Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
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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 ...
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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!
Defintions
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \...
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Why is the quantum capacity quantified by the coherent information?
Most types channel capacities associated to a given quantum channel are quantified using mutual informations (sometimes classical, sometimes quantum, sometimes regularised), which is not surprising ...
glS♦
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Physical interpretation of a 2-photon-qubit system occupying the antisymmetric Bell state
This question regards the reconciliation of QIT with what I have learned separately in lectures about quantum physics. My understanding is that indistinguishable bosons are always described by ...
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Does the quantum relative entropy have a direct operational interpretation?
Consider the quantum relative entropy, defined as
$$D(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma),$$
for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS♦
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Does the max-relative entropy satisfy $0 < D_{\max}(\rho \parallel I_A \otimes \sigma_B) < 1$?
The quantum conditional min-entropy is defined as
$$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 \...
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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
$$D_{\max}(\rho \parallel I_A \...
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Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$
I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent.
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B ...
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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 ...
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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 $\{ \...
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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 ...
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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 ...
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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 ...
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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)(|\...
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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 ...
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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 ...
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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 $2^n$ ...
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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 ...
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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, ...
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Prove $|η(r) − η(s)| ≤ η(|r − s|)$ when $|r − s| ≤ 1/2$ [closed]
Background
If $\rho$ and $\sigma$ are density matrices such that the trace distance between them satisfies $T(\rho,\sigma)\leq1/e$. Then the Fannes' inequality states that $$|S(\rho)-S(\sigma)|\leq T(...
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How to calculate the log of a density matrix?
In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a ...
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With $\vert\Psi^+\rangle$ the Bell state, can $\sqrt{\rho}\vert\Psi^+\rangle\langle\Psi^+\vert\sqrt{\rho}$ be simplified?
Let $\vert\Psi^+\rangle_{AB} = \frac{1}{\sqrt n}\sum_{i=1}^n\vert i\rangle_A\vert i\rangle_B$ be the maximally entangled state in Hilbert space $\mathcal{H}(AB)$ and $\rho_A$ be some state in Hilbert ...
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How to prove the strong convexity of the trace distance?
On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\...
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How many measurements are needed to distinguish two fixed density matrices?
Suppose there are two fixed density matrices $\rho_1$ and $\rho_2$ are prepared for equal probability. Can we say something about the minimum number of measurements required to distinguish the two ...
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How to derive the quantum Fisher information from the relative entropy?
The quantum relative entropy (QRE) between two states $\rho$ and $\sigma$ is given by
$$
S(\rho\|\sigma)=\operatorname{Tr}(\rho\ln\rho)-\operatorname{Tr}(\rho\ln\sigma)
$$
Now if $\rho$ and $\sigma$ ...
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Expectation value over random $k$-local Pauli operators for two random quantum states
Suppose we have a uniform distribution $D$ over $k$-local Pauli operators $P_{q_1}\otimes \dotsc \otimes P_{q_k} $, $P_{q_i} \in \{ X, Y, Z, I \}$. Is it possible to calculate $\mathbb{E}_{P_i \sim D} ...
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What is the quantum relative entropy between pure states?
Given two pure quantum state $\rho=|\psi_\rho\rangle\langle\psi_\rho\mid$ and $\sigma=\mid\psi_\sigma\rangle\langle\psi_\sigma\mid$ ($\rho\neq\sigma$).
We know that the fidelity between quantum ...
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Does proving $Q^{(1)}(\mathcal{N}\otimes\mathcal{N})=Q^{(1)}(\mathcal{N})+Q^{(1)}(\mathcal{N})$ imply additivity for arbitrary $n$?
I have been reading the proofs that are usually presented in order to proof the additivity of degradable and conjugate degradable channels, and they usually present that the coherent information is ...
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How is the von Neumann entropy of a state defined from its eigendecomposition?
The definition of the von Neumann entropy of a mixed state says that it can be calculated as the Shannon entropy of coefficients of the decomposition of the state into a sum of projectors.
My question:...
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Why the "Close Images" problem is QIP-complete
The following problem is known as the "close images" problem:
the input is two circuits $Q_0$, $!_1$, with the same number of input
and output qubits (The circuits are allowed to add ancilla ...
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How can one impliment Bennett's partial measurement onto a binomial subspace for state distillation?
I'm reading the seminal paper on entanglement distillation by Bennett et. al.
The idea is that Alice and Bob have $n$ identical copies of an imperfect (but pure) Bell state. The initial state is ...
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What are explicit examples of smoothed conditional min(max) entropies?
