Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
22
questions
0
votes
1answer
14 views
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 ...
1
vote
0answers
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 $\|\...
3
votes
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 ...
4
votes
2answers
69 views
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 ...
4
votes
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 ...
3
votes
0answers
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}(\...
3
votes
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 \...
3
votes
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 ...
4
votes
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 $...
3
votes
1answer
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? ...
4
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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 ...
5
votes
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)}$...
3
votes
0answers
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 ...
0
votes
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 ...
2
votes
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 ...
4
votes
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 ...
3
votes
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:
$$
...
1
vote
0answers
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 ...
