The f?f˜ Correspondence and Its Applications in Quantum Information Geometry. (2024)

This explains why it is so interesting to study the structure of F[sub.op]: any understanding in this field necessarily provides us with more insight into operator means and Quantum Fisher Information(s).

This is exactly what the f-f˜ correspondence will produce.

4. Thef?f˜Bijection for Operator Monotone Functions

As in Section 2, we divide the representing functions for operator means into two parts.

Definition 7.

For f?Fop, we define f(0)=limx?0f(x). We say that a function f?Fop is regular if f(0)?0 and non-regular if f(0)=0, cf. [8,9].

We introduce the sets of regular and non-regular functions, F[sub.op][sup.r]:={f?F[sub.op]|f(0)?0}, F[sub.op][sup.n]:={f?F[sub.op]|f(0)=0}, and notice that, trivially, F[sub.op] is the disjoint union of F[sub.op][sup.r] and F[sub.op][sup.n].

Definition 8.

For f?Fopr, we set f˜(x)=1/2[(x+1)-( x - 1 )[sup.2]f ( 0 )/f ( x )] x>0.

Set G(f)=f˜, cf. [5].

Theorem 3.

The correspondence f?f˜ is a bijection between Fopr and Fopn.

5. The Inversion Formula and Wigner–Yanase–Dyson Information

Definition 9.

For g?Fopn, we set (5)g?(x)={g[sup.?](1)·( x - 1 ) 2/2 g ( x ) - ( x + 1 ) , x?(0,1)?(1,8),1,x=1.

Define H(g)=g?.

Proposition 3.

If g is non-regular then g? is regular; namely, g??Fopr. Moreover, if f?Fopr and g?Fopn, then H(G(f))=f and G(H(g))=g.

The correspondence between the WYD information (see [10]), I[sub.?][sup.ß](A)=-1/2Tr([?[sup.ß],A][?[sup.1-ß],A]), 0<ß<1/2, and the Quantum Fisher Information depends on the operator monotonicity of the functions f[sub.ß](x)=ß(1-ß)(x-1)2/(xß-1)(x1-ß-1) 0<ß<1/2.

See [8,10,11] for the existing proofs. Indeed, Proposition 3 provides a new approach to the above result.

The function g[sub.ß](x)=xß+x1-ß/2 0<ß<1/2 is operator monotone and, moreover, g[sub.ß]?F[sub.op] and g[sub.ß] is non-regular. The calculations show that f˜[sub.ß]=g[sub.ß]. Therefore, the function f[sub.ß]?F[sub.op][sup.r] for 0<ß<1/2.

Here we provide the first examples of the correspondence (Table 2).

6. Regular QFI in Terms of Covariance

Quantum covariance is usually defined as (6)Cov[sub.?](A,B):=1/2Tr(?(AB+BA))-Tr(?A)·Tr(?B), where A=A[sup.*] and B=B[sup.*]. The above formula can be written using the arithmetic mean of the left and right multiplication operator as (7)Cov[sub.?](A,B):=Tr((L?+R?/2)(A[sub.0])B[sub.0]), where A[sub.0]=A-Tr(?A)·I. This simple remark led Petz to the following definition (see [12]):

Definition 10.

For any f?Fop, define the Quantum f-Covariance as (8)Cov[sub.?][sup.f](A,B):=Tr(m[sub.f](L[sub.?],R[sub.?])(A[sub.0])B[sub.0]).

As usual, Var[sub.?][sup.f](A):=Cov[sub.?][sup.f](A,A). If f(x)=(1+x)/2, then (9)Cov[sub.?][sup.f](A,B)=1/2Tr(?(AB+BA))-Tr(?A)·Tr(?B)=Cov[sub.?](A,B), which is the above given standard definition for the quantum covariance.

With this generalized notion of Petz covariance, we show that there is an unexpected relation between QFI and the covariance itself.

