## Inequalities for quantum entropy. A review with conditions with equality

Citations: | 34 - 7 self |

### BibTeX

@TECHREPORT{Ruskai_inequalitiesfor,

author = {Mary Beth Ruskai},

title = {Inequalities for quantum entropy. A review with conditions with equality},

institution = {},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A → Tr e K+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2) ≤ S(ρ12) + S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123 = log ρ12 − log ρ2 + log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s

### Citations

7283 |
A mathematical theory of communications
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- 1948
(Show Context)
Citation Context ...i-definite operator with Trρ = 1), into quantum theory defined its entropy as as S(ρ) ≡ −Tr(ρ log ρ) in 1927, well before the corresponding classical quantity was introduced in Shannon’s seminal work =-=[41]-=- on “The Mathematical Theory of Communication” in 1948. (Admittedly, von Neumann’s motivation was the extension of the classical theory of statistical mechanics, developed by Gibbs, Boltzman, et al to... |

3839 | Convex Analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ...hat both sides of (17) equal Tr ρ1 ⊗ρ23 when R = ρ1 ⊗ρ2 ⊗I3, S = I1 ⊗ρ2 ⊗I3, T = I1 ⊗ρ23, even when T does not commute with R or S. Proof: Lieb’s proof of (17) begins with the easily-established fact =-=[39]-=- that if F(A) is concave and homogeneous in the sense F(xA) = xF(A) , then F(A + xB) − F(A) lim ≥ F(B). (18) x→0 x Applying this to the functions in Theorem 2 with A = S, B = T, K = log R−log S yields... |

1343 | Information Theory and Statistics
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(Show Context)
Citation Context ... S[p(a, b, c)] + S[p(b)] ≤ S[p(a, b)] + S[p(b, c)]. (9) The classical relative entropy of the distribution q(a) with respect to p(a) is H[p(a), q(a)] = ∑ a p(a) log p(a) . It is well-known (see, e.g.,=-=[23]-=-) that the convexq(a) ity of the function f(x) = x log x implies that H[p(a), q(a)] ≥ 0 and its strict convexity implies that equality holds if and only if p(a) = q(a) ∀ a. (The generalization of this... |

464 |
Quantum Computation and Quantum Information. Cambridge Univ
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ... on accessible information . . . . . . . . . . . . . . 21 8 Concluding remarks 22 A Epstein’s proof of concavity of A → Tre K+log A 24 21 Introduction 1.1 Quantum Entropy Quantum information science =-=[32]-=- is the study of the information carrying and processing properties of quantum mechanical systems. Recent work in this area has generated renewed interest in the properties of the quantum mechanical e... |

120 |
Operator Algebras and Quantum
- Bratteli, Robinson
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(Show Context)
Citation Context ... immediately as a special case. However, Uhlmann’s approach, which has been extended by Petz [35, 33], was developed within the framework of the relative modular operator formalism developed by Araki =-=[5, 7, 33]-=- for much more general situations. Recently, Lesniewski and Ruskai [29] observed that within this relative modular operator framework, monotonicity can be established directly using an argument based ... |

110 |
Monotone matrix functions and analytic continuation
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Citation Context ...he upper half plane into the upper half plane. Functions with this property have been studied extensively under various names, including, “operator monotone”, “Herglotz” or “Pick”. (See, for example, =-=[3, 10, 33]-=-). It then follows that g has an integral representation of the form ∫ µ 1 g(z) = a + bz + dm(t) (64) −µ t − z for some positive measure µ(t). This yields (via the change of variables s = t−1 ) ∫ µ x ... |

75 |
Relative entropy of states of von Neumann algebras
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(Show Context)
Citation Context ... immediately as a special case. However, Uhlmann’s approach, which has been extended by Petz [35, 33], was developed within the framework of the relative modular operator formalism developed by Araki =-=[5, 7, 33]-=- for much more general situations. Recently, Lesniewski and Ruskai [29] observed that within this relative modular operator framework, monotonicity can be established directly using an argument based ... |

67 |
Convex trace functions and the Wigner-Yanase-Dyson conjecture
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(Show Context)
Citation Context ..., conjectured by Bauman [6], is equivalent to the joint concavity in A and B of the map (A, B) → Tr A s K † B (1−s) K for A, B > 0, 0 < s < 1 (8) (where † is used to denote the adjoint). Lieb’s proof =-=[25]-=- of the concavity of the WYD function (8) and his realization of a connection between SSA and Bauman’s concavity conjecture was a crucial breakthrough. However, concavity of the WYD function was only ... |

