Relating different quantum generalizations of the conditional Rényi entropy
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. Our result generalizes the well-known duality relation of the conditional von Neumann entropy for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies and derive a new entropic uncertainty relation.
Recently, there has been renewed interest in finding suitable quantum generalizations of Rényi’s  entropies and divergences. This is due to the fact that Rényi entropies and divergences have a wide range of applications in classical information theory and cryptography, see, e.g. .
We will review some of the recent progress here, but refer the reader to  for a more in-depth discussion. For our purposes, a quantum system is modeled by a finite dimensional Hilbert space. We denote by the set of positive semi-definite operators on that Hilbert space, and by the subset of density operators with unit trace.
The following natural quantum generalization of the Rényi divergence has been widely used and has found operational significance, for example, as a cut-off rate in quantum hypothesis testing  (see also ). It is usually referred to as quantum Rényi relative entropy and for all given as
for arbitrary , that satisfy . (The notation means that dominates , i.e. the kernel of lies inside the kernel of .)
While this definition has proven useful in many applications, it has a major drawback in that it does not satisfy the data-processing inequality (DPI) for . The DPI states that the quantum Rényi relative entropy is contractive under application of a quantum channel, i.e., for any completely positive trace-preserving map . Intuitively, this property is very desirable since we want to think of the divergence as a measure of how well can be distinguished from , and this can only get more difficult after a channel is applied.
for all and , that satisfy . The quantum Rényi divergence has found operational significance in the converse part of quantum hypothesis testing . As such, it satisfies the DPI for all as was shown by Frank and Lieb  and independently by Beigi  for . See also earlier work  where a different proof is given for . Furthermore, the quantum Rényi divergence has already proven an indispensable tool, for example in the study of strong converse capacities of quantum channels .
The definitions, and , are in general different but coincide when and commute. For , we define and as the corresponding limit. For it has been shown that :
with the eigenvalue decomposition , , and the projector on the support of . In the limit both expressions converge to the quantum relative entropy , namely
with the eigenvalue decompositions and .
It has been observed  that the relation
Finally, very recently Audenaert and Datta  defined a more general two parameter family of -z-relative Rényi entropies of the form
and explored some of its properties. We clearly have and .
2Quantum Conditional Rényi Entropies
We will in the following consider disjoint quantum systems, denoted by capital letters and . The sets and take on the expected meaning.
The conditional von Neumann entropy can be conveniently defined in terms of the quantum relative entropy as follows. For a bipartite state , we define
where is the usual von Neumann entropy. The last equality can be verified using the relation together with the fact that is positive definite.
In the case of Rényi entropies, it is not immediate which expression, , or , should be used to define the conditional Rényi entropies. It has been found in the study of the classical special case (see, e.g.  for an overview) that generalizations based on have severe limitations, for example they cannot be expected to satisfy a DPI. On the other hand, definitions based on the underlying divergence, as in or , have proven to be very fruitful and lead to quantities with operational significance. Together with the two proposed quantum generalizations of the Rényi divergence in and , this leads to a total of four different candidates for conditional Rényi entropies. For and , we define
The fully quantum entropy has first been studied in . For the classical and classical-quantum special case this quantity gives a generalization of the leftover hashing lemma  for the modified mutual information to Rényi entropies with .
The classical version of was introduced by Arimoto for an evaluation of the guessing probability . We note that he used another but equivalent expression for that we later explain in Lemma ?. Then, Gallager used (again in the form of Lemma ?) to upper bound the decoding error probability of a random coding scheme for data compression with side-information . The classical and classical-quantum special cases of were, for example, also investigated in  and realize another type of a generalization of the leftover hashing lemma for the -distinguishability in the study of randomness extraction to Rényi entropies with .
It follows immediately from the definition and the corresponding property of that these two entropies satisfy a data-processing inequality. Namely for any quantum operation with and any , we have
while their classical-quantum versions have been obtained in .
The conditional entropy was proposed in  and investigated in , whereas is first considered in this paper. (Since the relative entropies and are identical for commuting operators, we note that as well as for classical distributions.) Both definitions satisfy the above data-processing inequality for .
Furthermore, it is easy to verify that all entropies considered are invariant under applications of local isometries on either the or systems. Lastly, note that the optimization over can always be restricted to for .
We use up and down arrows to express the trivial observation that and by definition. Finally, gives us the additional relations and . These relations are summarized in Figure 1. Moreover, inheriting these properties from the corresponding divergences, all entropies are monotonically decreasing functions of
For , all definitions coincide with the usual von Neumann conditional entropy . For , two quantum generalizations of the conditional min-entropy emerge, which both have been studied by Renner . Namely,
(The notation and is widely used. However, we prefer our notation as it makes our exposition in this manuscript clearer.) For , we find a quantum generalization of the conditional collision entropy as introduced by Renner :
For , we find the quantum conditional max-entropy first studied by König et al. ,
where denotes the projector onto the support of .
It is well known that, for any tripartite pure state , the relation
holds. We call this a duality relation for the conditional entropy. To see this, simply write and and note that the spectra of and as well as the spectra of and agree. The significance of this relation is manifold — for example it turns out to be useful in cryptography where the entropy of an adversarial party, let us say , can be estimated using local state tomography by two honest parties, and . In the following, we are interested to see if such relations hold more generally for conditional Rényi entropies.
