Strong subadditivity of quantum entropy

In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems (or of one quantum system with three degrees of freedom). It is a basic theorem in modern quantum information theory. It was conjectured by D. W. Robinson and D. Ruelle[1] in 1966 and O. E. Lanford III and D. W. Robinson[2] in 1968 and proved in 1973 by E.H. Lieb and M.B. Ruskai,[3] building on results obtained by Lieb in his proof of the Wigner-Yanase-Dyson conjecture.[4]

The classical version of SSA was long known and appreciated in classical probability theory and information theory. The proof of this relation in the classical case is quite easy, but the quantum case is difficult because of the non-commutativity of the reduced density matrices describing the quantum subsystems.

Some useful references here include:

  • "Quantum Computation and Quantum Information"[5]
  • "Quantum Entropy and Its Use"[6]
  • Trace Inequalities and Quantum Entropy: An Introductory Course[7]

Definitions

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We use the following notation throughout the following: A Hilbert space is denoted by  , and   denotes the bounded linear operators on  . Tensor products are denoted by superscripts, e.g.,  . The trace is denoted by  .

Density matrix

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A density matrix is a Hermitian, positive semi-definite matrix of trace one. It allows for the description of a quantum system in a mixed state. Density matrices on a tensor product are denoted by superscripts, e.g.,   is a density matrix on  .

Entropy

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The von Neumann quantum entropy of a density matrix   is

 .

Relative entropy

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Umegaki's[8] quantum relative entropy of two density matrices   and   is

 .

Joint concavity

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A function   of two variables is said to be jointly concave if for any   the following holds

 

Subadditivity of entropy

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Ordinary subadditivity [9] concerns only two spaces   and a density matrix  . It states that

 

This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies   and   are both non-negative. In the quantum case, however, both can be negative, e.g.   can be zero while  . Nevertheless, the subadditivity upper bound on   continues to hold. The closest thing one has to   is the Araki–Lieb triangle inequality [9]

 

which is derived in [9] from subadditivity by a mathematical technique known as purification.

Strong subadditivity (SSA)

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Suppose that the Hilbert space of the system is a tensor product of three spaces:  . Physically, these three spaces can be interpreted as the space of three different systems, or else as three parts or three degrees of freedom of one physical system.

Given a density matrix   on  , we define a density matrix   on   as a partial trace:  . Similarly, we can define density matrices:  ,  ,  ,  ,  .

Statement

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For any tri-partite state   the following holds

 ,

where  , for example.

Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state  ,

 .

This can also be restated in terms of quantum mutual information,

 .

These statements run parallel to classical intuition, except that quantum conditional entropies can be negative, and quantum mutual informations can exceed the classical bound of the marginal entropy.

The strong subadditivity inequality was improved in the following way by Carlen and Lieb [10]

 ,

with the optimal constant  .

J. Kiefer[11][12] proved a peripherally related convexity result in 1959, which is a corollary of an operator Schwarz inequality proved by E.H.Lieb and M.B.Ruskai.[3] However, these results are comparatively simple, and the proofs do not use the results of Lieb's 1973 paper on convex and concave trace functionals.[4] It was this paper that provided the mathematical basis of the proof of SSA by Lieb and Ruskai. The extension from a Hilbert space setting to a von Neumann algebra setting, where states are not given by density matrices, was done by Narnhofer and Thirring .[13]

The theorem can also be obtained by proving numerous equivalent statements, some of which are summarized below.

Wigner–Yanase–Dyson conjecture

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E. P. Wigner and M. M. Yanase [14] proposed a different definition of entropy, which was generalized by Freeman Dyson.

The Wigner–Yanase–Dyson p-skew information

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The Wigner–Yanase–Dyson  -skew information of a density matrix  . with respect to an operator   is

 

where   is a commutator,   is the adjoint of   and   is fixed.

Concavity of p-skew information

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It was conjectured by E. P. Wigner and M. M. Yanase in [15] that  - skew information is concave as a function of a density matrix   for a fixed  .

Since the term   is concave (it is linear), the conjecture reduces to the problem of concavity of  . As noted in,[4] this conjecture (for all  ) implies SSA, and was proved for   in,[15] and for all   in [4] in the following more general form: The function of two matrix variables

  (1)

is jointly concave in   and   when   and  .

