Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
185
questions
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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 ...
- 179
2
votes
1
answer
35
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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 ...
- 127
2
votes
1
answer
60
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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♦
- 19.6k
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29
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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♦
- 19.6k
3
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1
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76
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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, ...
- 51
1
vote
1
answer
99
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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 ...
- 185
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2
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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. (...
- 123
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0
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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}.$$
If I restrict $\rho$ to be either $\vert0\rangle\langle 0\...
- 2,169
4
votes
2
answers
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Classical Information Theory vs. Quantum Information Theory
I am quite familiar with the basic concepts of information theory (sources, alphabets, simbols, strings, information, Shannon's entropy, noisy channels, Shannon's theorems, etc.). I always thought of ...
- 141
1
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2
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What is the conditional min-entropy for diagonal ("classical") matrices?
The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
glS♦
- 19.6k
2
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1
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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)}(\...
2
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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 ...
2
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contraction related Bit flip channel
Studying the bit flip channel using the Nielsen & Chuang's.
And ran into the picture with the caption stating y-z plane is uniformly contracted by a factor of 1-2p. I don't quite understand how ...
- 889
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2
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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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1
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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?
- 21
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2
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76
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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 ...
1
vote
1
answer
55
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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 ...
- 13
4
votes
2
answers
149
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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. ...
- 12.8k
3
votes
2
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Existence of a perturbed channel that achieves a perturbed output state
Consider a $d$-dimensional maximally entangled state $\vert\phi\rangle = \frac{1}{d}\sum_{i=1}^d\vert i\rangle_A\vert i\rangle_B$. Let $N_{A\rightarrow A'}$ be a quantum channel and consider $\rho_{A'...
- 2,169
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votes
2
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134
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maximization of trace between two operators with respect to different norm constraints
I want to maximize $\text{Tr}(XY)$ over $X$ for fixed $Y$, where $X$ and $Y$ are both hermitian (but doesn't necessarily positive) operators, and $X$ is constrained by its p-norm bounded by $1$, i.e. $...
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Is it possible to extract $x_1$ and $x_2$ from $|\phi\rangle=\frac1{\sqrt2}(|x_1,0^n\rangle+|0^n,x_2\rangle)$ with non-negligible probability?
Let $\left\vert \phi\right\rangle=\frac 1{\sqrt2}\left\vert x_1,0^n\right\rangle+\frac1{\sqrt2}\left\vert 0^n,x_2\right\rangle$ be a $2n$-bit quantum state for some unknown $x_1,x_2\in\{0,1\}^n$. My ...
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6
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1
answer
148
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Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$
Suppose I have a tripartite system $ABC$ in a pure state $|\psi_{ABC}\rangle$ with mutual information $I(A:C)=0$. This implies that the reduced density matrix $\rho_{AC}$ factorizes as $\rho_{AC} = \...
- 353
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1
answer
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How can classical bits be copied if qubits cannot be copied?
The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $|0\rangle |\psi\rangle\...
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vote
2
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Is it possible to efficiently measure outer products of quantum states, of the form $|a\rangle\langle b|$?
I am looking at a matrix reconstruction algorithm that, given singular values $\sigma_i$ and quantum states $|u_i\rangle$ and $|v_i\rangle$ that are efficiently prepared on a quantum computer, ...
- 51
5
votes
1
answer
73
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What is the quantum analogue of $P_{XY} = P_{Y|X}P_X$
A standard trick in probability manipulation is to take some joint distribution $P_{XY}$ and express it as $P_{Y|X}P_X$. This trick is useful because when one looks at things like the ratio of $\frac{...
- 2,169
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votes
1
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Quantum channel between two states with inaccessible reference - when can it be done?
Suppose I have a pair of bipartite states $\rho_{AR}$ and $\sigma_{BR}$. $R$ is a reference system that we do not have access to.
It is clear that we cannot always have a channel $N_{A\rightarrow B}$ ...
- 2,169
0
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0
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38
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The derivation of the quantum information no-hiding theorem, question 2
I am reading Samuel L. Braunstein, Arun K. Pati, Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. The paper purportedly proves ...
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The derivation of the quantum information no-hiding theorem, question 1
I am reading Samuel L. Braunstein, Arun K. Pati, Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. I am puzzling over the derivation ...
- 157
1
vote
0
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63
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How to understand Clifford+T from a quantum information theory perspective [closed]
I'd like to study the important operator set Clifford+T from a quantum information theory perspective.
The Clifford+T set is universal and it is important because allows for efficient error correction ...
- 911
4
votes
1
answer
157
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Threshold for quantum Repetition Code
I'm learning about the Threshold theorem but I struggle with the computations of the threshold which are usually presented (even the one in the book by Nielsen & Chuang). To clear my head, I would ...
2
votes
1
answer
52
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If information is increase in entropy, why does large entropy mean little information?
