Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Quantum mutual information
open-in-new
<h2 id="motivation">Motivation</h2> <p>For simplicity, it will be assumed that all objects in the article are finite-dimensional. </p><p>The definition of quantum mutual entropy is motivated by the classical case. For a probability distribution of two variables <i>p</i>(<i>x</i>, <i>y</i>), the two marginal distributions are </p> p ( x ) = ∑ y p ( x , y ) , p ( y ) = ∑ x p ( x , y ) . {\displaystyle p(x)=\sum _{y}p(x,y),\qquad p(y)=\sum _{x}p(x,y).} <p>The classical mutual information <i>I</i>(<i>X</i>:<i>Y</i>) is defined by </p> I ( X : Y ) = S ( p ( x ) ) + S ( p ( y ) ) − S ( p ( x , y ) ) {\displaystyle I(X:Y)=S(p(x))+S(p(y))-S(p(x,y))} <p>where <i>S</i>(<i>q</i>) denotes the <a href="/facts/Shannon_entropy/NLg4NLvt">Shannon entropy</a> of the probability distribution <i>q</i>. </p><p>One can calculate directly </p> S ( p ( x ) ) + S ( p ( y ) ) = − ( ∑ x p x log p ( x ) + ∑ y p y log p ( y ) ) = − ( ∑ x ( ∑ y ′ p ( x , y ′ ) log ∑ y ′ p ( x , y ′ ) ) + ∑ y ( ∑ x ′ p ( x ′ , y ) log ∑ x ′ p ( x ′ , y ) ) ) = − ( ∑ x , y p ( x , y ) ( log ∑ y ′ p ( x , y ′ ) + log ∑ x ′ p ( x ′ , y ) ) ) = − ∑ x , y p ( x , y ) log p ( x ) p ( y ) {\displaystyle {\begin{aligned}S(p(x))+S(p(y))&=-\left(\sum _{x}p_{x}\log p(x)+\sum _{y}p_{y}\log p(y)\right)\\&=-\left(\sum _{x}\left(\sum _{y'}p(x,y')\log \sum _{y'}p(x,y')\right)+\sum _{y}\left(\sum _{x'}p(x',y)\log \sum _{x'}p(x',y)\right)\right)\\&=-\left(\sum _{x,y}p(x,y)\left(\log \sum _{y'}p(x,y')+\log \sum _{x'}p(x',y)\right)\right)\\&=-\sum _{x,y}p(x,y)\log p(x)p(y)\end{aligned}}} <p>So the mutual information is </p> I ( X : Y ) = ∑ x , y p ( x , y ) log p ( x , y ) p ( x ) p ( y ) , {\displaystyle I(X:Y)=\sum _{x,y}p(x,y)\log {\frac {p(x,y)}{p(x)p(y)}},} <p>Where the logarithm is taken in basis 2 to obtain the mutual information in <a href="/facts/Bit/S972NSaD">bits</a>. But this is precisely the <a href="/facts/Relative_entropy/nh7SjlPE">relative entropy</a> between <i>p</i>(<i>x</i>, <i>y</i>) and <i>p</i>(<i>x</i>)<i>p</i>(<i>y</i>). In other words, if we assume the two variables <i>x</i> and <i>y</i> to be uncorrelated, mutual information is the <i>discrepancy in uncertainty</i> resulting from this (possibly erroneous) assumption. </p><p>It follows from the property of relative entropy that <i>I</i>(<i>X</i>:<i>Y</i>) ≥ 0 and equality holds if and only if <i>p</i>(<i>x</i>, <i>y</i>) = <i>p</i>(<i>x</i>)<i>p</i>(<i>y</i>). </p> <h2 id="definition">Definition</h2> <p>The quantum mechanical counterpart of classical probability distributions are modeled with <a href="/facts/Density_matrix/A1XdylJa">density matrices</a>. </p><p>Consider a quantum system that can be divided into two parts, A and B, such that independent measurements can be made on either part. The state space of the entire quantum system is then the <a href="/facts/Tensor_product_of_Hilbert_spaces/GSavLJfU">tensor product of the spaces for the two parts</a>. </p> H A B := H A ⊗ H B . {\displaystyle H_{AB}:=H_{A}\otimes H_{B}.} <p>Let <i>ρ</i><i>AB</i> be a density matrix acting on states in <i>H</i><i>AB</i>. The <a href="/facts/Von_Neumann_entropy/nzeJKowg">von Neumann entropy</a> of a density matrix S(<i>ρ</i>), is the quantum mechanical analogy of the Shannon entropy. </p> S ( ρ ) = − Tr ρ log ρ . {\displaystyle S(\rho )=-\operatorname {Tr} \rho \log \rho .