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| In [[quantum statistical mechanics]], the '''von Neumann entropy''', named after [[John von Neumann]], is the extension of classical [[Gibbs entropy]] concepts to the field of [[quantum mechanics]]. For a quantum-mechanical system described by a [[density matrix]] {{mvar|ρ}}, the von Neumann entropy is<ref name="bengtsson301">{{cite book|last1=Bengtsson|first1=Ingemar|last2=Zyczkowski|first2=Karol|title=Geometry of Quantum States: An Introduction to Quantum Entanglement|page=301|edition=1st}}</ref>
| | The author is recognized by the title of Figures Wunder. One of the issues he enjoys most is ice skating but he is struggling to discover time for it. Years ago we moved to Puerto Rico and my family enjoys it. Bookkeeping is my occupation.<br><br>Review my site; over the counter std test ([http://dns125.dolzer.at/?q=node/696 what google did to me]) |
| :<math> S = - \mathrm{tr}(\rho \ln \rho),</math>
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| where tr denotes the [[Trace (linear algebra)|trace]]. If {{mvar|ρ}} is written in terms of its eigenvectors |1〉, |2〉, |3〉, ... as
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| :<math> \rho = \sum_j \eta_j |j\rang \lang j | ~, </math>
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| then the von Neumann entropy is merely<ref name="bengtsson301" />
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| :<math> S = -\sum_j \eta_j \ln \eta_j.</math>
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| In this form, ''S'' can be seen to amount to the information theory [[Shannon entropy]].<ref name="bengtsson301" />
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| == Background ==
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| [[John von Neumann]] rigorously established the mathematical framework for quantum mechanics in his work ''Mathematical Foundations of Quantum Mechanics''−−''Mathematische Grundlagen der Quantenmechanik''.<ref>{{Cite book |last=Von Neumann |first=John |authorlink=John von Neumann |coauthors= |title=Mathematische Grundlagen der Quantenmechanik |year=1955 |publisher=Springer |location=Berlin |isbn=3-540-59207-5 }}; {{Cite book |last=Von Neumann |first=John |authorlink=John von Neumann |title=Mathematical Foundations of Quantum Mechanics | year=1996|publisher=Princeton University Press | ISBN= 978-0-691-02893-4 }} </ref> In it, he provided a theory of measurement, where the usual notion of wave-function collapse is described as an irreversible process (the so-called von Neumann or projective measurement).
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| The [[density matrix]] was introduced, with different motivations, by von Neumann and by [[Lev Landau]]. The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.<ref>{{cite doi|10.1007/BF01343064}}</ref> On the other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.
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| The density matrix formalism was developed to extend the tools of classical statistical mechanics to the quantum domain. In the classical framework we compute the [[Partition function (statistical mechanics)|partition function]] of the system in order to evaluate all possible thermodynamic quantities. Von Neumann introduced the density matrix in the context of states and operators in a Hilbert space. The knowledge of the statistical density matrix operator would allow us to compute all average quantities in a conceptually similar, but mathematically different way. Let us suppose we have a set of wave functions |''Ψ''〉 which depend parametrically on a set of quantum numbers ''n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>''. The natural variable which we have is the amplitude with which a particular wavefunction of the basic set participates in the actual wavefunction of the system. Let us denote the square of this amplitude by ''p(n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>)''. The goal is to turn this quantity ''p'' into the classical density function in phase space. We have to verify that ''p'' goes over into the density function in the classical limit, and that it has [[ergodic]] properties. After checking that ''p(n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>)'' is a constant of motion, an ergodic assumption for the probabilities ''p(n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>)'' makes ''p'' a function of the energy only .
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| After this procedure, one finally arrives at the density matrix formalism when seeking a form where ''p(n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>)'' is invariant with respect to the representation used. In the form it is written, it will only yield the correct expectation values for quantities which are diagonal with respect to the quantum numbers ''n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>''.