66 |
M.B.: Proof of the Strong Subadditivity of Quantum-Mechanical Entropy
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(Show Context)
Citation Context ... in [9] and [24]. (It should not be confused with the concavity S(xρ ′ + (1 − x)ρ ′′ ) ≥ xS(ρ ′ ) + (1 − x)S(ρ ′′ ) (2) which can actually be obtained from subadditivity by considering block matrices =-=[26, 28, 47]-=-). In the more complex situation in which the composite system is composed of three subsystems the following stronger inequality, known as strong subadditivity (SSA), holds. S(ρ123) + S(ρ2) ≤ S(ρ12) +... |

59 | Minimal entropy of states emerging from noisy quantum channels
- King, Ruskai
- 2001
(Show Context)
Citation Context ...e again write, e.g., ρ23 for I1 ⊗ ρ23. More generally, the relative entropy is monotone under completely positive, trace-preserving maps (also known as “quantum operations” [32] and “stochastic maps” =-=[1, 18]-=- and discussed in more detail in section 3.4), i.e., H[Φ(ρ), Φ(γ)] ≤ H(ρ, γ). (7) This monotonicity implies (6) when Φ = T3 is the partial trace operation; perhaps surprisingly, the converse is also t... |

58 |
General properties of entropy”, Rev
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Citation Context ...any fundamental properties of the quantum entropy were proved in a remarkable, but little-known, 1936 paper of Delbrück and Molèiere [9]. For further discussion of the history of quantum entropy, see =-=[33, 38, 47]-=- and the introductory remarks in [40]. One important class of inequalities relates the entropy of subsystems to that of a composite system, whose Hilbert space is a tensor product is H12 = H1 ⊗ H2 of ... |

52 | Distinguishability and Accessible Information in Quantum Theory
- Fuchs
- 1995
(Show Context)
Citation Context ...mute. Holevo’s original longer derivation [14] of the bound (48) also concluded that commutativity was necessary and sufficient for equality. Some simplifications of this argument were given by Fuchs =-=[12]-=- in his thesis. 207.4 Another bound on accessible information When ρ is a density matrix, the mapping A ↦→ ρ −1/2 Aρ −1/2 and its inverse gives a duality between ensembles and POVM’s. Hall [13] obser... |

50 |
Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory
- Uhlmann
- 1977
(Show Context)
Citation Context ...re real-valued. Indeed, Lieb’s original proof of the concavity of WYD entropy used a complex interpolation argument. In his influential book [42] on Trace Ideals, Simon (extracting ideas from Uhlmann =-=[44]-=-) gave a longer“elementary” proof using the Schwarz inequality, perhaps inadvertently reinforcing the notion that any complete proof of SSA is long and forbidding. Similar ideas are implicit in Ando [... |

42 |
Counterexample to an additivity conjecture for output purity of quantum channels”, preprint lanl:quantph/0203003
- Werner, Holevo
(Show Context)
Citation Context ... advocated by Amosov, Holevo and Werner [2], may be a promising avenue for studying entropy and capacity in quantum information. Despite the results mentioned above, many open conjectures remain; see =-=[2, 8, 20, 21, 48]-=- for further details. Acknowledgments The work of M.B. Ruskai was partially supported by the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) under Army Research Of... |

30 |
Quantum Entropy and Its Use (Springer-Verlag
- Ohya, Petz
- 1993
(Show Context)
Citation Context ...rove the monotonicity of relative entropy under completely positive trace-preserving maps. SSA then follows immediately as a special case. However, Uhlmann’s approach, which has been extended by Petz =-=[35, 33]-=-, was developed within the framework of the relative modular operator formalism developed by Araki [5, 7, 33] for much more general situations. Recently, Lesniewski and Ruskai [29] observed that withi... |

28 |
personal communication
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- 1994
(Show Context)
Citation Context ... implies that ∑ |Eb〉〈Eb| b TrEb projects onto the span({Zj}). However, this alone is not sufficient to imply that the Eb form a projective measurement. 197.3 Other approaches Chris King has observed =-=[22]-=- that when the POVM is a projective measurement of the form Eb = |b〉〈b|, one can obtain the Holevo bound from the joint convexity of relative entropy. Let β(˜ρ) = ∑ b |b〉〈b|TrEb ˜ρ. Then applying Theo... |