Some general discussion of smoothed entropic quantities is found for example in Watrous notes, and an overview and discussion on its operational interpretations in (Koenig et al. 2008). It seems the ...
glS♦
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In what sense is the "conditional min-entropy" a conditional entropy?
$\newcommand{\H}{\mathsf{H}}\newcommand{\Hmin}{\H_{\rm min}}\newcommand{\D}{\mathsf{D}}\newcommand{\Dmax}{\D_{\rm max}}$Consider the conditional min-entropy $\Hmin(A|B)_\rho$, discussed e.g. in this ...
glS♦
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Why are "smooth entropic quantities" useful/necessary?
Consider the $\epsilon$-smoothed relative max-entropy of $\rho$ with respect to $Q$, defined as (following Watrous' notation from these notes):
$$\mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in ...
glS♦
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References of group theory for quantum information theory
Now that quantum information theory (QIT) reaches the point that group theory have deeply combined with its applications and theoretical understandings, such as random benchmarking, quantum scrambling,...
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General Proof of the Statement that You Need $1$ Ebit and $2$ Bits to Teleport $1$ Qubit?
I understand from the standard teleportation protocol that 1 ebit is used up in teleporting 1 qubit and thus, cannot be used again -- and thus, we need 1 fresh ebit of shared entanglement between ...
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How much information can be stored in a system with synthetic dimensions? [closed]
Okay this is a completely serious question and keep in mind I have a PhD in theoretical condensed matter physics, in which I have somewhat of a specialization in Floquet physics. So as the title says, ...
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$2$ ebits $+$ $1$ bit $ = 2$ bits?
The Set-Up
Let's say Alice and Bob share $k$ ebits, i.e., they have one-qubit each of each of the $k$ Bell states $\frac{\vert 00\rangle+\vert 11\rangle}{\sqrt{2}}$. Now, Alice wants to send $2n$ bits ...
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Is it possible to prevent quantum communication detection?
From what I understood, one of the advantages of quantum communication is that one can (mathematically/physically) prove that the quantum communication/message has not been intercepted/tampered with. (...
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Entanglement-assisted communication ability of a quantum depolarizing channel vs. a classical binary symmetric channel
Consider a quantum qubit depolarizing channel which takes a quantum state $\rho$ to output
$$N(\rho) = (1-p)\rho + p\frac{\mathbb{1}_2}{2}.$$
If I restrict $\rho$ to be either $\vert0\rangle\langle 0\...
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Classical Information Theory vs. Quantum Information Theory
I am quite familiar with the basic concepts of information theory (sources, alphabets, simbols, strings, information, Shannon's entropy, noisy channels, Shannon's theorems, etc.). I always thought of ...
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What is the conditional min-entropy for diagonal ("classical") matrices?
The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
glS♦
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Additivity of degradable and anti-degradable quantum capacities
It is known that for degradable channels $\mathcal{N}$ and $\mathcal{M}$, the single-letter quantum capacity is aditive (Potential Capacities of Quantum Channels), i.e.
\begin{equation}
Q^{(1)}(\...
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What role does Landauer's principle play in quantum reversibility?
In section 3.2.5 of Nielsen and Chuang (starting page 153) they talk about Landauer’s principle, where they discuss the lower bound on the thermodynamic cost of erasing information.
In irreversible ...
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Why does the bit flip channel produce a uniform contraction of $1-2p$?
Studying the bit flip channel using the Nielsen & Chuang's.
And ran into the picture with the caption stating $yz$ plane is uniformly contracted by a factor of $1-2p$. I don't quite understand how ...
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Is there a non-deterministic protocol for entanglement generation between distant parties?
I'm aware that one can imperfectly clone entanglement that's shared between two parties (i.e. Bell pairs) using deterministic quantum cloning machines to produce two, lower fidelity entangled states.
...
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Is there a non one-way quantum computer?
Be it theoretical proposal or anything else, is there even a definition for non one-way (or non measurement-based) quantum computer/computation?
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Derivation of the linear cross entropy
I'm looking at cross-entropy benchmarks and there's much that I'm reading at the moment but I'm stuck on one detail: how to derive the linear cross-entropy formula from the cross-entropy formula.
The ...
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Can one define a Choi state for a a classical channel?
Suppose one has a classical channel $W(y|x)$ that is a conditional probability distribution. Can one define a Choi state for this channel?
My guess is that one should think of it as a special case of ...
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Tapering off qubits
Suppose you have a Hamiltonian of the form
$$ H = ZXXX + YXXX + XXXX $$
where $Z,X,Y$ are the usual Pauli matrices with $ZXXX = Z \otimes X \otimes X \otimes X$ and similar for the other two terms. ...
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