Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
61
questions
1
vote
0answers
21 views
Shared entanglement and classical resources
If Alice and Bob share only classical communication resources such as noisy or perfect classical channels, is shared entanglement always equivalent to shared randomness?
In other words, must there be ...
4
votes
0answers
38 views
Average secret key rate calculation
I am working on a problem related to average secret key rate calculation in satellite quantum key distribution (QKD). The problem is similar to a paper on arXiv:1712.09722 (P. No. 21).
In a single ...
4
votes
0answers
93 views
Calculating the entropy of a quantum state
Let $\rho_{AR}$ be some $d-$dimensional pure quantum state. Consider a channel $N_{A\rightarrow B}$ that outputs a constant state in $B$. We now consider the Stinespring dilation of this channel with ...
3
votes
0answers
32 views
Pure state ensembles achieving the Holevo $\chi$-quantity with at most $d^2$ states
Theorem 8.10 in Chapter 8 of Theory of Quantum Information asserts that the Holevo capacity of a quantum channel (between density operators on $\mathbb{C}^d$) can be achieved by an ensemble consisting ...
4
votes
0answers
58 views
A question in classical and quantum information
Let $\rho, \sigma \in \mathfrak{D}(A)$ with $\operatorname{supp}(\rho) \subseteq \operatorname{supp}(\sigma),$ and spectral decomposition
$$
\rho=\sum_{x} p_{x}\left|\psi_{x}\right\rangle\left\langle\...
2
votes
2answers
49 views
When is the Choi matrix of a channel pure?
For a quantum channel $\mathcal{E}$, the Choi state is defined by the action of the channel on one half of an unnormalized maximally entangled state as below:
$$J(\mathcal{E}) = (\mathcal{E}\otimes I)\...
5
votes
1answer
85 views
Prove that the conditional entropy of a classical-quantum state is non-negative
Let $\rho_{XA}$ be a classical-quantum state, i.e., $\rho_{XA} = \sum_{x} p(x) |x\rangle \langle x| \otimes \rho_A^x$.
How to prove that the conditional von Neumann entropy $S(X|A) = S(\rho_{XA}) - ...
2
votes
1answer
64 views
Additivity of Renyi entropy
The Renyi entropy of order $\beta$, for a discrete probability distribution $p$ is given by
\begin{equation}
H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right),
\end{...
5
votes
1answer
63 views
Stinespring dilation: Size of environment
Let $\mathcal{E}_{A\rightarrow B}$ be a quantum channel and consider its $n-$fold tensor product $\mathcal{E}^{\otimes n}_{A^n\rightarrow B^n}$.
Any isometry $V_{A\rightarrow BE}$ that satisfies $\...
3
votes
1answer
86 views
Fidelity of extensions of states
Given two states $\rho_A, \sigma_A$, Uhlmann's theorem states that the fidelity between them is achieved in the following way
$$F(\rho_A, \sigma_A) = \max_{U_{R'}}F(\rho_{AR'}, (I\otimes U_{R'})\...
5
votes
1answer
143 views
Closeness of purifications of states
Uhlmann's theorem states that if two states $\rho_A, \sigma_A$ satisfy $F(\rho_A, \sigma_A)\geq 1 - \varepsilon$, then there for any purification $\Psi_{AR}$ of $\rho_A$, one can find a purification $\...
3
votes
1answer
94 views
Question about Haar random quantum states
Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$,
$$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
2
votes
1answer
48 views
What is the Stinespring representation of the adjoint of a channel?
For any completely positive trace nonincreasing map $N_{A\rightarrow B}$, the adjoint map is the unique completely positive linear map $N^\dagger_{B\rightarrow A}$ that satisfies
$$\langle N^\dagger(\...
2
votes
1answer
37 views
What is meant with “reconciliation” in CV QKD?
I am working on a paper about Continuous Variable QKD. (https://arxiv.org/abs/1711.08500v2)
I read about direct and reverse reconciliation in this paper. I don't understand what exactly Reconciliation ...
2
votes
1answer
61 views
Diamond norm distance bound on Stinespring dilations of channels
The diamond distance between two channels $\Phi_0$ and $\Phi_1$ is defined in this answer.
$$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 ...
1
vote
1answer
49 views
Positive conditional quantum entropy for entangled state
The quantum conditional entropy $S(A|B)\equiv S(AB)-S(A)$, where $S(AB)=S(\rho_{\rm AB})$ and $S(B)=S(\rho_{\rm B})$ is known to be non-negative for separable states. For entangled states, it is known ...