26 | On capacities of quantum channels
- Ohya, Petz, et al.
- 1997
(Show Context)
Citation Context ...implies that all ˜ρj also commute since V † V = I. It should be noted that Petz was able to use his equality conditions to find the conditions for equality in the Holevo bound and this is sketched in =-=[34]-=-. Indeed, Petz’s analogue of (57) is ˜ρ itD−it = ˜ρ it j D−it j parts of ˜ρ, ˜ρj respectively. Then b ∀ j where D, Dj denotes the diagonal ˜ρ it j = ˜ρ it D −it D it j . (58) Since (58) holds for all ... |

25 |
Completely Positive Maps and Entropy
- Lindblad
- 1975
(Show Context)
Citation Context ... discussed in more detail in section 3.4), i.e., H[Φ(ρ), Φ(γ)] ≤ H(ρ, γ). (7) This monotonicity implies (6) when Φ = T3 is the partial trace operation; perhaps surprisingly, the converse is also true =-=[31]-=-. This, and other connections between strong subadditivity and relative entropy are discussed in Section 5.3 . The approach to SSA presented here can also be used to obtain conditions for equality in ... |

23 |
Some convexity and subadditivity properties of entropy
- Lieb
- 1975
(Show Context)
Citation Context ... in [9] and [24]. (It should not be confused with the concavity S(xρ ′ + (1 − x)ρ ′′ ) ≥ xS(ρ ′ ) + (1 − x)S(ρ ′′ ) (2) which can actually be obtained from subadditivity by considering block matrices =-=[26, 28, 47]-=-). In the more complex situation in which the composite system is composed of three subsystems the following stronger inequality, known as strong subadditivity (SSA), holds. S(ρ123) + S(ρ2) ≤ S(ρ12) +... |

22 |
Information theoretical aspects of quantum measurements
- Holevo
- 1973
(Show Context)
Citation Context ...ropy, including its joint convexity and monotonicity. The explicit statements are postponed to later sections. Since the monotonicity can be used to give a simple proof of the celebrated Holevo bound =-=[14, 32]-=- on accessible information, we show how our results can be used to recover the equality conditions in that bound. As discussed in section 2.3, Petz [33, 36] has also obtained several equality conditio... |

22 |
Thermodynamik quantenmechanischer Gesamtheiten
- Neumann
(Show Context)
Citation Context ...ssing properties of quantum mechanical systems. Recent work in this area has generated renewed interest in the properties of the quantum mechanical entropy. It is interesting to note that von Neumann =-=[44, 45]-=- introduced the notion of mixed state, represented by a density matrix ρ (a positive semi-definite operator with Trρ = 1), into quantum theory defined its entropy as as S(ρ) ≡ −Tr(ρ log ρ) in 1927, we... |

20 |
Trace Ideals and Their Applications (Cambridge
- Simon
- 1979
(Show Context)
Citation Context ...r e A+B ≤ Tre A e B with equality if and only if A and B commute. Although this inequality is extremely well-known, the conditions for equality do not appear explicitly in such standard references as =-=[16, 42, 47]-=-. However, one method of proof is based on the observation that Tr [eA/2keB/2k] 2k is monotone decreasing in k, yielding e A+B in the limit as k → ∞. The equality conditions then follow easily from th... |

20 |
Ultimate Information Carrying Limit of Quantum Systems
- Yuen, Ozawa
- 1993
(Show Context)
Citation Context ... which simultaneously diagonalize the density matrices ˜ρj. We wish to show that this condition is also necessary, i.e., equality can only be achieved in (48) if all the ˜ρj commute. j 17It is known =-=[19, 50]-=- that (48) can be obtained from (7). First, observe that S(˜ρ) − ∑ πjS(˜ρj) = ∑ πjH(˜ρj, ˜ρ) (49) j Now let ΩM be the map ΩM(A) = ∑ b |b〉〈b| Tr(AEb) where M = {Eb}. Then ΩM is a stochastic map of the ... |

