the tomographic tensors (2.9). To summarize, although individual tomographic tensors are degenerate, they can be put together by means of the tomographic map to produce a quantum metric on the manifold of parameters if a quorum has been assigned.
5
Remark: The metric (2.20) is in agreement with Petz classification theorem [11], indeed it can be obtained from the general formula gf =
1 w2 dw ⊗ dw + 1−w (θ1 ⊗ θ1 + θ2 ⊗ θ2 ) 1 − w2 (1 + w)f ( 1+w )
(2.23)
if we use as operator monotone function q(1 − q) (t − 1)2 f (t) = q , (t − 1)(t1−q − 1) where we pose t =
(2.24)
1−w 1+w .
To sum up, from these considerations we can deduce that all these concepts are related, according to the following diagram: Entropy o
/ Metric o O
/ Operator Monotone Function
(2.25)
Tomography Here the main point emerges. Since from the usual (linear) tomography (2.2) we can reconstruct only one quantum metric, by means of the reconstruction formulas , and such quantum metric is in turn associated by Petz classification theorem to one operator monotone function, it is clear that a choice must have been made somewhere to select a unique metric in the family of quantum metrics. Our claim is that this choice is actually the choice of the particular tomographic scheme we have used. Indeed, in general, one can define another (possibly non-linear) tomography as WF (u|m) = hm| uF (ρ)u† |mi ,
(2.26)
where the function F has to be invertible and such that F (ρ) is still a quantum state. In the following, we will show that the metric associated to this tomography is not related to the operator monotone function (2.24) but to another one. In other words, we are going to prove that in the previous diagram there exists a link between the tomography and the operator monotone function, say: Entropy o
/ Metric o O
/ Operator Monotone Function j4 j j j j j jt j
(2.27)
Tomography In particular, we will discuss the explicit example of a tomographic scheme provided by a thermal-like state for a two-level system and show that in this case our claim is correct. Then we will show that such a relation always exists and that it is bijective. Moreover, since the metric on the space of quantum states can be obtained via tomographic reconstruction and since we are interested in understanding the effect of a change of tomographic scheme on the corresponding quantum metric, it is convenient to write the Fisher metric defined 6
on tomographic probability distributions in the same coordinates as Petz’s quantum metrics, that is in polar coordinates (cfr. Eq. (2.23)). This amounts to locally rotate Pauli matrices by means of “dreibeine” so to have σw = sin θ cos φ σ1 + sin θ sin φ σ2 + cos θ σ3 (2.28)
σθ = cos θ cos φ σ1 + cos θ sin φ σ2 − sin θ σ3 σ = − sin φ σ + cos φ σ 1 2 φ
or in matrix form σw =
cos θ sin θe−iφ sin θeiφ − cos θ
! ,
− sin θ cos θe−iφ cos θeiφ sin θ
σθ =
! ,
σφ =
0 −i e−iφ i eiφ 0
! (2.29)
and write the tomograms in term of the eigeinstates of one of them. In particular, in what follows the reference frame in the Hilbert space of states will be given by the eigenstates {|mw i} of σw . In this basis, the matrices (2.29) assume the following expression ! ! ! 1 0 0 eiθ e−iφ 0 −i eiθ e−iφ σw = , σθ = , σφ = . (2.30) 0 −1 e−iθ eiφ 0 i e−iθ eiφ 0 With respect to the basis (σ0 , σθ , σφ , σw ) of u(2), the Bloch decomposition (2.5) for a generic density matrix ρ reads as ! 1+w 1 0 2 , (2.31) ρ = (σ0 + wσw ) = 1−w 2 0 2 where we have used the relation σw = π uθ = exp i σφ 4
~ y ·~ σ |~ y|
,
=
1 w
P3
k=1 yk σk .