This theorem is an essential part of the proof of SSA in.[3]

In their paper [15] E. P. Wigner and M. M. Yanase also conjectured the subadditivity of  -skew information for  , which was disproved by Hansen[16] by giving a counterexample.

First two statements equivalent to SSA

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It was pointed out in [9] that the first statement below is equivalent to SSA and A. Ulhmann in [17] showed the equivalence between the second statement below and SSA.

  •   Note that the conditional entropies   and   do not have to be both non-negative.
  • The map   is convex.

Both of these statements were proved directly in.[3]

Joint convexity of relative entropy

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As noted by Lindblad[18] and Uhlmann,[19] if, in equation (1), one takes   and   and   and differentiates in   at  , one obtains the joint convexity of relative entropy: i.e., if  , and  , then

  (2)

where   with  .

Monotonicity of quantum relative entropy

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The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations   on density matrices,

 .

This inequality is called Monotonicity of quantum relative entropy. Owing to the Stinespring factorization theorem, this inequality is a consequence of a particular choice of the CPTP map - a partial trace map described below.

The most important and basic class of CPTP maps is a partial trace operation  , given by  . Then

  (3)

which is called Monotonicity of quantum relative entropy under partial trace.

To see how this follows from the joint convexity of relative entropy, observe that   can be written in Uhlmann's representation as

 

for some finite   and some collection of unitary matrices on   (alternatively, integrate over Haar measure). Since the trace (and hence the relative entropy) is unitarily invariant, inequality (3) now follows from (2). This theorem is due to Lindblad [18] and Uhlmann,[17] whose proof is the one given here.

SSA is obtained from (3) with   replaced by   and   replaced  . Take      . Then (3) becomes

 

Therefore,

 

which is SSA. Thus, the monotonicity of quantum relative entropy (which follows from (1) implies SSA.

Relationship among inequalities

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All of the above important inequalities are equivalent to each other, and can also be proved directly. The following are equivalent:

  • Monotonicity of quantum relative entropy (MONO);
  • Monotonicity of quantum relative entropy under partial trace (MPT);
  • Strong subadditivity (SSA);
  • Joint convexity of quantum relative entropy (JC);

The following implications show the equivalence between these inequalities.

  • MONO   MPT: follows since the MPT is a particular case of MONO;
  • MPT   MONO: was shown by Lindblad,[20] using a representation of stochastic maps as a partial trace over an auxiliary system;
  • MPT   SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";
  • SSA   MPT: by choosing   to be block diagonal, one can show that SSA implies that the map

  is convex. In [3] it was observed that this convexity yields MPT;

  • MPT   JC: as it was mentioned above, by choosing   (and similarly,  ) to be block diagonal matrix with blocks   (and  ), the partial trace is a sum over blocks so that  , so from MPT one can obtain JC;
  • JC   SSA: using the 'purification process', Araki and Lieb,[9][21] observed that one could obtain new useful inequalities from the known ones. By purifying   to   it can be shown that SSA is equivalent to
 

Moreover, if   is pure, then   and  , so the equality holds in the above inequality. Since the extreme points of the convex set of density matrices are pure states, SSA follows from JC;

See,[21][22] for a discussion.

The case of equality

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Equality in monotonicity of quantum relative entropy inequality

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In,[23][24] D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:

For all states   and   on a Hilbert space   and all quantum operators  ,

 

if and only if there exists a quantum operator   such that

  and  

Moreover,   can be given explicitly by the formula

 

where   is the adjoint map of  .

D. Petz also gave another condition [23] when the equality holds in Monotonicity of quantum relative entropy: the first statement below. Differentiating it at   we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.

For all states   and   on   and all quantum operators  ,

 

if and only if the following equivalent conditions are satisfied:

  •   for all real  .
  •  

where   is the adjoint map of  .

Equality in strong subadditivity inequality

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P. Hayden, R. Jozsa, D. Petz and A. Winter described the states for which the equality holds in SSA.[25]

A state   on a Hilbert space   satisfies strong subadditivity with equality if and only if there is a decomposition of second system as

 

into a direct sum of tensor products, such that

 

with states   on   and   on  , and a probability distribution  .