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| Expectation values of operators which are not diagonal involve the phases of the quantum amplitudes. Suppose we encode the quantum numbers ''n<sub>1</sub>, n<sub>2</sub>, ..., n<sub>N</sub>'' into the single index ''i'' or ''j''. Then our wave function has the form
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| :<math>|\Psi \rangle \,=\,\sum_i a_i\, | \psi_i \rangle. </math>
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| The expectation value of an operator ''B'' which is not diagonal in these wave functions, so
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| :<math> \langle B\rangle \,=\,\sum_{i,j} a_i^{*}a_j\, \langle i| B |j\rangle.</math>
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| The role which was originally reserved for the quantities <math>|a_i|^2</math> is thus taken over by the density matrix of the system ''S''.
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| :<math> \langle j|\,\rho \, |i\rangle \,=\,a_j\, a_i^{*}.</math>
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| Therefore 〈''B''〉 reads
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| :<math> \langle B\rangle \,=\, Tr (\rho \, B) ~.</math>
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| The invariance of the above term is described by matrix theory. A mathematical framework was described where the expectation value of quantum operators, as described by matrices, is obtained by taking the trace of the product of the density operator ''ρ̂'' and an operator ''B̂'' (Hilbert scalar product between operators). The matrix formalism here is in the statistical mechanics framework, although it applies as well for finite quantum systems, which is usually the case, where the state of the system cannot be described by a [[Quantum state|pure state]], but as a statistical operator ''ρ̂'' of the above form. Mathematically, ''ρ̂'' is a positive, semidefinite hermitian matrix with unit trace.
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| ==Definition==
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| Given the density matrix ''ρ'', von Neumann defined the entropy<ref>[http://books.google.com/books?id=aA4vXMbuOTUC&pg=PA301 Geometry of Quantum States: An Introduction to Quantum Entanglement, by Ingemar Bengtsson, Karol Życzkowski, p301]</ref><ref name=Zachos/> as
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| :<math>S(\rho) \,=\,-\mathrm{Tr} (\rho \, {\rm \ln} \rho),</math>
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| which is a proper extension of the [[Entropy_(statistical_thermodynamics)#Gibbs_Entropy_Formula|Gibbs entropy]] (up to a factor {{math|''k''<sub>B</sub>}}) and the [[Shannon entropy]] to the quantum case. To compute ''S''(''ρ'') it is convenient (see [[logarithm of a matrix]]) to compute the [[Eigendecomposition_of_a_matrix|Eigendecomposition]] of <math>~\rho = \sum_j \eta_j |j\rangle\langle j|</math>. The von Neumann entropy is then given by
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| :<math>S(\rho) \,=\, - \sum_j \eta_j \ln \eta_j ~.</math>
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| Since, for a pure state, the density matrix is [[Idempotent matrix|idempotent]], ''ρ''=''ρ''<sup>2</sup>, the entropy ''S''(''ρ'') for it vanishes. Thus, if the system is finite (finite dimensional matrix representation), the entropy ''S''(''ρ'') quantifies ''the departure of the system from a pure state''. In other words, it codifies the degree of mixing of the state describing a given finite system.
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| Measurement [[decoherence|decoheres]] a quantum system into something noninterfering and [[Density_matrix#Entropy|ostensibly classical]]; so, e.g., the vanishing entropy of a pure state
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| ''Ψ''〉= (|0〉+|1〉) /√{{overline|2}}, corresponding to a density matrix
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| :<math>\rho = {1\over 2} \begin{pmatrix}
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| 1 & 1 \\
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| 1 & 1 \end{pmatrix} </math>
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| increases to ''S''=ln 2 =0.69 for the measurement outcome mixture
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| :<math>\rho = {1\over 2} \begin{pmatrix}
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| 1 & 0 \\
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| 0 & 1 \end{pmatrix} </math>
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| as the quantum interference information is erased.
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| ==Properties==
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| Some properties of the von Neumann entropy:
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| * {{math|''S''(''ρ'')}} is zero if and only if {{math|''ρ''}} represents a pure state.
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| * {{math|''S''(''ρ'')}} is maximal and equal to {{math|ln ''N''}} for a [[Quantum state|maximally mixed state]], {{math|''N''}} being the dimension of the [[Hilbert space]].
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| * {{math|''S''(''ρ'')}} is invariant under changes in the basis of {{math|''ρ''}}, that is, {{math|''S''(''ρ'') {{=}} ''S''(''UρU''<sup>†</sup>)}}, with {{math|''U''}} a unitary transformation.