In this case, the quorum is given by
π uφ = exp −i σθ 4
,
uw = 1
(2.32)
and the corresponding tomograms are Ww ≡ Wρ (mw = ±1|uw ) = hmw |uw ρu†w |mw i =
1±w , 2
1 , 2 1 Wφ ≡ Wρ (mw = ±1|uφ ) = hmw |uφ ρu†φ |mw i = . 2 Wθ ≡ Wρ (mw = ±1|uθ ) = hmw |uθ ρu†θ |mw i =
(2.33)
Therefore, according to Eqs. (2.9, 2.10), the unique non-zero tomographic tensor is the one associated to uw , say 1 Gw = Gww dw ⊗ dw = dw ⊗ dw . (2.34) 1 − w2 Finally, by means of the relation q w = ± 1 − G−1 ww
(2.35)
it is possible to reconstruct the quantum metric (2.23) by means of the tomographic tensors.
7
3
Changing the tomographic scheme
Let us analyze in details what the introduction of a new tomographic scheme means and what are its main consequences. Let us recall that a generic density state ρ of a N -level quantum system can be parametrized in terms of diagonal matrices with positive entries and unitary transformations as p1 . . . 0 0 p2 . . . † ρ(U, p~) = U (3.1) U = U ρ0 U † , . . . . . . . . . 0 . . . pN where U ∈ U (N ) and, being Tr ρ = 1, p~ ≡ (p1 , . . . , pN ) is a probability vector belonging to a (N − 1)-simplex of classical probabilities. This essentially amounts to “quantize” the classical simplex by associating a coadjoint orbit of the unitary group with each probability vector identified with the diagonal elements of a density matrix [14]. The union of all these i
D orbits is the space of all quantum states and, according to the map ρ 7−→ ρ(U, p~), is identified with the parameter space M . Clearly, the unitary matrices U in (3.1) are determined only up to unitary transformations in the isotropy group G0 of ρ0 and the parameter space M is thus given by the homogeneous space SU (N )/G0 times the open part of the simplex to which the diagonal part of the state belongs. As stressed in [14], since such a homogeneous space is not parallelizable, we shall consider the differential calculus as carried on the redundant SU (N ) times the interior of the simplex.
At the tomographic level, the spin-tomographic probability distribution defined in (2.2) can be similarly parametrized by means of unitary matrices and diagonal matrices. Indeed, Eq. (2.2) can be recast in the form Wρ (u|m) = Tr uρu† |mi hm| = Tr (uU )ρ0 (uU )† |mi hm| (3.2) = Tr vρ0 v † |mi hm| ≡ Wρ0 (v|m) , where U is the unitary matrix which diagonalizes ρ and we have defined the unitary matrix v as the product uU with u, U ∈ U (N ). Hence, spin-tomograms come to be probability distributions defined on a discrete space X consisting of 2j + 1 points identified by the possible values of m (e.g., two points for the qubit case) and, coherently with the parametrization (3.1) of the states, they are also parametrized in terms of v ∈ SU (N ) and ρ0 , namely Wρ0 (v|m) ∈ P(X ; M ). From a geometrical point of view, the parameter space M is embedded both in the space of quantum states D and in the space P(X ) of probability distributions on X according to the following factorization
M Dp D
iP / P(X ) O DD DD DD DD D W iD DD DD DD D"
D
where the embedding maps iD and iP are defined by Eqs. (3.1, 3.2), respectively. 8
(3.3)
Let us consider now a change of the tomographic scheme, say WF (u|m) = Tr uF (ρ)u† |mi hm| ,
(3.4)
where F is an analytic, invertible function transforming states into states. The invertibility requirement ensures no loss of information about the starting state. Moreover, since F is analytic, F (ρ) = F (U ρ0 U † ) = U F (ρ0 )U † , U being the unitary matrix which diagonalizes ρ. Thus, Eq. (3.4) can be written as P(X ; M ) 3 WF (ρ0 ) (v|m) = Tr vF (ρ0 )v † |mi hm| ,
(3.5)
i.e., the spin-tomographic probability distributions associated with the new tomograms are still parametrized by elements in M . More specifically, the analyticity requirement implies that F (ρ) is diagonalized by means of the same unitary matrix of ρ and therefore F affects only the probability vector associated with the elements of ρ0 . Indeed, F (ρ0 ) is still a quantum state in its diagonal form and can be thus identified with a diagonal density matrix ρ˜0 parametrized by a probability vector p~˜ depending on p~ through the following relation F ρ0 (~ p) = ρ˜0 (p~˜) .