17 |
A fundamental property of quantum-mechanical entropy
- Lieb, Ruskai
- 1973
(Show Context)
Citation Context ...tems the following stronger inequality, known as strong subadditivity (SSA), holds. S(ρ123) + S(ρ2) ≤ S(ρ12) + S(ρ23) (3) This inequality was conjectured by Lanford and Robinson in [24] and proved in =-=[27, 28]-=-. In this paper, we review its proof in a form that easily yields the following condition for equality. 3Theorem 1 Equality holds in strong subadditivity (3) if and only if log ρ123 − log ρ12 = log ρ... |

16 |
Mean Entropy of States in Quantum-Statistical Mechanics
- Lanford, Robinson
- 1965
(Show Context)
Citation Context ...es of the subsystems are given by the reduced density matrices, e.g., ρ1 = T2(ρ12), obtained by taking the partial trace. The subadditivity inequality S(ρ12) ≤ S(ρ1) + S(ρ2) (1) was proved in [9] and =-=[24]-=-. (It should not be confused with the concavity S(xρ ′ + (1 − x)ρ ′′ ) ≥ xS(ρ ′ ) + (1 − x)S(ρ ′′ ) (2) which can actually be obtained from subadditivity by considering block matrices [26, 28, 47]). I... |

13 |
Remarks on two theorems of
- Epstein
- 1973
(Show Context)
Citation Context ...ty of the relative entropy [27, 28, 30]. Although SSA is a deep theorem, a complete proof is not as forbidding as is sometimes implied. Therefore, for completeness, we include Epstein’s elegant proof =-=[11]-=- of Theorem 2 in Appendix A, and then follow the original strategies of Lieb and Ruskai [28] to show how it implies SSA. 1.4 Overview Although this paper grew out of questions about the conditions for... |

12 |
Sufficient subalgebras and the relative entropy of states of a von Neumann algebra
- Petz
- 1986
(Show Context)
Citation Context ...simple proof of the celebrated Holevo bound [14, 32] on accessible information, we show how our results can be used to recover the equality conditions in that bound. As discussed in section 2.3, Petz =-=[33, 36]-=- has also obtained several equality conditions in different, but equivalent, forms. However, Theorem 8, which applies to the most general form of monotonicity, appears to be new. 41.3 Lieb’s convex t... |

11 | A Minkowski type trace inequality and strong subadditivity of quantum entropy
- Carlen, Lieb
- 1999
(Show Context)
Citation Context ...concavity of the WYD function, yields another “short” proof of SSA, albeit one which does not appear to be wellsuited to establishing conditions for equality. Finally, we mention that Carlen and Lieb =-=[8]-=- obtained another proof of SSA by using Epstein’s technique to prove some Minkowski type inequalities for Lp trace norms. Using a different approach, King [20, 21] recently proved several additivity r... |

7 | Capacity of Quantum Channels Using Product Measurements” xxx.lanl.gov preprint quant-ph/0004062
- King, Ruskai
(Show Context)
Citation Context ... which simultaneously diagonalize the density matrices ˜ρj. We wish to show that this condition is also necessary, i.e., equality can only be achieved in (48) if all the ˜ρj commute. j 17It is known =-=[19, 50]-=- that (48) can be obtained from (7). First, observe that S(˜ρ) − ∑ πjS(˜ρj) = ∑ πjH(˜ρj, ˜ρ) (49) j Now let ΩM be the map ΩM(A) = ∑ b |b〉〈b| Tr(AEb) where M = {Eb}. Then ΩM is a stochastic map of the ... |

6 |
of capacity and p-norms for some product channels
- King, “Maximization
- 2002
(Show Context)
Citation Context ... Finally, we mention that Carlen and Lieb [8] obtained another proof of SSA by using Epstein’s technique to prove some Minkowski type inequalities for Lp trace norms. Using a different approach, King =-=[20, 21]-=- recently proved several additivity results for the minimal entropy and Holevo capacity of a noisy channel by using Lp inequalities in which Epstein’s technique provided a critical estimate. This sugg... |

6 |
Quasi-entropies for Finite Quantum
- Petz
- 1986
(Show Context)
Citation Context ...rove the monotonicity of relative entropy under completely positive trace-preserving maps. SSA then follows immediately as a special case. However, Uhlmann’s approach, which has been extended by Petz =-=[35, 33]-=-, was developed within the framework of the relative modular operator formalism developed by Araki [5, 7, 33] for much more general situations. Recently, Lesniewski and Ruskai [29] observed that withi... |