Carlen-Lieb Extension

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E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality,[10] namely,

 

If   and  , as is always the case for the classical Shannon entropy, this inequality has nothing to say. For the quantum entropy, on the other hand, it is quite possible that the conditional entropies satisfy   or   (but never both!). Then, in this "highly quantum" regime, this inequality provides additional information.

The constant 2 is optimal, in the sense that for any constant larger than 2, one can find a state for which the inequality is violated with that constant.

Operator extension of strong subadditivity

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In his paper [26] I. Kim studied an operator extension of strong subadditivity, proving the following inequality:

For a tri-partite state (density matrix)   on  ,

 

The proof of this inequality is based on Effros's theorem,[27] for which particular functions and operators are chosen to derive the inequality above. M. B. Ruskai describes this work in details in [28] and discusses how to prove a large class of new matrix inequalities in the tri-partite and bi-partite cases by taking a partial trace over all but one of the spaces.

Extensions of strong subadditivity in terms of recoverability

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A significant strengthening of strong subadditivity was proved in 2014,[29] which was subsequently improved in [30] and.[31] In 2017,[32] it was shown that the recovery channel can be taken to be the original Petz recovery map. These improvements of strong subadditivity have physical interpretations in terms of recoverability, meaning that if the conditional mutual information   of a tripartite quantum state   is nearly equal to zero, then it is possible to perform a recovery channel   (from system E to AE) such that  . These results thus generalize the exact equality conditions mentioned above.