(3.6)
In other words, the change of the tomographic scheme (3.4) identifies a diffeomorphism on the parameter space M acting non-trivially only on the simplex part, i.e.: ( p~ 7−→ p~˜ = p~˜(~ p) F ∈ Diff(M ) s.t. . (3.7) 1U (N ) In what follows we will focus on two-level systems for which the unitary transformations diagonalizing the state ρ belong to the group SU (2) and the corresponding probability vector p~ can be parametrized as p~ = (p1 , p2 ) = (1 + w)/2, (1 − w)/2 with −1 ≤ w ≤ 1. In other words, in this case, the parameter space M can be described in terms of SU (2) times the open interval (−1, 1). A natural way to realize the above-mentioned embedding (3.2) of M into the space of tomographic probabilities P(X ) is to consider the spin-tomographic probability distribution associated to the state ρ, calculated in the reference frame of eigenstates of σw , which reads as Wρ (u|mw ) = Tr uρu† |mw i hmw | (3.8) with ρ is given by (2.31). In such a reference frame, the change of the tomographic scheme (3.4) reads as WF (u|mw ) = Tr uF (ρ)u† |mw i hmw | = Tr u˜ ρu† |mw i hmw | = Wρ˜(u|mw ) ,
(3.9)
with
1 ˜ w) . ρ˜ = (σ0 + wσ 2 Hence, Eq. (3.6) relating the parameters of the states ρ and ρ˜ now becomes F (ρ(w)) = ρ˜(w) ˜ ,
9
(3.10)
(3.11)
and we see that the change of tomography amounts to the following diffeomorphism on M non-trivially acting only on the simplex part ( w 7−→ w ˜ = w(w) ˜ F : . (3.12) 1 We are interested in understanding how tomographic tensors behave under a diffeomorphism of the form (3.12), i.e., under a change of the tomographic scheme. As already stressed in the introduction, quantum tomograms are true probability distributions and hence, according to Chenstov’s theorem, a unique monotone Fisher-Rao metric can be defined on their parameter space M without ambiguities in complete analogy to the classical setting. According to Eq. (2.34), the tomographic tensor acquires the following expression in terms of the parameter w ˜ of ρ˜: GW f=
X 1 1 dWρ˜ ⊗ dWρ˜ = dw ˜ ⊗ dw ˜. Wρ˜ 1−w ˜2 m
(3.13)
w
The pull-back along the map (3.12) thus yields ∗
GWF = F GW f
1 = 1−w ˜ 2 (w)
dw(w) ˜ dw
2
1 − w2 dw ⊗ dw = 1−w ˜ 2 (w)
dw(w) ˜ dw
2 GW
(3.14)
where GW in the second line is given by (2.34). Therefore, the action of the diffeomorphism F on the tomographic metric essentially amounts to the following conformal transformation GWF = AF GW ,
(3.15)
where the conformal factor AF is defined as 1 − w2 AF ≡ A(w(w)) ˜ = 1−w ˜ 2 (w)
dw(w) ˜ dw
2 .
(3.16)
Remark: Let us notice that the appearance of the conformal factor does not contradict Chentsov’s theorem. Indeed, the latter states that the Fisher-Rao metric is the unique monotone metric invariant (up to a constant factor) under the action of stochastic maps, i.e., under diffeomorphisms on the space X of random variables [4], while in the case under examination the diffeomorphism F is on the parameter space M and not on X . To sum up, changing the tomographic scheme corresponds in general to perform a diffeomorphism of the form (3.7) on the parameter space whose action on the metric tensor defined on the space of tomographic probabilities maps Fisher-Rao metrics into conformal Fisher-Rao metrics, where the conformal factor is uniquely determined by the change of tomographic scheme.
4
Counterpart on the space of states
According to the diagram (3.3), the parameter space M is embedded both into the space of tomographic probabilities and into the space of density matrices. Since we are interested in showing a one-to-one correspondence between the choice of a tomographic scheme and operator monotone 10
functions, it is therefore reasonable to explore the consequences of such a diffeomorphism related to a tomography change on the metric tensors defined on the space of quantum states. What we do expect, in analogy with the classical case, is that this class of diffeomorphisms will map a quantum metric characterized by an operator monotone function to a different quantum metric characterized by another operator monotone function times a conformal factor which depends only on the change of tomographic scheme. To explicitly visualize this, we will first focus on the specific example of a non-linear exponential tomographic scheme, and then we will set up the general problem.
