# Questions tagged [information-theory]

The tag is used for questions connected with information theory in classical and/or quantum sense.

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### Does ${\rm tr}(\Pi_z\rho\Pi_z)\le p$ imply $\cal E(\rho)$ and $\cal E(\Pi_{-z}\rho\Pi_{-z})$ are close in trace distance?

Suppose I have a quantum operation $\mathcal{E}$ and a state $\rho$ such that: $$\operatorname{tr}(\Pi_z \rho \Pi_z) \le p$$ for some probability $p$ and some projection $\Pi_z$ onto some subspace ...
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### Prove that autocompatible LCPT maps are antidegradable

A LCPT map $\Phi$ is antidegradable if and only if it is compatible with itself. The proof that an antidegradable map (satisfying $\Phi=\Lambda_E\circ \tilde\Phi$ where $\tilde\Phi$ is the ...
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### How does the Kraus decomposition imply the Stinespring representation?

To show that the Kraus decomposition $\Phi(\rho)=\sum_{k=1}^D M_k\rho_S M_k^\dagger$ implies the Stinespring form $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
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### Why do we want the no error limit to be 1?

In a textbook by Nielsen and Chuang, there's the following paragraph: The idea of quantum data compression is that the compressed data should be recovered with very good fidelity. Think of the ...
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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, ...
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### Is the quantum mutual information variance bounded from above?

The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be $$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$ where $D(\rho\|\sigma)$...
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### How can the entropy of quantum states increase after projective measurements?

I'm reading Nielsen and chuang 11.3.3 Measurements and Entropy. It says after measurement, one's entropy increases. How is this possible? Shouldn't measurement decrease one's uncertainty?
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### Can one quantify entanglement between different parts of a system?

Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
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### 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 ...
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### Deriving the depolarizing channel

Consider a circuit built as follows: take two ancilla states and an operator $U$ made of a series of controlled gates which act on a pure state $\rho$ as follows: $X$ if the ancilla is in $|00\rangle$...
I recently came to know that there is a connection between Bures Fidelity $(F_B)$ and Quantum Fisher Information $(F_Q)$ given by [F_{B}(\rho, \rho_\theta)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + ...
Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a ...