We stated previously that there exists a natural identification of T[sub.?]D[sub.n][sup.1] with the space of self-adjoint traceless matrices; namely, for any ??D[sub.n][sup.1] T[sub.?]D[sub.n][sup.1]={A?M[sub.n]|A=A[sup.*] , Tr A=0}.

Moreover, the PKA theorem states that the Quantum Fisher Information(s) are given by the formula

Monotone metrics are usually normalized in such a way that [A,?]=0 implies g[sub.?](A,A)=Tr(?[sup.-1]A[sup.2]).

Remark 2.

Let us remember that T?:={A=A*|Tr(?A)=0}; the tangent space in ? to the state space has a natural orthogonal decomposition in terms of “commuting” and “noncommuting” parts as (10)T[sub.?]=T[sub.?][sup.c]?T[sub.?][sup.n],

where (11)T[sub.?][sup.c]={A=A[sup.*]|[?,A]=0}, T[sub.?][sup.n]={i[?,A]|A=A[sup.*]}.

Due to the Chentsov uniqueness theorem, the different QFI(s) are characterized from what they do on the noncommuting part of the tangent space; namely, on T?n that is on tangent vectors of the form i[?,A].

We are now ready to state the QFI(s) in terms of covariances.

Theorem 4.

Gibilisco, Imparato, and Isola (Proposition 6.3, page 11 in [13]).

If f?Fopr, then (12)f ( 0 )/2·< i [ ?, A ], i [ ?, B ] >[sub.?, f]=Cov[sub.?](A,B)-Cov[sub.?][sup.f ˜](A,B).

The above formula has many important consequences.

7. A Look at the Petz–Sudar Theorem

In the PKA classification theorem (Theorem 2 in Section 3), we see that the QFI is defined only for faithful states (?>0). It is Petz himself, in collaboration with Sudár, who understood how to define a radial extension of a QFI to pure states and how to prove that only regular QFIs possess such an extension (for all details, the reader can refer to [9] or to [13]). The statement is as follows:

Theorem 5

(Petz and Sudár in [9]).A QFI admits a radial extension iff it is regular (f(0)>0). In such a case (13)2f(0)<·,·>[sub.·,f]?<·,·>[sub.·,FS]where <·,·>·,FS is the Fubini-study metric on the space of pure states.

The fact that the radial limit of 2f(0)<·,·>[sub.·,f] does not depend on f is an immediate consequence of Theorem 4 in Section 6.

It is natural to ask, can the Petz–Sudár theorem be generalized and proven using Formula (22)? Here, generalization means using Formula (22) for states that are neither faithful nor pure.

8. Extension of Regular QFI and MASI for Non-Faithful States

A far-reaching generalization of the Wigner–Yanase Skew Information has been proposed by Hansen in [8].

Definition 11.

Metric Adjusted Skew Information (MASI).

For f?Fopr and ?>0, set (14)I[sub.?][sup.f](A):=f ( 0 )/2·< i [ ?, A ], i [ ?, A ] >[sub.?, f].

In the case where f(x)=(1+x)[sup.2]/4, we can see that the MASI coincides with the Wigner–Yanase Skew Information:(15)I[sub.?](A):=I[sub.?][sup.f](A)=-1/2Tr([?,A][sup.2]).

Note that recently, using MASI, it has been proven that the Local Quantum Uncertainty (LQU) and the Interferometric Power (IP), which are two important measures of quantum discord, are instances of a family of quantum discords parametrized by the function f?F[sub.op][sup.r]. This allows a unified study of the properties of LQU and IP (see [14]). Due to Theorem 4, we have the following:

Proposition 4.

(16)I[sub.?][sup.f](A)=Var[sub.?](A)-Var[sub.?][sup.f ˜](A).