4.1
An example of non-linear tomography for qubits
In this section we will show by means of an explicit example that the diffeomorphism induced by changing the tomographic scheme maps the metric associated to the von Neumann entropy to a different quantum metric parametrized by another operator monotone function times the same conformal factor (3.16). In particular, we will consider the tomography WF (u|mw ) = Tr uF (ρ)u† |mw i hmw | , (4.1) where we set F (ρ) =
e−βρ . T r(e−βρ )
(4.2)
Let us recall that, being both states, we have the following Bloch decompositions for ρ and F (ρ) 1 ρ = (σ0 + wσw ) 2
,
1 F ρ(w) ≡ ρ˜(w) ˜ = (σ0 + wσ ˜ w) 2
w, w ˜ ∈ (−1, 1)
(4.3)
which are diagonal in the basis of the eigenstates of σw . A straigthforward computation shows that the tomographic probabilities associated to the quorum (2.32), are given by 1 2 fρ˜(±, uθ ) = W fρ˜(±, uφ ) = 1 W 2
Wρ (±, uθ ) = Wρ (±, uφ ) =
, ,
1±w 2 1 ± w ˜ fρ˜(±, uw ) = W . 2 Wρ (±, uw ) =
(4.4)
Therefore, with respect to the tomographic scheme (4.1), the only non-vanishing parameter of the state ρ˜ is given by the usual formula: fρ˜(+|uw ) − 1. w ˜ = 2W
(4.5)
Comparing Eqs. (4.3), we obtain the relation w ˜ = w(w) ˜ between the parameter of ρ and the parameter of ρ˜. In the particular case of Eq. (4.2) we obtain3 the following relation: βw . (4.6) w(w) ˜ = − tanh 2 Since ρ˜ is still a quantum state, we can use, for example, as generating function of the quantum metric either the Von Neumann entropy ˜ SvN ρ˜, ζ˜ = Tr ρ˜(ln ρ˜ − ln ζ) (4.7) 3
For the explicit computation we refer to Appendix A.
11
or the (rescaled) Tsallis entropy ST ρ˜, ζ˜ =
1 1 − Tr (˜ ρq ζ˜1−q ) . q(1 − q)
(4.8)
We remark that the von Neumann entropy can be obtained from the Tsallis entropy in the limit q → 1. From them, we can compute the metric tensors 1+w ˜ 1 (θ1 ⊗ θ1 + θ2 ⊗ θ2 ) (4.9) dw ˜ ⊗ dw ˜ + 2w ˜ ln g(w) ˜ = 1−w ˜2 1−w ˜ and gq (w) ˜ =
1 2 dw ˜ ⊗ dw ˜+ (aq − bq )(a1−q − b1−q )(θ1 ⊗ θ1 + θ2 ⊗ θ2 ) 2 1−w ˜ q(1 − q)
(4.10)
respectively. In [14] it was proved that these two metrics are respectively related to the following operator monotone functions f (t) =
(t − 1) ln t
,
f (t) =
(q(1 − q))(t − 1)2 (tq − 1)(t1−q − 1)
(4.11)
where
1−w ˜ . 1+w ˜ To prove our claim, we will proceed through the following two steps: t=
(4.12)
1) We perform the pullback of the metric (4.9) (or of the metric (4.10)) with respect to the diffeomorphism (3.12) in which the non-trivial part is given by Eq. (4.6). In this way we obtain a new tensor written in terms of the parameter w of ρ; 2) We have to compare the coefficients of this new tensor with the general expression of quantum metric (2.23) written in terms of w and letting the operator monotone function as an unknown. If this operator monotone function exists, we have shown that the metric tensor associated to this operator monotone function is linked to the tomography (4.1). The first step4 , consists in replacing the expression (4.6) into the metric (4.9): 2 1 −β dw ⊗ dw+ g w(w) ˜ = 2 βw 2 βw 1 − tanh 2 cosh 2 2 1 − tanh βw 2 βw (θ1 ⊗ θ1 + θ2 ⊗ θ2 ). ln − 2 tanh βw 2 1 + tanh
(4.13)
2
In order to compare with the general expression (2.23), we need to factor out a common factor ¯ A(w) defined as: ¯ A(w) =
1−
w2
1 − tanh2
βw 2
2
−β β 2 (1 − w2 ) = . 2 βw 2 cosh2 βw 4 cosh 2 2
4
(4.14)
For simplicity, we will focus only on the metric obtained from the von Neumann entropy but the same considerations can be done also for the metric obtained from any other divergence function, e.g. the Tsallis entropy.
