Questions tagged [information-theory]

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

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What is the difference between classical-quantum and completely classical states?

States that are completly classical : $$ \begin{aligned} \tilde\rho_{A B} & =\sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p_{X, Y}(x, y)(|x\rangle \otimes|y\rangle)(\langle x| \otimes\langle ...
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Classical capacity of quantum channel - Holevo quantity vs accessible information of a channel

Just above Eq (20.7) in Mark Wilde's book, while discussing the classical capacity of a quantum channel, he says: These results then suggest that the ultimate classical capacity of the channel is the ...
user1936752's user avatar
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quantum generalisation of random variables

What is the quantum information equivalent of a classical probability random variable ? Is it a density matrix or an observable ? If so can someone show me how to write a random variable that follows ...
yosh's user avatar
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Is Classical-Classical-Quantum state equivalent to Classical-Quantum state?

Suppose we have the following CQ-state between two parties Alice & Bob $$ \rho_{A B}^{\otimes n}=\sum_{x^n} p^n\left(x^n\right)\left|x^n\right\rangle\langle\left. x^n\right|^A \otimes \rho_{x^n}^B ...
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How to compute the QFI of a thermal state?

Let $\rho=\frac{1}{Z}\exp(-\beta H)$ be the thermal state associated to the Hamiltonian $$H=\hbar\omega\sum_i\left( a_i^\dagger a_i+\frac12\right).$$ I wonder how the quantum Fisher information of ...
Noobgrammer's user avatar
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Rabi amplitude in theory vs practice

I am a bit confused with the definition of Rabi amplitude and how it relates to experimental values. If I understand correctly, we can show a rabi driving (waveform) with the following formula: $\...
Rex's user avatar
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What are necessary and sufficient conditions for the output of a parametrized unitary $U(\theta)$ to be smooth?

Let us consider a unitary $U$ parameterised by $\theta \in \mathbb{R}$, i.e, $U(\theta)$. What are the necessary and sufficient conditions for the output states of this unitary to be smooth? One ...
Song of Physics's user avatar
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Coherent information is a lower bound of channel capacity. What about coherent information based on Renyi entropies?

It is known that coherent information defined in terms of von Neumann entropies is a lower bound of quantum channel capacity. If we define coherent information in terms of $\alpha$-Renyi entropies, ...
MonteNero's user avatar
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An Inequality related to the Trace Norm

Let $\rho, \sigma$ be two states of a qubit, and let $U$ be the $CX_{12}$-gate (control on 1st qubit, target on 2nd qubit). Prove that, for an arbitrary CPTP Map $\mathcal{E}$, $$ ||\mathcal{E}(\rho -...
Sowmitra Das's user avatar
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How to discriminate between $N$ states drawn from one of two ensembles?

Consider the following quantum discrimination problem: Suppose, there are two sets of states, $P = \{ \rho_i \}$ and $Q = \{\sigma_i\}$. Both Alice and Bob know which states are in each set. We can ...
Sowmitra Das's user avatar
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Some mathematical problem of calculating distance

Suppose we have POVM $E = A^\dagger A$. $E$ satisfy: $\forall i,j$, $\lambda_i,\lambda_j$ are eigenvalues of $E$, and $\lambda_i\le e^\epsilon \lambda_j$ holds. Then we can decompose $E=U^\dagger DU$, ...
Zehong Fan's user avatar
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Increasing quantum channel capacity

I have a few confusions regarding quantum communication. Please bear with me as I ask what might appear to be basic or naive questions: In the case of classical communication the capacity of wireless ...
ZE1's user avatar
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Defining quantum memory mathematically

Is it possible that we may define the notion of quantum memory in abstract manner, that is, mathematically? A definition which is hardware independent that may be treated as a mathematical object. We ...
madeel's user avatar
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Quantum relative entropy between pre- and post-measurement states

The quantum relative entropy between the states $\rho$ and $\sigma$ is defined by $$D(\rho||\sigma)= \textrm{tr}\Big(\rho \big(\log\rho - \log \sigma \big) \Big)\,,$$ as long as the support of $\rho$ ...
quantum_theo's user avatar
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Is Klein's inequality due to Klein?

