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 renyi61 () 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. csiszar95 ().
We will review some of the recent progress here, but refer the reader to lennert13 () 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 mosonyi11 () (see also ogawa04 (); nagaoka06 ()). 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.
Recently, an alternative quantum generalization has been investigated martinthesis (); lennert13 (); wilde13 () (see also mytutorial12 ()). It is referred to as quantum Rényi divergence (or sandwiched Rényi relative entropy in wilde13 ()) and defined as
for all and , that satisfy . The quantum Rényi divergence has found operational significance in the converse part of quantum hypothesis testing mosonyiogawa13 (). As such, it satisfies the DPI for all as was shown by Frank and Lieb frank13 () and independently by Beigi beigi13 () for . See also earlier work martinthesis (); lennert13 () 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 wilde13 (); Gupta13 ().
with the eigenvalue decompositions and .
follows from the Araki-Lieb-Thirring trace inequality ariki90 (); liebthirring05 (). Furthermore, and are monotonically increasing functions. For the latter quantity, this was shown in lennert13 () and independently in beigi13 ().
Finally, very recently Audenaert and Datta audenaert13 () 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 .
Ii Quantum 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, (10), (11) or (12), should be used to define the conditional Rényi entropies. It has been found in the study of the classical special case (see, e.g. Iwamoto13 () for an overview) that generalizations based on (10) 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 (11) or (12), 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 (1) and (2), 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 tomamichel08 (). For the classical and classical-quantum special case this quantity gives a generalization of the leftover hashing lemma bennett95 () for the modified mutual information to Rényi entropies with Hayashi11_2 (); hayashi12 ().
The classical version of was introduced by Arimoto for an evaluation of the guessing probability Arimoto75 (). We note that he used another but equivalent expression for that we later explain in Lemma 1. Then, Gallager used (again in the form of Lemma 1) to upper bound the decoding error probability of a random coding scheme for data compression with side-information gallager79 (); Yagi12 (). The classical and classical-quantum special cases of were, for example, also investigated in hayashi12 (); Hayashi13_2 () 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 hayashi12 ().
The conditional entropy was proposed in mytutorial12 () and investigated in lennert13 (), 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, (8) 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 (11). For , two quantum generalizations of the conditional min-entropy emerge, which both have been studied by Renner renner05 (). 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 renner05 ():
For , we find the quantum conditional max-entropy first studied by König et al. koenig08 (),
where denotes the fidelity. (The alternative notation is often used.) For , we find a quantum conditional generalization of the Hartley entropy hartley28 () that was initially considered by Renner renner05 (),
where denotes the projector onto the support of .
Iii Duality Relations
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 (tomamichel08, , Lem. 6) 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 (14).
Let and . Then,
This generalizes a result by one of the current authors (hayashi12, , Lem. 7).
Recall the definition
This can immediately be lower bounded by the expression in (27) by substituting
Furthermore, if , we also have a reverse Hölder inequality which states that
For , we employ (31) for , , and to find
An alternative proof also follows rather directly from a quantum generalization of Sibson’s identity, which was introduced by Sharma and Warsi (sharma13, , Lem. 3 in Suppl. Mat.).
This allows us to show our main result.
Let with and let be pure. Then, .
Substituting and employing Lemma 1, 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 (37) 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 (23), whereas the case was also shown in (berta08, , Prop. 3.11). (See (tomamichel10, , Lem. 25) for a concise proof.) Again, note that the transformation maps the interval where data-processing holds for to where data-processing holds for .
For any pure , the following holds:111We use the convention that and .
Iv Some Inequalities Relating Conditional Entropies
Our first application yields relations between different conditional Rényi entropies for arbitrary mixed states. Recently, Mosonyi (mosonyi13, , Lem. 2.1) used a converse of the Araki-Lieb-Thirring trace inequality due to Audenaert Audenaert08 () 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 3.
Let . Then, the following inequalities hold for :
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 3, 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 (47) and (48), which were proven before by one of authors (Hayashi:2013aa, , Lem. 6). Other special cases of these inequalities are also well known and have operational significance. For example, (48) for states that , which relates the conditional min-entropy in (19) to the conditional collision entropy in (20). To understand this inequality more operationally we rewrite the conditional min-entropy as its dual semi-definite program koenig08 (),
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 berta13 (),
where denotes the pretty good recovery map of Barnum and Knill Barnum02 (). Also, (46) for yields , which relates the quantum conditional max-entropy in (21) to the quantum conditional generalization of the Hartley entropy in (22).