It is important to note that the two sides of Equation (16) are somehow different in nature. The MASI on the left side is defined only for faithful states (?>0), while the right-hand side always makes good sense since quantum covariance is defined for any state. Therefore, one can look to Equation (16) as a “definition” of the LHS, which solves the problem of extending the MASI with an approach that is different from the one proposed by Hansen in Theorem 3.8 in [8]. Motivated by the above consideration, it is natural to introduce the following sesquilinear form, which is the natural extension of MASI for two observables.

Definition 12.

I[sub.?][sup.f](A,B):=Cov[sub.?](A,B)-Cov[sub.?][sup.f ˜](A,B).

Another important remark is that, using the f-f˜ correspondence, it is possible to establish a relation between MASI and the quasi-entropy S[sub.F](·,·) introduced by Petz in [15]; S[sub.F](·,·) can be seen as a quantum version of Csiszar F-entropy in classical statistics and information theory (see [16]). Indeed, if Tr(?A)=0, Theorem 3.1 in [17] proves that ?2/?t?sS[sub.f˜](?+ti[?,A],?+si[?,A])|[sub.t=s=0]=2I[sub.?][sup.f](A).

9. Inequalities for the MASI and the Bloch Sphere Case

In this section, we discuss some basic properties of MASI and we will see how the f˜ function appears, for example, as a calculation tool. What follows is the generalization of the work in [18] that appears in [19].

(a) If a quantum evolution is given by a Hamiltonian H that commutes with the observable A, then the MASI is a constant of motion. Namely, if we set ?[sub.H](t):=e[sup.-itH]?e[sup.itH] and [A,H]=0, then the function I[sub.?H(t)][sup.f](A) is constant. Since the Quantum Fisher Information contracts under coarse graining, we can see that QFI is a unitary covariant, and this is the crucial ingredient of the proof.

(b) For any MASI, we have:(17)I[sub.?][sup.f](A)=I[sub.?][sup.SLD](A)=1/2f(0)I[sub.?][sup.f](A).

(c) The constant 1/2f(0) is optimal in inequality (17). Namely, if 1=k<1/2f(0), the inequality I[sub.?][sup.SLD](A)=kI[sub.?][sup.f](A) is false and a counterexample can be found in the elementary 2×2 case, namely on the Bloch sphere.

Let us see how this can be proven by means of the f˜ function.

Let {f[sub.i]} be a complete orthonormal base composed of eigenvectors of ?, and {?[sub.i]} the corresponding eigenvalues. Set a[sub.ij]=

Recall that the Pauli matrices are as follows:s[sub.1]=(0110), s[sub.2]=(0-ii0), s[sub.3]=(100-1).

A generic 2×2 density matrix in the Stokes parameterization is written as ?=1/2(1+xy+izy-iz1-x)=1/2(I+xs[sub.1]+ys[sub.2]+zs[sub.3]), where (x,y,z)?R[sup.3] and x[sup.2]+y[sup.2]+z[sup.2]=1. Let r:=[square root of x[sup.2]+y[sup.2]+z[sup.2]]?[0,1]. The eigenvalues of ? are ?[sub.1]=1-r/2 and ?[sub.2]=1+r/2.

Proposition 5.

I[sub.?][sup.f](A)=[1-m[sub.f ˜](1-r,1+r)]·| a 12 |[sup.2].

Corollary 1.

If r?0 then I[sub.?][sup.S L D](A)=[r 2/1 - m f ˜ ( 1 - r, 1 + r )]·I[sub.?][sup.f](A).

Proposition 6.

If f is regular, then limr?0r 2/1 - m f ˜ ( 1 - r, 1 + r )=-1/2 f ˜ ? ( 1 )=1/2 f ( 0 ).

From this last result, the optimality of the constant follows.

10. The Dynamical Uncertainty Principle

From Equation (16), one has Var[sub.?](A)=I[sub.?][sup.f](A).