12
Figure 1: Graph of the operator monotone function (4.18) for β = 0.5.
Indeed, in this way, we have that the tensor factorizes as: ¯ g w(w) ˜ = A(w)h w(w) ˜ ,
(4.15)
where 1 dw ⊗ dw+ 1 − w2 βw 4 βw 2 βw 1 − tanh 2 4 cosh 1 − tanh 2 2 βw (θ1 ⊗ θ1 + θ2 ⊗ θ2 ). − 2 tanh ln 2 2 2 1−w β 1 + tanh βw 2
h w(w) ˜ =
(4.16) A straightforward computation shows that the conformal factor is exactly of the form (3.16), i.e., the same of the tomographic case, when the transformation w(w) ˜ is given by Eq. (4.6). Therefore, if the analogy with the classical case holds, we have to check that the tensor h w(w) ˜ can be recast in the form of the general metric (2.23). By using the properties 1+x βw ln = 2 arctanh(x), with x = − tanh 1−x 2 (4.17) 2 2 sinh (α) − cosh (α) = 1 we obtain the following expression for the operator monotone function: 1−w β w(1 − w) f = . 1+w 4 sinh(βw)
(4.18)
It is an operator monotone function if the parameter β satisfies 0 < β ≤ 1 as shown for instance in fig. 1 for the value β = 0.5. It remains to check the symmetry property t f (1/t) = f (t). By setting 1−w 1−t t= =⇒ w = w(t) = , (4.19) 1+w 1+t the function (4.18) becomes: 1−t 1 − β1−t 1+t t(1 − t) β = . f (t) = f w(t) = (4.20) 4 1 + t sinh β 1−t 2 (1 + t)2 sinh β 1−t 1+t
13
1+t
Then
β 1−t 1−t 1 1 β = , =− f 2 1−t t 2 (1 + t) sinh −β 2 (1 + t)2 sinh β 1−t 1+t 1+t
Finally, by comparing equations (4.20) and (4.21), we have that: 1 f (t) = tf . t
(4.21)
(4.22)
This proves that the operator monotone function (4.18) characterizes a quantum metric in the sense of the Petz’s classification. Therefore, we have explicitly shown that to the tomography (4.1), with F (ρ) expressed as in Eq. (4.2), is associated the operator monotone function (4.18) and, in turn, an unique quantum metric tensor in the Petz sense.
4.2
The inverse problem
From previous considerations it results that a change of tomographic scheme induces a diffeomorphism on the parameters space of quantum states which maps the quantum metric obtained from the von Neumann entropy to another quantum metric characterized by the operator monotone function (4.18) times the same conformal factor which arises in the tomographic case described in Section 3. Now we have to show that this is a general behaviour of any quantum metric for any change of tomographic scheme which satisfies the invertibility and analyticy requirements. In order to prove it, we are going to state an inverse problem that will lead us to a non-linear first order, ordinary differential equation whose solutions identify the change of tomographic scheme which allows to maps any starting quantum metric into another one. We have seen that any change of tomography gives rise to a relation between the parameter of ρ, say w, and the parameter of F (ρ) = ρ˜, say w, ˜ namely a relation w ˜ = w(w). ˜ When we replace this relation in the expression of the quantum metric5 g(w), ˜ we obtain the following expression, for any w ˜ = w(w): ˜ g w(w) ˜ =
1 2 1 − w(w) ˜
dw(w) ˜ dw
2
dw ⊗ dw + 2w(w) ˜ ln
1 + w(w) ˜ 1 − w(w) ˜
(θ1 ⊗ θ1 + θ2 ⊗ θ2 ). (4.23)
In order to compare with the general expression (2.23), we factorizes the pulled-back tensor as g w(w) ˜ = A w(w) ˜ h w(w) ˜ ,
(4.24)
where 1 dw ⊗ dw+ 1 − w2 −2 2 1 + w(w) ˜ 1 − w(w) ˜ dw(w) ˜ + 2w(w) ˜ ln (θ1 ⊗ θ1 + θ2 ⊗ θ2 ) 1 − w(w) ˜ 1 − w2 dw
h w(w) ˜ =
and
(1 − w2 ) A w(w) ˜ = 2 1 − w(w) ˜
5
dw(w) ˜ dw
2
Let us focus again only on the metric obtained from the von Neumann entropy.
