Petz–Rényi relative entropy of thermal states and their displacements

In this letter, we obtain the precise range of the values of the parameter \(\alpha \) such that Petz–Rényi \(\alpha \)-relative entropy \(D_{\alpha }(\rho ||\sigma )\) of two faithful displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states \(\rho \) and \(\sigma \) with inverse temperature parameters \(r_1, r_2,\ldots , r_n\) and \(s_1,s_2, \ldots , s_n\), respectively, \(0<r_j,s_j<\infty \), for all j, we have

where we adopt the convention that the minimum of an empty set is equal to infinity. This result is particularly useful in the light of operational interpretations of the Petz–Rényi \(\alpha \)-relative entropy in the regime \(\alpha >1 \). Along the way, we also prove a special case of a conjecture of Seshadreesan et al. (J Math Phys 59(7):072204, 2018. https://doi.org/10.1063/1.5007167).

The data that supports the findings of this study are available within the article.

The authors are grateful to the referees whose suggestions greatly improved the article. The second author wishes to acknowledge the Army Research Office MURI award ‘Theory and Engineering of Large-Scale Distributed Entanglement Quantum Network Science—QNS’ awarded under grant number W911NF2110325—ARO MURI. The second author also thanks the Fulbright Scholar Program and the United States-India Educational Foundation for providing funding to conduct part of this research through a Fulbright-Nehru Postdoctoral Fellowship (Award No. 2594/FNPDR/2020).

Authors and Affiliations

1. University of South Carolina, Columbia, SC, USA

   George Androulakis

2. The University of Arizona, Tucson, AZ, USA

   Tiju Cherian John

Corresponding author

Correspondence to Tiju Cherian John.

Conflict of interest

The authors do not have any Conflict of interest.

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Both authors contributed equally in the article.

A Appendix

1.1 Single mode Weyl unitary operators

The classical Laguerre polynomials play an important role in our analysis of Weyl unitary operators in 1-mode. The Laguerre polynomials are defined as

The following proposition is available in [20] with a different proof.

Proposition A.1

[20] Let \(L_j\) denote the j-th Laguerre polynomial, \(j\in \mathbb {Z}_{\ge 0}\). Then the 1-mode Weyl unitary operator at \(u\in \mathbb {C}\), satisfy

Proof

For clarity of writing in the proof, we denote the particle basis vector of \(\Gamma (\mathbb {C})\) as \(f_j\). Notice that \(f_i\otimes f_j = f_{i+j}\) for \(i,j\ge 0\). Note also that in the one-mode case \(u=uf_1\) for all \(u\in \mathbb {C}\). So, we have

Now, by the definition of exponential vectors and the action of Weyl unitary operators on the exponential vectors (2.15),

Identifying the coefficient of \(t^j\) on both sides of the equation above and writing \(m+k=j\), we have

Hence, from (A.2), we get

Thus from (A.1), we get

\(\square \)

Lemma A.2

For every \(u\in \mathbb {C}\), there is an infinite sequence of positive integers j such that

Proof

Observe that

Notice that

Since both \(\frac{m \pi ^2}{4|u|^2}+\frac{\pi ^2}{16| u|^2} \rightarrow \infty \) and \(\left( \frac{m \pi }{2|u|}\right) ^2 \rightarrow \infty \) as \(m\rightarrow \infty \), we can find an infinite sequence of positive integers m such that the intervals

are pairwise disjoint and have lengths bigger than one. Therefore, for every such interval we can choose a positive integer j in \(I_m\). Thus we have an infinite sequence of j’s such that

\(\square \)

Proposition A.3

For every \(u\in \mathbb {C}\), there exists \(C>0\) such that

for infinitely many \(j\in \mathbb {Z}_{\ge 0}\).

Proof

By Proposition A.1 it is enough to show that for every \(u\in \mathbb {C}\), there exists \(C>0\) such that

for infinitely many \(j\in \mathbb {Z}_{\ge 0}\). By Fejér’s formula [21, Theorem 8.22.1] (alternatively [22] taking \(\alpha =0\)), we have for large j’s

Taking , we get

Using Lemma A.2 we see that there exists positive constants \(C_1\) and \(C_2\) (depending only on u) such that for infinitely many j’s

Thus for every \(\epsilon >0\), there exists \(C>0\) such that

for infinitely many j’s. We conclude the proof by choosing \(\epsilon = 1/8.\) \(\square \)

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