See also

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References

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  1. ^ Robinson, Derek W.; Ruelle, David (1967). "Mean entropy of states in classical statistical mechanics". Communications in Mathematical Physics. 5 (4). Springer Science and Business Media LLC: 288–300. Bibcode:1967CMaPh...5..288R. doi:10.1007/bf01646480. ISSN 0010-3616. S2CID 115134613.
  2. ^ Lanford, Oscar E.; Robinson, Derek W. (1968). "Mean Entropy of States in Quantum‐Statistical Mechanics". Journal of Mathematical Physics. 9 (7). AIP Publishing: 1120–1125. Bibcode:1968JMP.....9.1120L. doi:10.1063/1.1664685. ISSN 0022-2488.
  3. ^ a b c d e Lieb, Elliott H.; Ruskai, Mary Beth (1973). "Proof of the strong subadditivity of quantum‐mechanical entropy" (PDF). Journal of Mathematical Physics. 14 (12). AIP Publishing: 1938–1941. Bibcode:1973JMP....14.1938L. doi:10.1063/1.1666274. ISSN 0022-2488.
  4. ^ a b c d Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-X. ISSN 0001-8708.
  5. ^ M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambr. U. Press, (2000)
  6. ^ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer (1993)
  7. ^ E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).
  8. ^ Umegaki, Hisaharu (1962). "Conditional expectation in an operator algebra. IV. Entropy and information". Kodai Mathematical Seminar Reports. 14 (2). Tokyo Institute of Technology, Department of Mathematics: 59–85. doi:10.2996/kmj/1138844604. ISSN 0023-2599.
  9. ^ a b c d e Araki, Huzihiro; Lieb, Elliott H. (1970). "Entropy inequalities". Communications in Mathematical Physics. 18 (2): 160–170. Bibcode:1970CMaPh..18..160A. doi:10.1007/BF01646092. ISSN 0010-3616. S2CID 189832417.
  10. ^ a b Carlen, Eric A.; Lieb, Elliott H. (2012). "Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy". Letters in Mathematical Physics. 101 (1): 1–11. arXiv:1203.4719. Bibcode:2012LMaPh.101....1C. doi:10.1007/s11005-012-0565-6. S2CID 119317605.
  11. ^ Kiefer, J. (July 1959). "Optimum Experimental Designs". Journal of the Royal Statistical Society, Series B (Methodological). 21 (2): 272–310. doi:10.1111/j.2517-6161.1959.tb00338.x.
  12. ^ Ruskai, Mary Beth. "Evolution of a Fundemental [sic] Theorem on Quantum Entropy". youtube.com. World Scientific. Retrieved 20 August 2020. Invited talk at the Conference in Honour of the 90th Birthday of Freeman Dyson, Institute of Advanced Studies, Nanyang Technological University, Singapore, 26–29 August 2013. The note on Kiefer (1959) is at the 26:40 mark.
  13. ^ Narnhofer, H. (1985). "From Relative Entropy to Entropy". Fizika. 17: 258–262.
  14. ^ Wigner, E. P.; Yanase, M. M. (1 May 1963). "Information Content of Distributions". Proceedings of the National Academy of Sciences. 49 (6): 910–918. Bibcode:1963PNAS...49..910W. doi:10.1073/pnas.49.6.910. ISSN 0027-8424. PMC 300031. PMID 16591109.
  15. ^ a b c Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. 16. Canadian Mathematical Society: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X.
  16. ^ Hansen, Frank (18 January 2007). "The Wigner-Yanase Entropy is not Subadditive". Journal of Statistical Physics. 126 (3). Springer Nature: 643–648. arXiv:math-ph/0609019. Bibcode:2007JSP...126..643H. doi:10.1007/s10955-006-9265-x. ISSN 0022-4715. S2CID 119667187.
  17. ^ a b A. Ulhmann, Endlich Dimensionale Dichtmatrizen, II, Wiss. Z. Karl-Marx-University Leipzig 22 Jg. H. 2., 139 (1973).
  18. ^ a b Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2): 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/BF01608390. ISSN 0010-3616. S2CID 120760667.
  19. ^ Uhlmann, A. (1977). "Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory". Communications in Mathematical Physics. 54 (1): 21–32. Bibcode:1977CMaPh..54...21U. doi:10.1007/BF01609834. ISSN 0010-3616. S2CID 15800519.
  20. ^ Lindblad, Göran (1975). "Completely positive maps and entropy inequalities". Communications in Mathematical Physics. 40 (2). Springer Science and Business Media LLC: 147–151. Bibcode:1975CMaPh..40..147L. doi:10.1007/bf01609396. ISSN 0010-3616. S2CID 121650206.
  21. ^ a b Lieb, E. H. (1975). "Some Convexity and Subadditivity Properties of Entropy". Bull. Am. Math. Soc. 81: 1–13. doi:10.1090/s0002-9904-1975-13621-4.
  22. ^ Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292. erratum 46, 019901 (2005)
  23. ^ a b Petz, Dénes (1986). "Sufficient subalgebras and the relative entropy of states of a von Neumann algebra". Communications in Mathematical Physics. 105 (1). Springer Science and Business Media LLC: 123–131. Bibcode:1986CMaPh.105..123P. doi:10.1007/bf01212345. ISSN 0010-3616. S2CID 18836173.
  24. ^ D. Petz, Sufficiency of Channels over von Neumann Algebras, Quart. J. Math. Oxford 35, 475–483 (1986).
  25. ^ P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of States which Satisfy Strong Subadditivity of Quantum Entropy with Equality, Comm. Math. Phys. 246, 359–374 (2003).
  26. ^ I. Kim, Operator Extension of Strong Subadditivity of Entropy, arXiv:1210.5190 (2012).
  27. ^ Effros, E. G. (2009). "A Matrix Convexity Approach to Some Celebrated Quantum Inequalities". Proc. Natl. Acad. Sci. USA. 106 (4): 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. PMC 2633548. PMID 19164582.
  28. ^ M. B. Ruskai, Remarks on Kim’s Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions, arXiv:1211.0049 (2012).
  29. ^ O. Fawzi, R. Renner. Quantum conditional mutual information and approximate Markov chains. Communications in Mathematical Physics: 340, 2 (2015)
  30. ^ M. M. Wilde. Recoverability in quantum information theory. Proceedings of the Royal Society A, vol. 471, no. 2182, page 20150338 October 2015
  31. ^ Marius Junge, Renato Renner, David Sutter, Mark M. Wilde, Andreas Winter. Universal recovery maps and approximate sufficiency of quantum relative entropy. Annales Henri Poincare, vol. 19, no. 10, pages 2955--2978, October 2018 arXiv:1509.07127
  32. ^ Carlen, Eric A.; Vershynina, Anna (2017-10-06). "Recovery map stability for the Data Processing Inequality". arXiv:1710.02409 [math.OA].