This is the case n=1 of the Dynamical Uncertainty Principle, which reads (18)det{Cov[sub.?](A[sub.j],A[sub.k])}=det{I[sub.?][sup.f](A[sub.j],A[sub.k])},

or equivalency (19)det{Cov[sub.?](A[sub.j],A[sub.k])}=det{f(0)·1/2·[sub.?,f]}, where f?F[sub.op][sup.r]. On the left-hand side, we have the Generalized Variance of the random vector (A[sub.1],...,A[sub.n]). Please note that, in this case, the right-hand side depends on the state-observables’ non-commutativity, and this is strictly related to a non-trivial dynamic induced by the observables according to the Schrödinger equation.

To understand the terminology, recall that the Standard Uncertainty Principle (SUP) in the Robertson version reads (20)det{Cov[sub.?](A[sub.j],A[sub.k])}=det{-i·1/2·Tr(?[A[sub.j],A[sub.k]])}, where A[sub.1],...,A[sub.n] is an arbitrary number of observables (self-adjoint matrices) and ? is a state. For n=2, one obtains the Schrödinger uncertainty principle from which the Heisenberg uncertainty principle follows. The bound in the right-hand side depends on the non-commutativity among the observables (see [20,21]).

Now, let n=2m+1 be odd; in this case the right-hand side is the determinant of an antisymmetric matrix and therefore is zero; for an odd number of observables the SUP does not say anything “quantum”.

Therefore, using the QFI and the f?f˜ correspondence, a new uncertainty principle has been proven, which is also not trivial for an odd number of observables. Moreover, SUP and DUP have been generalized for an arbitrary g-covariance; see [22,23,24].

If we set V(f):=det{I[sub.?][sup.f](A[sub.j],A[sub.k])}, one can see that (Theorem 4.4 in [22]) f˜=g˜ ? V(f)=V(g).

This implies, for example, that we have the biggest bound in the DUP for f(x)=(1+x)/2. Indeed, in this case, f˜(x)=2x/(1+x)=g˜ for any regular g, and this provides the conclusion.

11. Semplification of Kosaki’s Work

To see how the f-f˜ correspondence sheds light on certain subjects, consider the paper by Kosaki [25]. In this paper, the author’s aim is to study how the RHS of the DUP (for n=2) depends on the function f. The main result by Kosaki is as follows. Remember that f[sub.ß](x)=ß(1-ß)(x-1)2/(xß-1)(x1-ß-1) 0<ß<1/2, f˜[sub.ß](x)=1/2(x[sup.ß]+x[sup.1-ß])

Let ?,A[sub.1],A[sub.2] be fixed and set F(f):=det{Cov[sub.?](A[sub.i],A[sub.j])}-det{I[sub.?][sup.f](A[sub.i],A[sub.j])}. F(ß):=F(f[sub.ß]).

The main result in [25] is Theorem 5, which reads as follows: F(ß) is decreasing in (0,1/2), F(1/2)=0 so that F(ß)=0. The result was the final output of a rather complicated tour de force of calculations.

Look how simple the approach is using the f?f˜ correspondence. First of all, it is straightforward that f˜=g˜?F(f˜)=F(g˜).

For x fixed, the function ß?f˜[sub.ß](x)=1/2(x[sup.ß]+x[sup.1-ß])

is decreasing in (0,1/2) so that ß[sub.1]=ß[sub.2]?f˜[sub.ß1]=f˜[sub.ß2]?F(ß[sub.1])=F(ß[sub.2]), and the Kosaki’s conclusion follows. One should read the complicated proof in [25] to fully appreciate the efficiency and clarity of the f-f˜ correspondence.

12. Refinements of Heisenberg Uncertainty Relations

In the literature, several quantities appear with the same aim: to measure quantum uncertainty. We will discuss some examples in this paper. For example, to quantify such uncertainty Luo introduced the following state-observable quantity, U[sub.?](A):=[square root of V[sub.?](A)[sup.2]-(V?(A)-I?(A))[sup.2]], where V[sub.?](A):=Var[sub.?](A). Furthermore, he was able to prove the following inequality:U[sub.?](A)·U[sub.?](B)=1/4|Tr?[A,B]|[sup.2].