} <p>For a probability distribution <i>p</i>(<i>x</i>,<i>y</i>), the marginal distributions are obtained by integrating away the variables <i>x</i> or <i>y</i>. The corresponding operation for density matrices is the <a href="/facts/Partial_trace/ra7RhkkT">partial trace</a>. So one can assign to <i>ρ</i> a state on the subsystem <i>A</i> by </p> ρ A = Tr B ρ A B {\displaystyle \rho ^{A}=\operatorname {Tr} _{B}\;\rho ^{AB}} <p>where Tr<i>B</i> is partial trace with respect to system <i>B</i>. This is the reduced state of <i>ρAB</i> on system <i>A</i>. The reduced von Neumann entropy of <i>ρAB</i> with respect to system <i>A</i> is </p> S ( ρ A ) . {\displaystyle \;S(\rho ^{A}).} <p><i>S</i>(<i>ρB</i>) is defined in the same way. </p><p>It can now be seen that the definition of quantum mutual information, corresponding to the classical definition, should be as follows. </p> I ( A : B ) := S ( ρ A ) + S ( ρ B ) − S ( ρ A B ) . {\displaystyle \;I(A\!:\!B):=S(\rho ^{A})+S(\rho ^{B})-S(\rho ^{AB}).} <p>Quantum mutual information can be interpreted the same way as in the classical case: it can be shown that </p> I ( A : B ) = S ( ρ A B ‖ ρ A ⊗ ρ B ) {\displaystyle I(A\!:\!B)=S(\rho ^{AB}\|\rho ^{A}\otimes \rho ^{B})} <p>where S ( ⋅ ‖ ⋅ ) {\displaystyle S(\cdot \|\cdot )} denotes <a href="/facts/Quantum_relative_entropy/cdoXwgDh">quantum relative entropy</a>. Note that there is an alternative generalization of mutual information to the quantum case. The difference between the two for a given state is called <a href="/facts/Quantum_discord/9FN0NUH4">quantum discord</a>, a measure for the quantum correlations of the state in question. </p> <h2 id="properties">Properties</h2> <p>When the state ρ A B {\displaystyle \rho ^{AB}} is pure (and thus S ( ρ A B ) = 0 {\displaystyle S(\rho ^{AB})=0} ), the mutual information is twice the <a href="/facts/Entanglement_entropy/yLSXsIMA">entanglement entropy</a> of the state: </p> I ( A : B ) = S ( ρ A ) + S ( ρ B ) − S ( ρ A B ) = S ( ρ A ) + S ( ρ B ) = 2 S ( ρ A ) {\displaystyle I(A\!:\!B)=S(\rho ^{A})+S(\rho ^{B})-S(\rho ^{AB})=S(\rho ^{A})+S(\rho ^{B})=2S(\rho ^{A})} <p>A positive quantum mutual information is not necessarily indicative of entanglement, however. A classical mixture of <a href="/facts/Separable_state/j0k8VWmF">separable states</a> will always have zero entanglement, but can have nonzero QMI, such as </p> ρ A B = 1 2 ( | 00 ⟩ ⟨ 00 | + | 11 ⟩ ⟨ 11 | ) {\displaystyle \rho ^{AB}={\frac {1}{2}}\left(|00\rangle \langle 00|+|11\rangle \langle 11|\right)} I ( A : B ) = S ( ρ A ) + S ( ρ B ) − S ( ρ A B ) = S ( 1 2 ( | 0 ⟩ ⟨ 0 | + | 1 ⟩ ⟨ 1 | ) ) + S ( 1 2 ( | 0 ⟩ ⟨ 0 | + | 1 ⟩ ⟨ 1 | ) ) − S ( 1 2 ( | 00 ⟩ ⟨ 00 | + | 11 ⟩ ⟨ 11 | ) ) = log 2 + log 2 − log 2 = log 2 {\displaystyle {\begin{aligned}I(A\!:\!B)&=S(\rho ^{A})+S(\rho ^{B})-S(\rho ^{AB})\\&=S\left({\frac {1}{2}}(|0\rangle \langle 0|+|1\rangle \langle 1|)\right)+S\left({\frac {1}{2}}(|0\rangle \langle 0|+|1\rangle \langle 1|)\right)-S\left({\frac {1}{2}}(|00\rangle \langle 00|+|11\rangle \langle 11|)\right)\\&=\log 2+\log 2-\log 2=\log 2\end{aligned}}} <p>In this case, the state is merely a <a href="/facts/Correlation/egkluAEm">classically correlated</a> state. </p>