0
votes
1answer
42 views
Prove that for a general tri-partite state $\rho_{ABE}$, $H(AB) = H(E)$
How do I prove that for a general tri-partite state $\rho_{ABE}$, the following holds:
$$
H(\rho_{AB}) = H(\rho_{E}), H(\rho_{AE}) = H(\rho_{B}),
$$
where, $H$ is the Von Neumann entropy. Would ...
3
votes
0answers
41 views
Teleportation followed by measurement: Lowering communication cost
Suppose Alice and Bob have access to shared entanglement and a classical channel and wish to simulate the following quantum protocol. Alice sends over some $n-$qubit state to Bob. The state is not ...
4
votes
1answer
53 views
Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?
The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
2
votes
1answer
36 views
Pseudoinverse of a quantum state
The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
2
votes
0answers
41 views
Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?
Below is a question and an answer.
How does quantum information relate to, diverge from or reduce to
Shannon information, which used log probabilities?
What people are more often interested in are ...
2
votes
1answer
58 views
Relating quantum max-relative entropy to classical maximum entropy
The quantum max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \...
2
votes
1answer
57 views
Wheeler's “information theoretic” derivation of quantum information
1980s. John Wheeler at the University of Texas would tell his students, “Give an information theoretic derivation of quantum theory!” Information theoretic is an adjective for Claude Shannon's ...
2
votes
1answer
48 views
Is “classical information” the same as “Shannon information”?
does Shannon meet Feynman?
Bits underlie classical information measurements in information theory, while qubits underlie quantum information measurements in, what I can only assume to be called, ...
4
votes
1answer
53 views
Alternative definition of the coherent information of a quantum channel
Let $T: M_n \to M_n$ be a quantum channel. If I understand Definition 13.5.1 of the book "Quantum information theory" of Wilde, the coherent information $Q(T)=\max_{\phi_{AA'}} I(A \rangle B)...
2
votes
1answer
72 views
von Neumann entropy in a limiting case
I am stuck with a question from the book Quantum theory by Asher Peres.
Excercise (9.11):
Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix}
...
2
votes
1answer
29 views
What is the relationship between these two definitions for the max-entropy?
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 ...
3
votes
1answer
87 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
87 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
81 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 ...
6
votes
2answers
93 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 ...
5
votes
0answers
30 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
$$D(\mathcal{E}(\rho_A)\|\mathcal{E}(\sigma_A))\geq D(\mathcal{F}(\rho_A)\|\mathcal{F}(\...
3
votes
1answer
57 views
The proof of monotonicity of fidelity for channels and its meaning
I have two questions regarding the exercise 9.2.8 of Quantum information by Wilde, which is as follows:
Let $\rho,\sigma \in \mathcal{D}(\mathcal{H}_A)$ and let $\mathcal{N: L(H}_A)\rightarrow \...
5
votes
1answer
61 views
Non-lockability of quantum max-entropy
Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The ...
5
votes
3answers
163 views
Prove that Shannon and von Neumann entropies satisfy $H(P)\ge S(\rho)$ with $P$ diagonal of $\rho$
Suppose there is some $n$-qubit state $\rho$. It is well known fact that, given some orthonormal basis $U = \{|u_i\rangle\}$, if $p_i = \langle u_i| \rho |u_i \rangle$ (that is, measuring $\rho$ with $...
5
votes
0answers
91 views
Connection between smooth max-relative entropy and smooth max-information
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
3
votes
1answer
53 views
How can one estimate the von Neumann entropy of an unknown quantum state?
Given many copies of some unknown quantum state $\rho$, I would like to compute its von Neumann entropy $S(\rho)$. What algorithm could be used for this that minimizes the number of copies required? ...
5
votes
2answers
643 views
Can classical linear algebra solvers implement quantum algorithms with similar speed-ups?
A quantum algorithm begins with a register of qubits in an initial state, a unitary operator (the algorithm) manipulates the state of those qubits, and then the state of the qubits is read out (or at ...
4
votes
1answer
149 views
How can the Holevo bound be used to show that $n$ qubits cannot transmit more than $n$ classical bits?
The inequality $\chi \le H(X)$ gives the upper bound on accessible information.
This much is clear to me. However, what isn't clear is how this tells me I cannot transmit more than $n$ bits of ...
2
votes
2answers
59 views
How is a bit field represented in Quantum Computing?
For example, a computer represent a variable named "A" as 01000001. How does a quantum computer represent "A"?
I am a newbie having difficulty understanding quantum computers. I ...
4
votes
1answer
73 views
Definition of quantum min-relative entropy
In John Watrous' lectures, he defines the quantum min-relative entropy as
$$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$
where $F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$. Here, I use this ...
4
votes
1answer
97 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
53 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)}$...
4
votes
0answers
75 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
104 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
162 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
229 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
162 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:
$$
...
2
votes
1answer
70 views
Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
1
vote
0answers
35 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 ...