14
(4.25)
(4.26)
which is exactly the conformal factor (3.16). We now consider the relation w ˜ = w(w) ˜ as unknown and compare the coefficient of (4.25) with the expression gf¯ =
1 w2 dw ⊗ dw + 1−w (θ1 ⊗ θ1 + θ2 ⊗ θ2 ) 1 − w2 (1 + w)f¯( 1+w )
(4.27)
where a particular operator monotone function f¯ has been chosen. If we call Cf¯(w) =
w2 , (1 + w)f¯( 1−w 1+w )
(4.28)
we can then write an equation for the inverse problem, i.e. for the particular relation w ˜ = w(w) ˜ such that the metric g w(w) ˜ transforms into the quantum metric associated to the operator ¯ monotone function f (t), namely: s 2 1 − w(w) ˜ 1 1 + w(w) ˜ dw(w) ˜ = 2w(w) ˜ ln . (4.29) dw Cf¯(w) 1 − w(w) ˜ 1 − w2 The same procedure can be carried out for the quantum metric obtained for instance from the Tsallis entropy and we obtain the following equations s 2 dw(w) ˜ 1 1 − w(w) ˜ 2 = (˜ aq − ˜bq )(˜ a1−q − ˜b1−q ) , (4.30) dw q(1 − q) Cf¯(w) 1 − w2 where
a ˜α =
1 + w(w) ˜ 2
α ,
˜bα =
1 − w(w) ˜ 2
α α = q, 1 − q.
,
(4.31)
They are non-linear, first-order, ordinary differential equations and therefore, under the hypothesis of Lipschitzian continuity, they admit an unique solution. It means that there is always a particular tomography that gives rise to a quantum metric related to a particular operator monotone function. Remark: The result of the example of the previous section satisfies these equations as it can be easily checked out by fixing as f¯( 1−w 1+w ) Eq. (4.18) and by replacing Eq. (4.6) into the inverse problem equation (4.29). It yields an identity. For the sake of completeness, we report here the explicit expressions of the coefficient Cf¯(w) relative to some of the operator monotone functions listed in [11]: f¯(t) = f¯( 1−w 1+w )
Cf¯(w)
2t 1+t
w2 1−w2
t−1 ln(t)
− w2 ln
2(t−1) (1+t) ln2 (t)
w − 2(1+w) ln2
√ 2(t−1) t (1+t) ln(t)
w − 2√1−w ln 2)
15
1−w 1+w
1−w 1+w
1−w 1+w
5
Conclusions
In this work we have proved that, in the case of a qubit, there exists a one-to-one correspondence between the choice of tomographic schemes and the operator monotone functions entering the relative coefficient of the quantum metric. Indeed, we established such a correspondence by exhibiting a non-linear, first order, ordinary differential equation which, given an operator monotone function, always admits a unique solution for the functional relation w(w) ˜ between the parameters associated with the new and the linear tomography, respectively. Vice-versa, once a particular tomography has been fixed, i.e., given the function w ˜ = w(w), ˜ there exists a unique expression for the coefficient of the “tangent” part of the quantum metric tensor and hence, according to the Petz classification theorem, a unique operator monotone function associated with it. In order to visualize such a correspondence, we focused on the qubit case and selected the tomography associated with a thermal-like state. Up to a conformal factor, the metric tensor generated from the von Neumann relative entropy was actually recast in a form in agreement with the Petz classification theorem and a new operator monotone function was found. Moreover, as a further confirmation, such an operator monotone function satisfied the above non-linear, first order, ordinary differential equation for that particular choice of tomography, i.e., for the specific relation between the parameters characterizing the density matrices which identify the two different