6 |
A variational expression for the relative entropy
- Petz
- 1988
(Show Context)
Citation Context ...ltiple of D implies e B/2 e A/2 = e A/2 e B/2 which holds if and only if A and B commute. One reference [33] that does discuss equality does so by making the interesting observation that (as shown in =-=[37]-=-) Theorem 4 and its equality conditions, can be derived as a consequence of the monotonicity of relative entropy, Theorem 7. The natural extension to three matrices Tr e A+B+C ≤ |Tre A e B e C |, fail... |

5 |
On Some Additivity
- Amosov, Holevo, et al.
- 2000
(Show Context)
Citation Context ...y of a noisy channel by using Lp inequalities in which Epstein’s technique provided a critical estimate. This suggests that connections with Lp inequalities, as advocated by Amosov, Holevo and Werner =-=[2]-=-, may be a promising avenue for studying entropy and capacity in quantum information. Despite the results mentioned above, many open conjectures remain; see [2, 8, 20, 21, 48] for further details. Ack... |

5 |
Zur Quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre
- Klein
- 1931
(Show Context)
Citation Context ... tools 3.1 Klein’s inequality The fact that the relative entropy is positive, i.e., H(ρ, γ) ≥ 0 when Tr ρ = Tr γ is an immediate consequence of the following fundamental convexity result due to Klein =-=[17, 32, 47]-=-. Theorem 3 (Klein’s Inequality) For A, B > 0 with equality if and only if A = B. Tr A(log A − log B) ≥ Tr(A − B), (16) The closely related Peierls-Bogoliubov inequality [33, 47] is sometimes used ins... |

5 |
Relative Entropy and Monotone Riemannian Metrics on Non-Commutative Probability
- Lesniewski, Ruskai
- 1999
(Show Context)
Citation Context ...y under stochastic maps Conditions for equality in the general monotonicity inequality (7) may be more subtle since it is not always possible to achieve equality. Indeed, it was noted in H[Φ(ρ),Φ(γ)] =-=[29]-=- that supρ̸=γ can be strictly less than 1. Using the reformulation H(ρ,γ) (38) above, we prove the following result. Theorem 8 Equality holds in (7), H[Φ(ρ), Φ(γ)] ≤ H(ρ, γ), if and only if log ρ − lo... |

5 | von Neumann and the von Neumann entropy
- Petz
(Show Context)
Citation Context ...any fundamental properties of the quantum entropy were proved in a remarkable, but little-known, 1936 paper of Delbrück and Molèiere [9]. For further discussion of the history of quantum entropy, see =-=[33, 38, 47]-=- and the introductory remarks in [40]. One important class of inequalities relates the entropy of subsystems to that of a composite system, whose Hilbert space is a tensor product is H12 = H1 ⊗ H2 of ... |

4 |
Convexity inequalities for estimating free energy and relative entropy
- Ruskai, Stillinger
(Show Context)
Citation Context ...ropy were proved in a remarkable, but little-known, 1936 paper of Delbrück and Molèiere [9]. For further discussion of the history of quantum entropy, see [33, 38, 47] and the introductory remarks in =-=[40]-=-. One important class of inequalities relates the entropy of subsystems to that of a composite system, whose Hilbert space is a tensor product is H12 = H1 ⊗ H2 of the Hilbert spaces for the subsystems... |

3 |
M.M.Yanase, Information content of distribution, Proc
- Wigner
(Show Context)
Citation Context ...b’s convex trace functions One of the most frequently cited approaches to strong subadditivity is to present it as a consequence of the concavity of a quantity known as the Wigner-YanaseDyson entropy =-=[49]-=-. This property, conjectured by Bauman [6], is equivalent to the joint concavity in A and B of the map (A, B) → Tr A s K † B (1−s) K for A, B > 0, 0 < s < 1 (8) (where † is used to denote the adjoint)... |

3 |
Expectations and Entropy
- Lindblad
- 1974
(Show Context)
Citation Context ...on A ↦→ F(A) = Tr e K+logA is concave in A > 0. This result played a fundamental role in the original proof [26, 27] of SSA and the closely related property of joint concavity of the relative entropy =-=[26, 27, 29]-=-. Although SSA is a deep theorem, a complete proof is not as forbidding as is sometimes implied. Therefore, for completeness, we include Epstein’s elegant proof [11] of Theorem 2 in Appendix A, and th... |