You may be familiar with "Klein's inequality"; one form of it is $$ -\operatorname{tr}(\rho \log \sigma) + \operatorname{tr}(\rho \log \rho) \ge 0, $$ stating that relative entropy is ...
echinodermata's user avatar
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What is the exact mathematical content of the no-teleportation theorem?

My question concerns the following theorem (not to be confused with quantum teleportation protocols): what is the exact mathematical content of this theorem? After all any (pure) quantum state (say in ...
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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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What is the classical Fisher information of a parametrized coherent state $|\alpha_\theta\rangle$?

Suppose $|\alpha_\theta\rangle$ is a coherent state depending on the real parameter $\theta$. What is the classical Fisher information it carries? There are explicit formulae for the quantum Fisher ...
Bentanglement's user avatar
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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) +...
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How to justify the conclusion $|E_{sq}(\rho)-E_{sq}(\sigma)|\le f(\epsilon)$, when proving the continuity of the squashed entanglement?

I am following the paper by Christandl and Winter introducing squashed entanglement. My question is particularly on the continuity proof of squshed entanglement mentioned after conjecture 14 and ...
Abir's user avatar
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G-twisted Pauli twirl circuit

Pauli twirls are obtained by taking a unitary $U$, and finding some Pauli gates $P_1, P_2$ such that $P_1 U P_2$. So, for example, one possible twirl of the $S$ gate would be $YSX$. In the paper ...
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Average-case error and worst-case error in entangment-assisted classical communication

I am reading this paper where one considers a quantum channel $\mathcal{N}$. We are interested in the maximum rate at which we can send classical messages over this channel in the one-shot setting i.e....
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What are the antisymmetric terms in $\sigma_{mn}$ in the expression for the Fisher information?

Given a parameter-dependent density operator $\hat\rho^\lambda$ and its spectral decomposition $\{\rho_m^\lambda, |\psi_n^\lambda\rangle\}$, Eq. $(17)$ from this review shows that one can compute its ...
Bentanglement's user avatar
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Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?

In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says: A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i, The ...
Guangliang's user avatar
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Dense coding for PR boxes

We can think of an entangled state as a resource for accomplishing various tasks. For example, passing a CHSH test, performing a dense coding operation (when enhanced by transmission of a single qubit)...
DaftWullie's user avatar
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What is the conditional min-entropy of a pure bipartite state?

In this paper, it is stated that the conditional min-entropy $H(A|B)_{\rho_{AB}}$ of $A$ conditioned on $B$ for any $\textbf{pure}$ quantum system $\rho_{AB}=|\psi_{AB} \rangle \langle \psi_{AB} |$ is ...
quantum_theo's user avatar
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Clarification about inverses in sandwiched Renyi divergence

The sandwiched Renyi divergence is defined as in $$ \tilde{D}_\alpha(\rho\|\sigma):=\frac{1}{\alpha−1}\log tr[(\sigma^{\frac{1−\alpha}{2\alpha}}\rho \sigma^{\frac{1−\alpha}{2 \alpha }})^\alpha] $$ The ...
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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 $\...
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Twirling of quantum states: Maximally entangled states

I have been reading the paper "Resource theory of unextendibility and non-asymptotic quantum capacity" (https://arxiv.org/pdf/1803.10710.pdf) by Kaur et.al, I have two questions ...
Newuser7's user avatar
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Optimal one-shot input for $\mathcal{N}^{\otimes n}$ and optimal one-shot input for $\mathcal{N}^{\otimes k}$ where $k< n$?

Suppose I have a classical or quantum channel $\mathcal{N}$. I wish to use it for some communication task in the one-shot setting i.e. where I have $n$ i.i.d. copies of $\mathcal{N}$ denoted by $\...
user1936752's user avatar
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Which quantum entropies are meaningful with respect to continuous distributions of states?

When using a quantum channel to transmit classical information, we consider an ensemble $\mathcal{E} = \{(\rho_x, p(x))\}$ consisting of states $\rho_x$ labelled with a symbol $x$ from a finite ...
forky40's user avatar
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Quantum Relative entropy- the math and intuition

I am new to quantum information theory and have been reading Mark Wilde's notes on quantum relative entropy. http://www.markwilde.com/teaching/2015-fall-qit/lectures/lecture-19.pdf I have three basic ...
Newuser7's user avatar
2 votes
0 answers
81 views

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-...
user1936752's user avatar
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2 votes
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Why is the quantum capacity quantified by the coherent information?