tomograms (the linear and the non-linear one). This essentially provides us with an explicit example in which a change of tomography reflects into a different operator monotone function and, in turn, into a different metric tensor on the space of quantum states. Let us notice that, for a generic invertible qudit state, the spectrum will be parametrized by d − 1 parameters. Then, since the diffeomorphism induced by a change of tomographic scheme is always of the form (3.7), ordinary differential equations such as (4.29), (4.30) will be replaced by partial differential equations. As concerns the conformal factor, we argued that it naturally arises in the tomographic setting since a change of the tomographic scheme uniquely identifies a diffeomorphism on the parameter space whose action on the metric tensor defined on tomographic probabilities maps Fisher metrics into conformal Fisher metrics. Let us stress once again that this does not contradict Chenstov’s uniqueness theorem since the diffeomorphism involves only the parameters of the tomographic probabilities and not the sample space. Similarly, on the states space quantum metrics manifest an analogue behavior under the action of such a diffeomorphism as sketched in fig. 2.
Figure 2: The diffeomorphism (3.7) maps an element of the surface Σ of Petz quantum metrics, associated to a generating function S and an operator monotone function f , to a different quantum metric, characterized by another generating function S¯ and another operator monotone function f¯, times a conformal factor A w(w) ˜ which depends only on the change of the tomographic scheme.
16
Indeed, we have seen that the diffeomorphism identified by the change of tomography maps quantum Petz metrics into a different quantum Petz metric multiplied by the same conformal factor. Therefore, since the conformal factor depends only on the specific diffeomorphism and the latter is uniquely determined by the selected tomography, we conclude that the choice of a tomographic scheme selects an operator monotone function and hence a unique element in the family of Petz metrics according to the following diagram: e = AF G(Fisher) e = G(Fisher) _ _ _F ∗_ _ _/ F ∗ G G W f W
√
q e −1 1−G w=± ˜ w ˜w ˜
w=±
g˜ =
A
(Petz) g˜S,f
(5.1)
1−G−1 ww
_ _ _ _ _ _ _/ F ∗ g˜ = AF h(Petz) ∗ ¯ f¯ S, F
Explicit computation of relation (4.6)
We want to obtain the relation (4.6), w ˜ = w(w), ˜ in the case of a tomography defined as WF (u|mw ) = hmw |uF (ρ)u† |mw i where F (ρ) =
(A.1)
e−βρ . T r(e−βρ )
(A.2)
Since both F (ρ) and ρ are states, they admit a Bloch decomposition 1 ρ = (σ0 + wσw ) 2
1 F (ρ) = (σ0 + wσ ˜ w ). 2
,
(A.3)
Then, we have e−βρ = e
−βσ0 2
e
−βwσw 2
=e
− β2
βw − sinh 0 cosh βw 2 2 . βw 0 cosh βw + sinh 2 2
The trace of this matrix is T r(e
−βρ
− β2
) = 2e
cosh
βw 2
(A.4)
.
(A.5)
Therefore, by dividing Eq. (A.4) for its trace, we obtain the matrix expression for F (ρ) βw 1 − tanh 0 1 2 . F (ρ) = (A.6) 2 0 1 + tanh βw 2
By comparing it with its Bloch decomposition 1 F (ρ) = 2
1+w ˜ 0 0 1−w ˜
17
! ,
(A.7)
it follows that 1 − tanh
βw 2
=1+w ˜
=⇒
w(w) ˜ = − tanh
βw 2
,
(A.8)
which is actually Eq. (4.6). Acknowledgements G.M. would like to acknowledge the grant “Santander-UC3M Excellence Chairs 2016”. P.V. acknowledges support by COST (European Cooperation in Science and Technology) in the framework of COST Action MP1405 QSPACE.
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