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| * {{math|''S''(''ρ'')}} is ''concave, that'' is, given a collection of positive numbers {{math|''λ''<sub>''i''</sub>}} which sum to unity (<math>\Sigma_i \lambda_i = 1</math>) and density operators {{math|''ρ''<sub>''i''</sub>}}, we have
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| ::<math> S\bigg(\sum_{i=1}^k \lambda_i \, \rho_i \bigg) \,\geq\, \sum_{i=1}^k \lambda_i \, S(\rho_i). </math>
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| * {{math|''S''(''ρ'')}} is additive for independent systems. Given two density matrices {{math| ''ρ''<sub>''A''</sub> , ''ρ''<sub>''B''</sub>}} describing independent systems ''A'' and ''B'', we have
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| ::<math>S(\rho_A \otimes \rho_B)=S(\rho_A)+S(\rho_B)</math>. | |
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| * {{math|''S''(''ρ'')}} is strongly subadditive for any three systems ''A'', ''B'', and ''C'':
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| ::<math>S(\rho_{ABC}) + S(\rho_{B}) \leq S(\rho_{AB}) + S(\rho_{BC}).</math>.
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| :This automatically means that {{math|''S''(''ρ'')}} is subadditive:
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| ::<math>S(\rho_{AC}) \leq S(\rho_{A}) +S(\rho_{C}).</math>
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| Below, the concept of subadditivity is discussed, followed by its generalization to strong subadditivity.
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| ===Subadditivity=== | |
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| If {{math| ''ρ''<sub>''A''</sub> , ''ρ''<sub>''B''</sub>}} are the reduced density matrices of the general state {{math| ''ρ''<sub>''AB''</sub>}}, then
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| :<math> |S(\rho_A)\,-\,S(\rho_B)|\,\leq \, S(\rho_{AB}) \, \leq \, S(\rho_A)\,+\,S(\rho_B) ~. </math>
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| This right hand inequality is known as ''[[subadditivity]]''. The two inequalities together are sometimes known as the ''[[triangle inequality]].'' They were proved in 1970
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| by [[Huzihiro Araki]] and [[Elliott H. Lieb]].<ref> Huzihiro Araki and Elliott H. Lieb, ''Entropy Inequalities,''
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| Communications in Mathematical Physics, vol 18, 160–170 (1970).</ref> While in Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case, i.e., it is possible that {{math| ''S''(''ρ''<sub>''AB''</sub>) {{=}} 0}}, while {{math| ''S''(''ρ''<sub>''A''</sub>) {{=}} ''S''(''ρ''<sub>''B''</sub>) > 0}}.
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| Intuitively, this can be understood as follows: In quantum mechanics, the entropy of the joint system can be less than the sum of the entropy of its components because the components may be [[Quantum entanglement|entangled]]. For instance, as seen explicitly, the [[Bell state]] of two spin-½s,
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| :<math>| \psi \rangle = | \uparrow \downarrow \rangle + | \downarrow \uparrow \rangle </math> ,
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| is a pure state with zero entropy, but each spin has maximum entropy when considered individually in its [[Quantum_entanglement#Reduced_density_matrices|reduced density matrix]]. <ref>{{cite doi|10.1103/RevModPhys.75.715}}</ref> The entropy in one spin can be "cancelled" by being correlated with the entropy of the other. The left-hand inequality can be roughly interpreted as saying that entropy can only be canceled by an equal amount of entropy.
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| If system {{mvar|A}} and system {{mvar|B}} have different amounts of entropy, the lesser can only partially cancel the greater, and some entropy must be left over. Likewise, the right-hand inequality can be interpreted as saying that the entropy of a composite system is maximized when its components are uncorrelated, in which case the total entropy is just a sum of the sub-entropies. This may be more intuitive in the [[phase space formulation]], instead of Hilbert space one, where the Von Neumann entropy amounts to minus the expected value of
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| the <small>★</small>-logarithm of the [[Wigner quasi-probability distribution|Wigner function]] up to an offset shift.<ref name=Zachos>{{Cite journal
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| | last1 = Zachos | first1 = C. K. | title = A classical bound on quantum entropy | doi = 10.1088/1751-8113/40/21/F02 | journal = Journal of Physics A: Mathematical and Theoretical
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| | volume = 40 | issue = 21 | pages = F407 | year = 2007 |arxiv = hep-th/0609148 |bibcode = 2007JPhA...40..407Z }}</ref> Up to this normalization offset shift, the entropy is majorized by that of its [[classical limit]].