2 |
Topics on operator
- Ando
(Show Context)
Citation Context ...]) gave a longer“elementary” proof using the Schwarz inequality, perhaps inadvertently reinforcing the notion that any complete proof of SSA is long and forbidding. Similar ideas are implicit in Ando =-=[3]-=- who restates the result in terms of tensor product spaces and block matrices. Uhlmann [44] again demonstrated the power of complex interpolation by using it to prove the monotonicity of relative entr... |

2 |
Quantum information and correlation bounds, Phys
- Hall
- 1997
(Show Context)
Citation Context ...Fuchs [12] in his thesis. 207.4 Another bound on accessible information When ρ is a density matrix, the mapping A ↦→ ρ −1/2 Aρ −1/2 and its inverse gives a duality between ensembles and POVM’s. Hall =-=[13]-=- observed that this duality can be used to give another upper bound on the accessible information (47) in terms of the POVM and average density ρ, i.e., I(E, M) ≤ = ∑ S(ρ) − ∑ b b τb H ( 1 τb τb S ( 1... |

2 |
Neumann Matheatische Grundlagen der Quantenmechanik
- von
- 1932
(Show Context)
Citation Context ...ssing properties of quantum mechanical systems. Recent work in this area has generated renewed interest in the properties of the quantum mechanical entropy. It is interesting to note that von Neumann =-=[45, 46]-=- introduced the notion of mixed state, represented by a density matrix ρ (a positive semi-definite operator with Trρ = 1), into quantum theory defined its entropy as as S(ρ) ≡ −Tr(ρ log ρ) in 1927, we... |

2 | Coding Theorems for Quantum Channels” preprint (lanl:quant-ph/9809023) 25 - Holevo - 1991 |

2 |
Completely Positive Maps and Entropy Inequalities” Commun
- Lindblad
- 1975
(Show Context)
Citation Context ... discussed in more detail in section 3.4), i.e., H[Φ(ρ), Φ(γ)] ≤ H(ρ, γ). (7) This monotonicity implies (6) when Φ = T3 is the partial trace operation; perhaps surprisingly, the converse is also true =-=[30]-=-. This, and other connections between strong subadditivity and relative entropy are discussed in Section 5.3 . The approach to SSA presented here can also be used to obtain conditions for equality in ... |

1 |
Bemerkungen Über Quantenmechanische Entropie Ungleichungen
- Bauman
- 1971
(Show Context)
Citation Context ...frequently cited approaches to strong subadditivity is to present it as a consequence of the concavity of a quantity known as the Wigner-YanaseDyson entropy [49]. This property, conjectured by Bauman =-=[6]-=-, is equivalent to the joint concavity in A and B of the map (A, B) → Tr A s K † B (1−s) K for A, B > 0, 0 < s < 1 (8) (where † is used to denote the adjoint). Lieb’s proof [25] of the concavity of th... |

1 |
Statistische Quantenmechanik und Thermodynamik
- Delbrürk, Molèiere
- 1936
(Show Context)
Citation Context ...ather than the development of a theory of quantum communication.) Many fundamental properties of the quantum entropy were proved in a remarkable, but little-known, 1936 paper of Delbrück and Molèiere =-=[9]-=-. For further discussion of the history of quantum entropy, see [33, 38, 47] and the introductory remarks in [40]. One important class of inequalities relates the entropy of subsystems to that of a co... |

1 |
Additivity for a class of unital qubit channels” quant-ph/0103156
- King
(Show Context)
Citation Context ... Finally, we mention that Carlen and Lieb [8] obtained another proof of SSA by using Epstein’s technique to prove some Minkowski type inequalities for Lp trace norms. Using a different approach, King =-=[20, 21]-=- recently proved several additivity results for the minimal entropy and Holevo capacity of a noisy channel by using Lp inequalities in which Epstein’s technique provided a critical estimate. This sugg... |

1 |
Endlich Dimensionale Dichtmatrizen
- Uhlmann
- 1973
(Show Context)
Citation Context ...is map (with A+xB = ρ12 +xγ12), yields (32). This shows that SSA ⇒ MPT so that we have the chain of implications MONO ⇒ MPT ⇐⇒ SSA ⇒ JC. (35) One can show that JC ⇒ MPT by using Uhlmann’s observation =-=[43]-=- that the partial trace can be written as a convex combination of unitary transformations. One can also show directly that JC ⇒ SSA by using the purification process described in section 3.3 to show t... |