Clearly, this can be seen as a refinement of the Heisenberg uncertainty principle because Var[sub.?](A)=U[sub.?](A).

After some failed attempt to generalize this result (see Kosaki [25], Remarks 3.2 and 3.3), Yanagi (see [26]) was able to prove a generalization that makes sense for the WYD information. He introduced the following quantity, U[sub.?][sup.ß](A):=[square root of V[sub.?](A)[sup.2]-(V?(A)-I?ß(A))[sup.2]], and was able to prove this inequality:U[sub.?][sup.ß](A)·U[sub.?][sup.ß](B)=ß(1-ß)|Tr?[A,B]|[sup.2].

Note that ß(1-ß)=f[sub.ß](0) where f[sub.ß](x)=ß(1-ß)(x-1)2/(xß-1)(x1-ß-1) 0<ß<1/2, which is the function associated with the WYD information. It is straightforward to propose an f-depending quantity, U[sub.?][sup.f](A):=[square root of V[sub.?](A)[sup.2]-(V?(A)-I?f(A))[sup.2]], as a measure of quantum uncertainty and try to prove the following inequality:U[sub.?][sup.f](A)·U[sub.?][sup.f](B)=f(0)|Tr?[A,B]|[sup.2] f?F[sub.op][sup.r].

Unfortunately, this inequality, in general, is false. Yanagi proved that the theorem holds true under a condition involving f˜; namely, we have the following:

Proposition 7.

For f?Fopr if x + 1/2+f˜(x)=2f(x) x>0 then it holds U[sub.?][sup.f](A)·U[sub.?][sup.f](A)=f(0)·Tr(?[A,B]) f?F[sub.op][sup.r]

On the other hand, one can prove the following:

Proposition 8.

For any f?Fopr and x>0 f˜( x )[sup.2]=1/4( x + 1 )[sup.2]-f( 0 )[sup.2]( x - 1 )[sup.2].

This has the following as a consequence:

Corollary 2.

f( 0 )[sup.2]( x - y )[sup.2]=1/4( x + y )[sup.2]-m[sub.f ˜]( x, y )[sup.2]

From this, an unconditional inequality follows: if we switch from the constant f(0) to the constant f(0)[sup.2] (see [27]), we obtain the following:

Proposition 9.

For f?Fopr and A,B?Mn,sa, it holds that U[sub.?][sup.f](A)·U[sub.?][sup.f](A)=f( 0 )[sup.2]·Tr(?[A,B]).

13. State Quantum Uncertainty Based on MASI

Luo proposed a notion of quantum uncertainty depending only on the state ?. In the paper [28], starting from the Wigner–Yanase information and from an orthonormal basis {H[sub.j]}, he introduced the quantity Q[sup.WY](?):=?jI[sub.?][sup.WY](H[sub.j]) as a measure of such uncertainty. First, Luo proved that Q[sup.WY](?) is basis independent, and after that Q[sup.WY](?)=?j<k?j-?k[sup.2], where {?[sub.j]} is the spectrum of ?. Applications of the function Q[sup.WY](?) also appear in paper [29].

If we remember that the WY information is the QFI associated with the functions f[sub.WY](x):=1+x2[sup.2], f˜[sub.WY]=[square root of x], we obtain Q[sup.WY](?)=2?j<k[?j+?k/2-[square root of ?[sub.j]?[sub.k]]]=2?j<k[m[sub.a](?[sub.j],?[sub.k])-m[sub.f˜WY](?[sub.j],?[sub.k])].

The above considerations lead naturally to the following questions:

(i) For a regular f?F[sub.op][sup.r], does the definition Q[sup.f](?):=?jI[sub.?][sup.f](H[sub.j])

produce a basis-independent function of the state ??

(ii) Imagine that we obtain a positive answer for (i). We may also ask if Q[sup.f](?)=2?j<k[m[sub.a](?[sub.j],?[sub.k])-m[sub.f˜](?[sub.j],?[sub.k])].