Most types channel capacities associated to a given quantum channel are quantified using mutual informations (sometimes classical, sometimes quantum, sometimes regularised), which is not surprising ...
glS's user avatar
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Physical interpretation of a 2-photon-qubit system occupying the antisymmetric Bell state

This question regards the reconciliation of QIT with what I have learned separately in lectures about quantum physics. My understanding is that indistinguishable bosons are always described by ...
symmetry-question-haver's user avatar
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Does the quantum relative entropy have a direct operational interpretation?

Consider the quantum relative entropy, defined as $$D(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma),$$ for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS's user avatar
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1 answer
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Does the max-relative entropy satisfy $0 < D_{\max}(\rho \parallel I_A \otimes \sigma_B) < 1$?

The quantum conditional min-entropy is defined as $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right)$$ where in general $$D_{\max}(\rho \...
Josh's user avatar
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Why can the max-relative entropy be written as $D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$?

The quantum conditional min-entropy is defined as $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right),$$ where $$D_{\max}(\rho \parallel I_A \...
Josh's user avatar
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1 answer
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Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$

I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent. $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B ...
Josh's user avatar
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1 vote
2 answers
107 views

How to measure an unknown state produced by a source of qubits?

What kind of experiment can allow me to measure an unknown state produced by a source of qubits? For example: the state of photon polarization. But it can be another one. I have no information about ...
Dimitri's user avatar
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Numerical optimization over separable measurements

For a set of bipartite density operators $\{\rho_a\}_{a=1}^m \subset D(\mathcal{X} \otimes \mathcal{Y})$ each associated with a probability $p(a)$, an optimal separable measurement is a POVM $\{ \...
forky40's user avatar
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4 votes
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Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy

The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is $$ S(A,B)\ge|S(A)-S(B)| $$ this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
Sooraj S's user avatar
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1 answer
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Why can't Quantum Fisher Information be negative?

Quantum Fisher Information is proportional to Fidelity susceptibility. Mathematically the equation is: $QFI=-\frac{\partial^2 d_B(\epsilon) }{\partial \epsilon^2}$ where above equation shows QFI is ...
Chetan Waghela's user avatar
3 votes
1 answer
191 views

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 ...
Josh's user avatar
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1 vote
0 answers
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Deriving the choi matrix definition of the quantum depolarizing channel

I was reading up on the quantum depolarizing channel (Preskill's Notes) (stack exchaange explanation), and I'm failing to see how the form \begin{align} \sigma &= (\mathcal E \otimes \mathbb I)(|\...
Apple Meson's user avatar
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35 views

Understanding adversarial Channels

The paper here (Definition 1) defines adversarial channels as $N(\rho)= \sum_i A_i \rho A_i$ with the mention that the $A_i$ is chosen only after a communication strategy is decide. This gives the ...
Root's user avatar
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5 votes
1 answer
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Why is error correction very different for circuits compared to channels?

Background Suppose one wishes to communicate information using a noisy channel $N$ instead of an ideal channel $I$. The general framework to communicate reliably is Take $n$ copies of $N$. For some ...
user1936752's user avatar
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6 votes
1 answer
312 views

Can $2^n$ bits be sent with $n$ instances of quantum teleportation?

So, right now these are two pieces of information I've been told are correct: Quantum teleportation can send a single qubit from Alice to Bob, with two classical bits $n$ qubits can store $2^n$ ...
Radvylf Programs's user avatar
2 votes
1 answer
112 views

Under what conditions does entanglement distillation work?

The picture below shows the two protocols to distill EPR pairs and enhance the fidelity. EDIT: I am trying to understand under what conditions such a protocol achieve the task, when repetead a number ...
Daniele Cuomo's user avatar
2 votes
2 answers
186 views

How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?

The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order. In the proof given in Box 11.2, ...
Sooraj S's user avatar
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