It was shown in  that indeed satisfies a duality relation, namely
Note that the map maps the interval , where data-processing holds, onto itself. This is not surprising. Indeed, consider the Stinespring dilation of a quantum channel . Then, for pure, is also pure and the above duality relation implies that
Hence, data-processing for holds if and only if data-processing for holds.
As expected, the map maps the interval , where data-processing holds, onto itself.
The purpose of the following is thus to show if a similar relation holds for the remaining two candidates, and . First, we find the following alternative expression for by determining the optimal in the definition .
This generalizes a result by one of the current authors .
Recall the definition
This can immediately be lower bounded by the expression in by substituting
for . It remains to show that this choice is optimal. We employ the following Hölder and reverse Hölder inequalities (cf. Lemma ? in Appendix A). For any , the Hölder inequality states that
Furthermore, if , we also have a reverse Hölder inequality which states that
For , we employ for , , and to find
which yields the desired upper bound since . For , we instead use . This leads us to upon the same substitutions, concluding the proof.
An alternative proof also follows rather directly from a quantum generalization of Sibson’s identity, which was introduced by Sharma and Warsi .
This allows us to show our main result.
Substituting and employing Lemma ?, it remains to show that
is equal to
In the following we show something stronger, namely that the operators
are unitarily equivalent. This is true since both of these operators are marginals — on and — of the same tripartite rank- operator,
To see that this is indeed true, note the first operator in can be rewritten as
The last equality can be verified using the Schmidt decomposition of with regards to the partition :. This concludes the proof.
The relation can readily be extended for all and . The limiting case is simply the duality of the conditional von Neumann entropy , whereas the case was also shown in . (See  for a concise proof.) Again, note that the transformation maps the interval where data-processing holds for to where data-processing holds for .
4Some Inequalities Relating Conditional Entropies
Our first application yields relations between different conditional Rényi entropies for arbitrary mixed states. Recently, Mosonyi  used a converse of the Araki-Lieb-Thirring trace inequality due to Audenaert  to find a converse to the ordering relation , namely
Here we follow a different approach and show that inequalities of a similar type for the conditional entropies are a direct corollary of the duality relations in Theorem ?.
Note that the first inequality on each line follows directly from the relations depicted in Figure 1. Next, consider an arbitrary purification of . The relations of Figure 1, for any , applied to the marginal are given as
We then substitute the corresponding dual entropies according to Theorem ?, which yields the desired inequalities upon appropriate new parametrization.
We note that the fully classical (commutative) case of all these inequalities is trivial except for the second inequalities in and , which were proven before by one of authors . Other special cases of these inequalities are also well known and have operational significance. For example, for states that , which relates the conditional min-entropy in to the conditional collision entropy in . To understand this inequality more operationally we rewrite the conditional min-entropy as its dual semi-definite program ,
where is a copy of , the infimum is over all quantum channels , denotes the dimension of , and is the maximally entangled state on . Now, the above inequality becomes apparent since the conditional collision entropy can be written as ,
where denotes the pretty good recovery map of Barnum and Knill . Also, for yields , which relates the quantum conditional max-entropy in to the quantum conditional generalization of the Hartley entropy in .
We believe that the sandwich relations – for close to will prove useful in applications in quantum information theory as they allow to switch between different definitions of the conditional Rényi entropy.
5Entropic Uncertainty Relations
A series of papers  culminating in  established a general technique to derive uncertainty relations for quantum conditional entropies based on two main ingredients: (1) a duality relation, and (2) a data-processing inequality for the underlying divergence. It is evident that all our definitions of conditional Rényi entropies fit the framework of , which then immediately yields the following entropic uncertainty relations:
We want to point out that the first and second inequality were first shown in  and , respectively; the third inequality is novel. To verify it, we apply  to and note that has the required form. Furthermore, it is already pointed out in  that the underlying divergence, for , satisfies the required properties for the application of their theorem. As such, comparing to the corresponding duality relation , we see that in order to derive the uncertainty relation we need to restrict to to be in the regime where data-processing holds.
It is noteworthy that even for the case of classical side information (if the systems and are classical), the three relations are genuinely different. The first inequality bounds the sum of two -entropies, the second the sum of two -entropies, and the third inequality the sum of a - and an -entropy. Let us further specialize these inequalities for the case where both and are trivial. It was already noted in  that specializes to the well-known Maassen-Uffink relation . We have
evaluated for the marginals of the states in . It is also easy to verify that and specialize to strictly weaker uncertainty relations when and are trivial.
Acknowledgments. MT is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant “Random numbers from quantum processes” (MOE2012-T3-1-009). MB thanks the Center for Quantum Technologies, Singapore, for hosting him while this work was done. MH is partially supported by a MEXT Grant-in-Aid for Scientific Research (A) No. 23246071 and the National Institute of Information and Communication Technology (NICT), Japan. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.
We prove the following Hölder and reverse Hölder inequalities for traces of operators.
The first statement also immediately follows from a Hölder inequality for unitarily invariant norms (the trace norm in this case), e.g. in . However, we believe that the following reduction of the proof to the commutative case is noteworthy.
For commuting and , the above result immediately follows from the corresponding classical Hölder and reverse Hölder inequalities. Now, let be a pinching in the eigenbasis of . Since commutes with , we have
under the respective constraints. Now, note that for , we have by the pinching inequality for the Schatten -norm  and follows. On the other hand, for , we use , which implies that . This yields and concludes the proof.
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