These questions both received a positive answer from Cai in their paper [30]. Once again, the f˜ function shows up when one has to look at a general scheme for Quantum Fisher Information.

14. The Average Coherence of a Quantum State

In a recent paper [31], Fan, Li, and Luo attempted to study quantum coherence (an important feature of a quantum system) by eliminating the influence of a reference basis. They introduced the average quantum coherence using three procedures: (1) average over all orthonormal basis; (2) average over all elements of operator orthonormal basis; (3) average over a complete family of MUBs (Mutually Unbiased Bases). The result of the paper was that these three different procedures produce the same quantity. The basic ingredient of the proof is the f-f˜ correspondence.

Indeed, if E is a quantum channel and {E[sub.j]} are the Kraus operators of E, the authors define a channel-depending coherence as C[sub.f](?,E):=?jI[sub.?][sup.f](E[sub.j]).

In the first case, they consider the channel as induced by a von Neumann measurement or equivalently by an orthonormal basis. Averaging on this reference basis is equivalent to integrating over the unitary orbit of a fixed basis, which amounts to using the normalized Haar measure over the unitary group, U, of the system Hilbert space. They set C[sub.f][sup.U](?)=?[sub.U]C[sub.f](?|U?U[sup.†])dU, where U?U[sup.†]={U|i>
In the third case, they define the C[sub.f][sup.mub](?) coherence averaging on MUBs, which surely exist if d is a power of a prime number.

Finally, they proved the following result:

Theorem 6.

For any state ? of any prime power dimensional system and for any regular operator monotone function f, one has that C[sub.f][sup.U](?)=C[sub.f][sup.ob](?)=C[sub.f][sup.mub](?)=d - Tr [ m f ˜ ( L ?, R ? ) ]/d + 1.

Note that if ?[sub.i] are the eigenvalues of the state ?, then Tr[m[sub.f˜](L[sub.?],R[sub.?])]=?[sub.ij]m[sub.f˜](?[sub.i],?[sub.j]).

15. Conclusions

The notion of means has its roots deeply situated in the history of Western mathematics; the Greeks themselves knew eleven different types of means. Still, the subject is currently undergoing strong developments. As an example, starting from the work of Rao [32,33] and Prakasa–Rao [34,35], the Jensen inequality for numerical and operator means has been proven, and more generalizations seem to be on their way [4,36,37].

The f?f˜ correspondence is indeed a correspondence between means. From the work by Petz, we know that two of the basic objects of Quantum Probability and Quantum Statistics, namely Quantum Covariance and Quantum Fisher Information, are indeed necessarily built on the notion of operator mean, and this explains why we find the manifold of different applications of the f?f˜ correspondence described in this paper.

It is rational to expect that this is not the end of the story and that many other applications will appear in the coming years.

At the moment, the most promising area is that of the extension of the Petz–Kubo–Ando theorem for states that are neither faithful nor pure. We expect this could be performed using the formula (21)f(0)/2·[sub.?,f]=Cov[sub.?](A,B)-Cov[sub.?][sup.f˜](A,B) for regular f. The right-hand side makes sense for any state and, on the other hand, by a continuity-approximation argument the formula is, somehow, forced to be unique; indeed, we can approximate any state by faithful states. A fully satisfying theorem would certainly deduce the scalar product (22)Cov[sub.?](A,B)-Cov[sub.?][sup.f˜](A,B) from first principles, as in the Petz proof of the PKA theorem. At the moment, a similar theorem has not yet been proven.

Conflicts of Interest

The authors declare no conflicts of interest.

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Tables

Table 1: Means and representing function.

Table 2: f?f˜ Correspondence.

where ß?(0,1/2).

Author Affiliation(s):

Department of Economics and Finance, Tor Vergata University of Rome, Via Columbia 2, 00133 Rome, Italy; [emailprotected]

DOI: 10.3390/e26040286