V Entropic Uncertainty Relations
A series of papers berta10 (); tomamichel11 () culminating in colbeck11 () 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 colbeck11 (), which then immediately yields the following entropic uncertainty relations:
Let and let and be two positive operator-valued measures. We define the overlap and consider the post-measurement states
Then, the following relations hold:
We want to point out that the first and second inequality were first shown in colbeck11 () and lennert13 (), respectively; the third inequality is novel. To verify it, we apply (colbeck11, , Thm. 1) to and note that has the required form. Furthermore, it is already pointed out in colbeck11 () that the underlying divergence, for , satisfies the required properties for the application of their theorem. As such, comparing (57) to the corresponding duality relation (44), 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 lennert13 () that (56) specializes to the well-known Maassen-Uffink relation maassen88 (). We have
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.
Appendix A Hölder Inequalities
We prove the following Hölder and reverse Hölder inequalities for traces of operators.
Let and let , such that . Then, the following Hölder and reverse Hölder inequalities hold:
Here, is evaluated on the support of by convention.
The first statement also immediately follows from a Hölder inequality for unitarily invariant norms (the trace norm in this case), e.g. in (bhatia97, , Cor. IV.2.6). 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 (bhatia97, , Eq. (IV.52)) and (59) follows. On the other hand, for , we use (bhatia97, , Thm. V.2.1), which implies that . This yields (60) and concludes the proof. ∎
- (1) H. Araki. On an inequality of Lieb and Thirring. Letters in Mathematical Physics, 19(2):167–170, Feb. 1990. DOI: 10.1007/BF01045887.
- (2) S. Arimoto. Information Measures and Capacity of Order alpha for Discrete Memoryless Channels. Colloquia Mathematica Societatis János Bolya, 16:41–52, 1975.
- (3) K. M. R. Audenaert. On the Araki-Lieb-Thirring inequality. Int. J. of Inf. and Syst. Sci., 4(1):78–83, Jan. 2008. arXiv: math/0701129.
- (4) K. M. R. Audenaert and N. Datta. -z-relative Renyi entropies. Oct. 2013. arXiv: 1310.7178.
- (5) H. Barnum and E. Knill. Reversing Quantum Dynamics with Near-Optimal Quantum and Classical Fidelity. J. Math. Phys., 43(5):2097, 2002. DOI: 10.1063/1.1459754.
- (6) S. Beigi. Sandwiched Rényi Divergence Satisfies Data Processing Inequality. J. Math. Phys., 54(12):122202, June 2013. DOI: 10.1063/1.4838855.
- (7) C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer. Generalized Privacy Amplification. IEEE Trans. on Inf. Theory, 41(6):1915–1923, 1995. DOI: 10.1109/18.476316.
- (8) M. Berta. Single-Shot Quantum State Merging. Master’s thesis, ETH Zurich, 2008. arXiv: 0912.4495.
- (9) M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner. The Uncertainty Principle in the Presence of Quantum Memory. Nat. Phys., 6(9):659–662, July 2010. DOI: 10.1038/nphys1734.
- (10) M. Berta, P. J. Coles, and S. Wehner. An equality between entanglement and uncertainty. page 5, Feb. 2013. arXiv: 1302.5902.
- (11) R. Bhatia. Matrix Analysis. Graduate Texts in Mathematics. Springer, 1997.
- (12) P. J. Coles, R. Colbeck, L. Yu, and M. Zwolak. Uncertainty Relations from Simple Entropic Properties. Phys. Rev. Lett., 108(21):210405, May 2012. DOI: 10.1103/PhysRevLett.108.210405.
- (13) I. Csiszár. Generalized Cutoff Rates and Renyi’s Information Measures. IEEE Trans. on Inf. Theory, 41(1):26–34, 1995. DOI: 10.1109/18.370121.
- (14) N. Datta and F. Leditzky. A Limit of the Quantum Renyi Divergence. Aug. 2013. arXiv: 1308.5961.
- (15) R. L. Frank and E. H. Lieb. Monotonicity of a Relative Rényi Entropy. J. Math. Phys., 54(12):122201, June 2013. DOI: 10.1063/1.4838835.
- (16) R. G. Gallager. Source Coding with Side Information and Universal Coding. In Proc. IEEE ISIT, volume 21, Ronneby, Sweden,, June 1979. IEEE.
- (17) M. K. Gupta and M. M. Wilde. Multiplicativity of Completely Bounded p-Norms Implies a Strong Converse for Entanglement-Assisted Capacity. Oct. 2013. arXiv: 1310.7028.
- (18) R. V. L. Hartley. Transmission of Information. Bell Syst. Tech. J., 7(3):535–563, July 1928.
- (19) M. Hayashi. Exponential decreasing rate of leaked information in universal random privacy amplification. Information Theory, IEEE Transactions on, 57(6):3989–4001, 2011. DOI: 10.1109/TIT.2011.2110950.
