Questions tagged [information-theory]

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

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Max-entropy - relationship between two definitions

On Wikipedia, the max-entropy for classical systems is defined as $$H_{0}(A)_{\rho}=\log \operatorname{rank}\left(\rho_{A}\right)$$ The term max-entropy in quantum information is reserved for the ...
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34 views

Quantum marginal problem - constructing a global state from reduced states

Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
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2answers
78 views

Checking whether a state is almost orthogonal to permutation invariant states

Let us consider \begin{equation} |T\rangle = |\psi \rangle^{\otimes m} \end{equation} for an $n$-qubit quantum state $|\psi\rangle$. Let $\mathcal{V}$ be the space of all $(m + 1)$-partite states that ...
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What is the difference between no signaling and non locality at operational and ontological level?

I understand the basic definitions. Locality means Alice's measurements do not affect Bob's and system and that no-signalling means a party can't send information faster than light. I also know that ...
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2answers
70 views

Quantum relative entropy with respect to a pure state

I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state ...
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25 views

Relative entropy inequality for many copies of a channel

Suppose we have two quantum channels $\mathcal{E}_{A\rightarrow B}, \mathcal{F}_{A\rightarrow B}$ that satisfy $$D(\mathcal{E}(\rho_A)\|\mathcal{E}(\sigma_A))\geq D(\mathcal{F}(\rho_A)\|\mathcal{F}(\...
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1answer
42 views

The proof of monotonicity of fidelity for channels and its meaning

I have two questions regarding the exercise 9.2.8 of Quantum information by Wilde, which is as follows: Let $\rho,\sigma \in \mathcal{D}(\mathcal{H}_A)$ and let $\mathcal{N: L(H}_A)\rightarrow \...
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1answer
50 views

Non-lockability of quantum max-entropy

Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The ...
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3answers
158 views

Prove that Shannon and von Neumann entropies satisfy $H(P)\ge S(\rho)$ with $P$ diagonal of $\rho$

Suppose there is some $n$-qubit state $\rho$. It is well known fact that, given some orthonormal basis $U = \{|u_i\rangle\}$, if $p_i = \langle u_i| \rho |u_i \rangle$ (that is, measuring $\rho$ with $...
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47 views

How can one estimate the von Neumann entropy of an unknown quantum state?

Given many copies of some unknown quantum state $\rho$, I would like to compute its von Neumann entropy $S(\rho)$. What algorithm could be used for this that minimizes the number of copies required? ...
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2answers
616 views

Can classical linear algebra solvers implement quantum algorithms with similar speed-ups?

A quantum algorithm begins with a register of qubits in an initial state, a unitary operator (the algorithm) manipulates the state of those qubits, and then the state of the qubits is read out (or at ...
4
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1answer
135 views

How can the Holevo bound be used to show that $n$ qubits cannot transmit more than $n$ classical bits?

The inequality $\chi \le H(X)$ gives the upper bound on accessible information. This much is clear to me. However, what isn't clear is how this tells me I cannot transmit more than $n$ bits of ...
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2answers
55 views

How is a bit field represented in Quantum Computing?

For example, a computer represent a variable named "A" as 01000001. How does a quantum computer represent "A"? I am a newbie having difficulty understanding quantum computers. I ...
4
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1answer
69 views

Definition of quantum min-relative entropy

In John Watrous' lectures, he defines the quantum min-relative entropy as $$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$ where $F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$. Here, I use this ...
4
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1answer
78 views

Mutual information of Choi state=0, what would that imply about the quantum 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 ...
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2answers
52 views

Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$

Given two 1-qubit rotations $U=R_n (\theta)$ and $V=R_m(\phi)$ with $n$ and $m$ vectors defining a rotation and $\theta, \phi$ angles, define $D(U,V)=Tr(|U-V|)$ where $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
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54 views

Schumacher compression - comparing with Shannon compression

Background Shannon's source coding theorem tells us the following. We shall consider a binary alphabet for simplicity. Suppose Alice has $n$ independent and identically distributed instances of a ...
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1answer
101 views

Can every process in nature be simulated by a Turing Machine or a quantum computer?

Given any initial condition or value A, A leads to B after a procedure of physics or nature P. Now is there any turing machine or quantum computer that can simulates P,converting A into B? In other ...
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1answer
151 views

What is the Von Neumann entropy of $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$?

Let $\overline{p}$ be a probability distribution on $\{1,....,d\}$. Then let $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$. How should I take the Von-Neumann entropy of $\rho$? I know that ...
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2answers
210 views

Nielsen and Chuang ex 2.73

I've been trying to solve exercise 2.73 (p.g 105), and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'm just wrong! Ex ...
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2answers
147 views

What does it mean to take the Choi-Jamiolkowski of a quantum channel?

The Choi-Jamiolkowski of a channel $\newcommand{\on}[1]{\operatorname{#1}}\Lambda : \on{End}(\mathcal{H_A}) \xrightarrow{} \on{End}(\mathcal{H_B})$ is obtained through an isomorphism of the form: $$ ...
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29 views

Impossibility of QRAC with random question, or more general proof of impossibility

TL;DR: Is it known that QRAC where the "question" is random instead of chosen are impossible? More generally what are the methods to prove impossibility games? I often face problems for which I want ...