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Questions tagged [mutual-information]

For questions about the quantum mutual information, a measure of correlations between subsystems of a quantum state. Can also be used for questions about classical mutual information, as long as the general question is of relevance to quantum information science.

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What are examples of states saturating the strong subadditivity of the von Neumann entropy?

A well-known property of classical distribution is that they satisfy strong subadditivity, meaning that for any tripartite joint probability distribution $p(x,y,z)$, we have the inequality $$H(AB)+H(...
glS's user avatar
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Mutual information between Alice and Eve in a BB84 intercept resend attack

I'm new to information theory and i need to calculate $I(A,E)$. To calculate it I need conditional entropy $H(A|E)$. I assume the BB84 protocol standard states $\{ |0\rangle,|1\rangle \},\{|+\rangle,|-...
forgetfuled's user avatar
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Mutual information of shared state is larger than expectation values

Im trying to prove the following identity for a special case: Alice and Bob share the Bell state \begin{align*} |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle). \end{align*} Consider the ...
user27450's user avatar
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1 answer
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Does quantum mutual information encompass information only about quantum correlations, or does it encompass both classical and quantum correlations?

I am confused about what quantum mutual information gives us. Does it give all kinds of quantum correlations? Or does it give all kinds of quantum and classical correlations? If it consists of ...
reza's user avatar
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1 answer
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Is quantum mutual information an entanglement measure?

For a bipartite system, the quantum mutual information is defined via the Von Neumann entropy as follows: $$I(A:B)=S(A)+S(B)-S(AB).$$ It's always positive. Is it an entanglement measurement? Also how ...
reza's user avatar
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2 votes
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General structure of the state with $I(A:B|C)_{\rho}{=}2 \log_2 \{\min (d_A, d_B)\}$

The conditional quantum mutual information (CQMI) of a state $\rho^{ABC}$ respects the dimension bound $I(A:B|C)_{\rho}{\leq}2 \log_2 \{\min (d_A, d_B)\}$ (Mark Wilde's book, exercise 11.7.9). One ...
Abir's user avatar
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2 answers
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How to evaluate the mutual information of a classical-quantum state?

Suppose I have the state $\sum_i p_i \vert i\rangle\langle i\vert \otimes\rho_i$ on subsystems $X$ (classical part) and $B$ (quantum part) and I wish to evaluate the mutual information $I(X:B) = S(X) +...
JRT's user avatar
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How can $\chi(\hat{A},\hat{B}:C) \le \chi(\hat{A},B:C)$ be true?

The holevo information of $\rho_{ABC}$ w.r.t to measurements on A and B (for the sake of this we'll assume local measurements suffice), is given by $$\chi(\hat{A},\hat{B}:C)$$ where $\hat{A}$ and $\...
GaussStrife's user avatar
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Partially smoothed max-information and AEP - where's the flaw in my logic?

Sorry for the defintional overload but I promise there's an interesting question at the end! Please skip to the last section if you're familiar with one-shot information theory. Defintions The max-...
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Comparison of Quantum Mutual Information and Coherent Information with Classical Mutual Information

Between quantum mutual information and coherent information, which one is more similar to classical mutual information? I understand that both measures have some similarities to classical mutual ...
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What are examples where the quantum discord is achieved by a non-projective POVM?

Consider the (asymmetric) quantum discord, defined as (borrowing notation from Eq. 4.13c of Zurek's review): $$\mathcal D(\mathcal S:\mathcal A) = I(\mathcal S:\mathcal A) - \chi(\rho_{\mathcal A}),$$ ...
glS's user avatar
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1 answer
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Question regarding the monotonicity of Mutual Information of a tripartite state under multiple non-local commuting unitaries

Given a system $\rho_{AB}\otimes\rho_{C}$, and a unitary interaction $U_{BC}$, due to the monotonicity of the relative entropy under the actions of the partial trace map, $$I(A:B)=I(A:U_{BC}(B,C))\ge ...
GaussStrife's user avatar
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Can separable states have quantum mutual information larger than one?

Consider bipartite (qubit) systems. The classical mutual information between a pair of binary registers, $$I(X:Y)\equiv H(X)+H(Y)-H(X,Y),$$ is always lesser than $1$ (and non-negative). On the other ...
glS's user avatar
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399 views

Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$

Suppose I have a tripartite system $ABC$ in a pure state $|\psi_{ABC}\rangle$ with mutual information $I(A:C)=0$. This implies that the reduced density matrix $\rho_{AC}$ factorizes as $\rho_{AC} = \...
nervxxx's user avatar
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How to understand Clifford+T from a quantum information theory perspective [closed]

I'd like to study the important operator set Clifford+T from a quantum information theory perspective. The Clifford+T set is universal and it is important because allows for efficient error correction ...
Daniele Cuomo's user avatar
2 votes
0 answers
573 views

What does a quantum mutual information larger than its classical upper bound represent?

Let $\rho$ be a bipartite state. Its quantum mutual information is defined as $$\newcommand{\tr}{\operatorname{tr}}I(\rho) = S(\tr_B(\rho)) + S(\tr_A(\rho)) - S(\rho),$$ where $S(\sigma)$ is the von ...
glS's user avatar
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How is $I(\rho^{QC})=I_{CC}(\rho^{QC})$

On page 3 of this paper, for the proof of theorem 1, it states that, using Lemma 2 from the previous page, that if $$I(\Lambda_{A}\otimes\Gamma_{B})[\rho]=I(\rho))$$ then there exists $\Lambda_{A}^{*}$...
GaussStrife's user avatar
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How to prove that the mutual information is subadditive?

Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
user15135's user avatar
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2 answers
231 views

How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?

I was reading Wilde's 'Quantum Information Theory' and saw the following theorem at chapter 11 $(11.7.2)$: $$ I(A; B | C) \ge 0, $$ where, $$ I(A;B|C) := H(A|C) + H(B | C) - H(AB|C). $$ I know that ...
QuestionEverything's user avatar
4 votes
1 answer
204 views

What does vanishishing mutual information of the Choi imply about the channel?

Classically, if the mutual information between the input and output of some channel or circuit $= 0$, it means the output is independent of the input, and the circuit is in a way 'useless'. For the ...
Dina Abdelhadi's user avatar
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1 answer
612 views

What exactly is the relation between the Holevo quantity and the mutual information?

On this page, it is stated that the Holevo quantity is an upper bound to the accessible information of a quantum state. In the scenario where Alice encodes classical information into a quantum state ...
user1936752's user avatar
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6 votes
1 answer
493 views

Does computing the quantum mutual information $I(\rho^{AB})$ require full tomographic information of $\rho^{AB}$?

In the discussions about quantum correlations, particularly beyond entanglement (discord, dissonance e.t.c), one can often meet two definitions of mutual information of a quantum system $\rho^{AB}$: ...
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