- (20) M. Hayashi. Large Deviation Analysis for Quantum Security via Smoothing of Renyi Entropy of Order 2. Feb. 2012. arXiv: 1202.0322.
- (21) M. Hayashi. Security analysis of epsilon-almost dual universal2 hash functions. Sep 2013. Available online: http://arxiv.org/abs/1309.1596.
- (22) M. Hayashi. Tight exponential analysis of universally composable privacy amplification and its applications. Information Theory, IEEE Transactions on, 59(11):7728–7746, 2013. DOI: 10.1109/TIT.2013.2278971.
- (23) M. Iwamoto and J. Shikata. Information Theoretic Security for Encryption Based on Conditional Renyi Entropies. 2013. Available online: http://eprint.iacr.org/2013/440.
- (24) R. König, R. Renner, and C. Schaffner. The Operational Meaning of Min- and Max-Entropy. IEEE Trans. on Inf. Theory, 55(9):4337–4347, Sept. 2009. DOI: 10.1109/TIT.2009.2025545.
- (25) E. H. Lieb and W. E. Thirring. Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities. In The Stability of Matter: From Atoms to Stars, chapter III, pages 205–239. Springer Berlin Heidelberg, 2005. DOI: 10.1007/3-540-27056-6_16.
- (26) H. Maassen and J. Uffink. Generalized Entropic Uncertainty Relations. Phys. Rev. Lett., 60(12):1103–1106, Mar. 1988. DOI: 10.1103/PhysRevLett.60.1103.
- (27) M. Mosonyi. Rényi Divergences and the Classical Capacity of Finite Compound Channels. 2013. arXiv: 1310.7525.
- (28) M. Mosonyi and F. Hiai. On the Quantum Rényi Relative Entropies and Related Capacity Formulas. IEEE Trans. on Inf. Theory, 57(4):2474–2487, Apr. 2011. DOI: 10.1109/TIT.2011.2110050.
- (29) M. Mosonyi and T. Ogawa. Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Renyi Relative Entropies. Sept. 2013. arXiv: 1309.3228.
- (30) M. Müller-Lennert. Quantum Relative Rényi Entropies. Master thesis, ETH Zurich, Apr. 2013.
- (31) M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel. On Quantum Rényi Entropies: A New Generalization and Some Properties. J. Math. Phys., 54(12):122203, June 2013. DOI: 10.1063/1.4838856.
- (32) H. Nagaoka. The Converse Part of The Theorem for Quantum Hoeffding Bound. Nov. 2006. arXiv: quant-ph/0611289.
- (33) We use the convention that and .
- (34) T. Ogawa and M. Hayashi. On Error Exponents in Quantum Hypothesis Testing. IEEE Trans. on Inf. Theory, 50(6):1368–1372, June 2004. DOI: 10.1109/TIT.2004.828155.
- (35) R. Renner. Security of Quantum Key Distribution. PhD thesis, ETH Zurich, Dec. 2005. arXiv: quant-ph/0512258.
- (36) A. Rényi. On Measures of Information and Entropy. In Proc. Symp. on Math., Stat. and Probability, pages 547–561, Berkeley, 1961. University of California Press.
- (37) N. Sharma and N. A. Warsi. Fundamental Bound on the Reliability of Quantum Information Transmission. Phys. Rev. Lett., 110(8):080501, Feb. 2013. DOI: 10.1103/PhysRevLett.110.080501.
- (38) M. Tomamichel. Smooth entropies—A tutorial: With focus on applications in cryptography, Sept. 2012. Available online: http://2012.qcrypt.net/docs/slides/Marco.pdf.
- (39) M. Tomamichel, R. Colbeck, and R. Renner. A Fully Quantum Asymptotic Equipartition Property. IEEE Trans. on Inf. Theory, 55(12):5840–5847, Dec. 2009. DOI: 10.1109/TIT.2009.2032797.
- (40) M. Tomamichel and R. Renner. Uncertainty Relation for Smooth Entropies. Phys. Rev. Lett., 106(11):110506, Mar. 2011. DOI: 10.1103/PhysRevLett.106.110506.
- (41) M. Tomamichel, C. Schaffner, A. Smith, and R. Renner. Leftover Hashing Against Quantum Side Information. IEEE Trans. on Inf. Theory, 57(8):5524–5535, Aug. 2011. DOI: 10.1109/TIT.2011.2158473.
- (42) M. M. Wilde, A. Winter, and D. Yang. Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels. June 2013. arXiv: 1306.1586.
- (43) H. Yagi. Finite Blocklength Bounds for Multiple Access Channels with Correlated Sources. In Proc. IEEE ISITA, pages 377–381, 2012.