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| The '''joint quantum entropy''' generalizes the classical [[joint entropy]] to the context of [[quantum information theory]]. Intuitively, given two [[quantum state]]s <math>\rho</math> and <math>\sigma</math>, represented as [[density operator]]s that are subparts of a quantum system, the joint quantum entropy is a measure of the total uncertainty or [[entropy]] of the joint system. It is written <math>S(\rho,\sigma)</math> or <math>H(\rho,\sigma)</math>, depending on the notation being used for the [[von Neumann entropy]]. Like other entropies, the joint quantum entropy is measured in [[bit]]s, i.e. the logarithm is taken in base 2.
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| In this article, we will use <math>S(\rho,\sigma)</math> for the joint quantum entropy.
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| ==Background==
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| In [[information theory]], for any classical [[random variable]] <math>X</math>, the classical [[Shannon entropy]] <math>H(X)</math> is a measure of how uncertain we are about the outcome of <math>X</math>. For example, if <math>X</math> is a probability distribution concentrated at one point, the outcome of <math>X</math> is certain and therefore its entropy <math>H(X)=0</math>. At the other extreme, if <math>X</math> is the uniform probability distribution with <math>n</math> possible values, intuitively one would expect <math>X</math> is associated with the most uncertainty. Indeed such uniform probability distributions have maximum possible entropy <math>H(X) = \log_2(n)</math>.
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| In [[quantum information theory]], the notion of entropy is extended from probability distributions to quantum states, or [[density matrix|density matrices]]. For a state <math>\rho</math>, the [[von Neumann entropy]] is defined by
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| :<math>- \operatorname{Tr} \rho \log \rho.</math> | |
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| Applying the [[spectral theorem]], or [[Borel functional calculus]] for infinite dimensional systems, we see that it generalizes the classical entropy. The physical meaning remains the same. A [[maximally mixed state]], the quantum analog of the uniform probability distribution, has maximum von Neumann entropy. On the other hand, a [[pure state]], or a rank one projection, will have zero von Neumann entropy. We write the von Neumann entropy <math>S(\rho)</math> (or sometimes <math>H(\rho)</math>.
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| == Definition ==
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| Given a quantum system with two subsystems ''A'' and ''B'', the term '''joint quantum entropy''' simply refers to the von Neumann entropy of the combined system. This is to distinguish from the entropy of the subsystems.
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| In symbols, if the combined system is in state <math>\rho^{AB}</math>,
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| the joint quantum entropy is then
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| :<math>S(\rho^A,\rho^B) = S(\rho^{AB}) = -\operatorname{Tr}(\rho^{AB}\log(\rho^{AB})).</math>
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| Each subsystem has it own entropy. The state of the subsystems are given by the [[partial trace]] operation.
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| ==Properties== | |
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| The classical joint entropy is always at least equal to the entropy of each individual system. This is not the case for the joint quantum entropy. If the quantum state <math>\rho^{AB}</math> exhibits [[quantum entanglement]], then the entropy of each subsystem may be larger than the joint entropy. This is equivalent to the fact that the conditional quantum entropy may be negative, while the classical conditional entropy may never be.
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| Consider a [[maximally entangled state]] such as a [[Bell state]]. If <math>\rho^{AB}</math> is a Bell state, say,
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| :<math>\left| \Psi \right\rangle = \frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right),</math> | |
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| then the total system is a pure state, with entropy 0, while each individual subsystem is a maximally mixed state, with maximum von Neumann entropy <math>\log 2 = 1</math>. Thus the joint entropy of the combined system is less than that of subsystems. This is because for entangled states, definite states cannot be assigned to subsystems, resulting in positive entropy.
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| Notice that the above phenomenon cannot occur if a state is a separable pure state. In that case, the reduced states of the subsystems are also pure. Therefore all entropies are zero.
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| ==Relations to other entropy measures== | |
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| The joint quantum entropy <math>S(\rho^{AB})</math> can be used to define of the [[conditional quantum entropy]]:
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| :<math>S(\rho^A|\rho^B) \ \stackrel{\mathrm{def}}{=}\ S(\rho^A,\rho^B) - S(\rho^B)</math>
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| and the [[quantum mutual information]]:
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| :<math>I(\rho^A:\rho^B) \ \stackrel{\mathrm{def}}{=}\ S(\rho^A) + S(\rho^B) - S(\rho^A,\rho^B)</math>
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| These definitions parallel the use of the classical [[joint entropy]] to define the [[conditional entropy]] and [[mutual information]].
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| == See also ==
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| *[[Quantum relative entropy]]
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| *[[Quantum mutual information]]
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| ==References==
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| * Nielsen, Michael A. and Isaac L. Chuang, ''Quantum Computation and Quantum Information''. Cambridge University Press, 2000. ISBN 0-521-63235-8
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| {{DEFAULTSORT:Joint Quantum Entropy}}
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| [[Category:Quantum mechanical entropy]]
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| [[Category:Quantum information theory]]
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