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| ===Strong subadditivity===
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| {{main|Strong Subadditivity of Quantum Entropy}}
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| The von Neumann entropy is also ''[[Strong Subadditivity of Quantum Entropy|strongly subadditive]].'' Given three [[Hilbert space]]s, ''A,B,C'',
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| :<math>S(\rho_{ABC}) \, + \, S(\rho_{B}) \, \leq \, S(\rho_{AB}) \,+\, S(\rho_{BC}).</math>
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| This is a more difficult theorem and was proved in 1973 by
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| [[Elliott H. Lieb]] and Mary Beth Ruskai,<ref> Elliott H. Lieb and
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| Mary Beth Ruskai, ''Proof of the Strong Subadditivity of Quantum-Mechanical Entropy,'' Journal of Mathematical Physics, vol 14, 1938–1941 (1973).</ref> using a
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| matrix inequality of [[Elliott H. Lieb]]<ref>
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| Elliott H. Lieb, ''Convex Trace Functions and the Wigner–Yanase–Dyson''
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| ''Conjecture,''
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| Advances in Mathematics, vol 67, 267–288 (1973).</ref> proved in
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| 1973. By using the proof technique that establishes the left side of the triangle inequality
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| above, one can show that the strong subadditivity inequality is equivalent to
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| the following inequality.
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| :<math>S(\rho_{A}) \, + \, S(\rho_{C}) \, \leq \, S(\rho_{AB}) \,+\, S(\rho_{BC})</math>
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| when {{math| ''ρ''<sub>''AB''</sub>}}, etc. are the reduced density matrices of a density matrix {{math| ''ρ''<sub>''ABC''</sub>}}. If we apply ordinary subadditivity to the left side of this inequality, and consider all permutations of ''A,B,C'', we obtain the ''[[triangle inequality]]'' for {{math| ''ρ''<sub>''ABC''</sub>}}: Each of the three numbers {{math|''S''( ''ρ''<sub>''AB''</sub>), ''S''( ''ρ''<sub>''BC''</sub>), ''S''( ''ρ''<sub>''AC''</sub>)}} is less than or equal to the sum of the other two.
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| ==Uses==
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| The von Neumann entropy is being extensively used in different forms ([[conditional entropy|conditional entropies]], [[relative entropy|relative entropies]], etc.) in the framework of quantum information theory.<ref>{{cite book|last=Nielsen|first=Michael A. and Isaac Chuang|title=Quantum computation and quantum information|year=2001|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0-521-63503-5|pages=700|edition=Repr.}}</ref> Entanglement measures are based upon some quantity directly related to the von Neumann entropy. However, there have appeared in the literature several papers dealing with the possible inadequacy of the [[Shannon information]] measure, and consequently of the von Neumann entropy as an appropriate quantum generalization of Shannon entropy. The main argument is that in classical measurement the Shannon information measure is a natural measure of our ignorance about the properties of a system, whose existence is independent of [[measurement]].
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| Conversely, quantum measurement cannot be claimed to reveal the properties of a system that existed before the measurement was made.<ref> Pluch, P. (2006). ''Theory for Quantum Probability,'' PhD Thesis, Klagenfurt University.</ref> This controversy has encouraged some authors to introduce the non-[[Additive function|additivity]] property of [[Tsallis entropy]] (a generalization of the standard Boltzmann–Gibbs entropy) as the main reason for recovering a true [[quantum information|quantal information]] measure in the quantum context, claiming that non-local correlations ought to be described because of the particularity of Tsallis entropy.
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| ==See also==
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| * [[Entropy (information theory)]]
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| * [[Linear entropy]]
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| * [[Partition function (mathematics)]]
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| * [[Quantum conditional entropy]]
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| * [[Quantum mutual information]]
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| * [[Quantum entanglement]]
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| * [[Strong Subadditivity of Quantum Entropy|Strong subadditivity of quantum entropy]]
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| ==References==
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| <references/>
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| [[Category:Quantum mechanical entropy]]
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