Excuse me since this is an elementary question in information theory.
I am asking this question here since the statement "large entropy means little information" is mentioned in the first ...
- 165
4
votes
1
answer
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Data processing inequality for relative entropy in the presence of an amplitude damping channel
Consider the single qubit quantum depolarizing channel, given by
$$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}. $$
For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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votes
0
answers
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Proof of upper and lower bound (Gilbert-Varshamov bound) for linear code
I am trying to prove the following bounds for a $[n, k]$ code that can correct $t$ errors
\begin{align}
1-H\left(\frac{t}{n}\right)\geq \frac{k}{n}\geq 1-H\left(\frac{2t}{n}\right)
\end{align}
where
\...
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votes
1
answer
170
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Entanglement entropy and depth
I wanted to verify two intuitions about the entanglement entropy of quantum states.
Consider an $n$ qubit quantum state, prepared by a depth $d$ circuit acting on $|0\rangle^{\otimes n}$ and a ...
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votes
1
answer
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Is the composition of two extremal channels also extremal?
In this question, I follow the terminology and notation of the book of Watrous, most notably chapter two.
Extremal channels
An extremal channel $\Phi(X) \in C(\mathcal{X},\mathcal{Y})$ is a channel ...
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How many minimum Quantum Rats are needed to figure out which bottle contains poison?
For the classical Poison and Rat puzzle, we need at least $\lceil\log_2({\rm bottles})\rceil$ rats to figure out the poisoned bottle.
If we have Schrödinger’s quantum rats, can we use fewer rats(...
- 217
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votes
1
answer
32
views
Properties of the generalized fidelity for subnormalized states
The generalized fidelity for quantum states that may be sub-normalized is given by (Defn 3.12)
$$F_{*}(\rho, \tau):=\left(\operatorname{Tr}|\sqrt{\rho} \sqrt{\tau}|+\sqrt{(1-\operatorname{Tr} \rho)(1-\...
- 33
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votes
0
answers
22
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Max-relative entropy quasi-convexity inequality under partial trace
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. ...
- 2,169
5
votes
1
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252
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What is the root of the non-trace-preserving bit-flip map
I have a quantum channel defined by the Kraus operators:
$$
U_1 =
\begin{bmatrix}
p & 0 \\
0 & p
\end{bmatrix},\quad
U_2 =
\begin{bmatrix}
0 & p \\
p & 0
\end{bmatrix}
$$
i.e.
$$
U_1\...
- 157
3
votes
1
answer
44
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Quasi concavity of max-relative entropy?
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$
It is known that the max-relative entropy is quasi-convex. ...
- 2,169
3
votes
1
answer
80
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Special properties of a channel whose Kraus decomposition contains Identity
I would like to know if there are any special properties of channels that permit a Kraus representation that includes an identity? That is, if I am given a Kraus representation of a CPTP map $\Phi$ ...
- 5,067
5
votes
1
answer
310
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No-cloning theorem and distinguishing between two non-orthogonal quantum states revisited
There are a couple of posts on this question, but I think they are not satisfactory. The question is Nielsen and Chuang's QCQI, Exercise 1.2, page 57, which asks "Explain how a device which, upon ...
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0
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265
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What does a quantum mutual information larger than its classical upper bound represent?
Let $\rho$ be a bipartite state. Its quantum mutual information is defined as
$$\newcommand{\tr}{\operatorname{tr}}I(\rho) = S(\tr_B(\rho)) + S(\tr_A(\rho)) - S(\rho),$$
where $S(\sigma)$ is the von ...
glS♦
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2
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1
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Does ${\rm tr}(\Pi_z\rho\Pi_z)\le p$ imply $\cal E(\rho)$ and $\cal E(\Pi_{-z}\rho\Pi_{-z})$ are close in trace distance?
Suppose I have a quantum operation $\mathcal{E}$ and a state $\rho$ such that:
$$ \operatorname{tr}(\Pi_z \rho \Pi_z) \le p $$
for some probability $p$ and some projection $\Pi_z$ onto some subspace ...
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1
vote
1
answer
64
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In Schumacher’s noiseless channel coding theorem, how do we get the exponents in $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\otimes n(1−p)/2}$?
On pg. 55 in Nielsen and Chuang, it's said that:
the $|0\rangle + |1\rangle$ product can be well approximated by a superposition of states of the form $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\...
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What is the quantum capacity of the combined amplitude and phase damping channel?
Quantum capacity for the amplitude damping channel and the pure dephasing channel have closed-form formulas as it can be seen in section 24.7.2 of From Classical to Quantum Shannon Theory. However, I ...
5
votes
1
answer
153
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What is a maximal entangled multipartite state?
We know the four Bell states are the maximal entangled states for two-qubit states, and we know if a state cannot be written as the tensor product by its subsets, then it is a entangled state, so is ...
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