Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
244
questions
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What is the difference between classical-quantum and completely classical states?
States that are completly classical :
$$
\begin{aligned}
\tilde\rho_{A B} & =\sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p_{X, Y}(x, y)(|x\rangle \otimes|y\rangle)(\langle x| \otimes\langle ...
- 433
2
votes
1
answer
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Classical capacity of quantum channel - Holevo quantity vs accessible information of a channel
Just above Eq (20.7) in Mark Wilde's book, while discussing the classical capacity of a quantum channel, he says:
These results then suggest that the ultimate classical capacity of the
channel is the ...
- 2,597
2
votes
1
answer
42
views
quantum generalisation of random variables
What is the quantum information equivalent of a classical probability random variable ? Is it a density matrix or an observable ? If so can someone show me how to write a random variable that follows ...
- 21
3
votes
1
answer
75
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Is Classical-Classical-Quantum state equivalent to Classical-Quantum state?
Suppose we have the following CQ-state between two parties Alice & Bob
$$
\rho_{A B}^{\otimes n}=\sum_{x^n} p^n\left(x^n\right)\left|x^n\right\rangle\langle\left. x^n\right|^A \otimes \rho_{x^n}^B ...
- 433
2
votes
1
answer
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How to compute the QFI of a thermal state?
Let $\rho=\frac{1}{Z}\exp(-\beta H)$ be the thermal state associated to the Hamiltonian $$H=\hbar\omega\sum_i\left( a_i^\dagger a_i+\frac12\right).$$
I wonder how the quantum Fisher information of ...
- 21
0
votes
1
answer
101
views
Rabi amplitude in theory vs practice
I am a bit confused with the definition of Rabi amplitude and how it relates to experimental values.
If I understand correctly, we can show a rabi driving (waveform) with the following formula:
$\...
- 3
0
votes
1
answer
63
views
What are necessary and sufficient conditions for the output of a parametrized unitary $U(\theta)$ to be smooth?
Let us consider a unitary $U$ parameterised by $\theta \in \mathbb{R}$, i.e, $U(\theta)$. What are the necessary and sufficient conditions for the output states of this unitary to be smooth? One ...
- 168
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0
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44
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Coherent information is a lower bound of channel capacity. What about coherent information based on Renyi entropies?
It is known that coherent information defined in terms of von Neumann entropies is a lower bound of quantum channel capacity. If we define coherent information in terms of $\alpha$-Renyi entropies, ...
- 2,369
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0
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40
views
An Inequality related to the Trace Norm
Let $\rho, \sigma$ be two states of a qubit, and let $U$ be the $CX_{12}$-gate (control on 1st qubit, target on 2nd qubit).
Prove that, for an arbitrary CPTP Map $\mathcal{E}$,
$$
||\mathcal{E}(\rho -...
- 39
2
votes
1
answer
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views
How to discriminate between $N$ states drawn from one of two ensembles?
Consider the following quantum discrimination problem:
Suppose, there are two sets of states, $P = \{ \rho_i \}$ and $Q = \{\sigma_i\}$. Both Alice and Bob know which states are in each set. We can ...
- 39
3
votes
0
answers
33
views
Some mathematical problem of calculating distance
Suppose we have POVM $E = A^\dagger A$. $E$ satisfy:
$\forall i,j$, $\lambda_i,\lambda_j$ are eigenvalues of $E$, and $\lambda_i\le e^\epsilon \lambda_j$ holds.
Then we can decompose $E=U^\dagger DU$, ...
- 141
1
vote
1
answer
77
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Increasing quantum channel capacity
I have a few confusions regarding quantum communication. Please bear with me as I ask what might appear to be basic or naive questions:
In the case of classical communication the capacity of wireless ...
- 11
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0
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38
views
Defining quantum memory mathematically
Is it possible that we may define the notion of quantum memory in abstract manner, that is, mathematically? A definition which is hardware independent that may be treated as a mathematical object. We ...
- 311
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vote
1
answer
48
views
Quantum relative entropy between pre- and post-measurement states
The quantum relative entropy between the states $\rho$ and $\sigma$ is defined by
$$D(\rho||\sigma)= \textrm{tr}\Big(\rho \big(\log\rho - \log \sigma \big) \Big)\,,$$
as long as the support of $\rho$ ...
- 55
2
votes
1
answer
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views
Is Klein's inequality due to Klein?
You may be familiar with "Klein's inequality"; one form of it is
$$
-\operatorname{tr}(\rho \log \sigma) + \operatorname{tr}(\rho \log \rho) \ge 0,
$$
stating that relative entropy is ...
- 123
1
vote
1
answer
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What is the exact mathematical content of the no-teleportation theorem?
My question concerns the following theorem (not to be confused with quantum teleportation protocols): what is the exact mathematical content of this theorem? After all any (pure) quantum state (say in ...
- 133
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0
answers
47
views
General structure of the state with $I(A:B|C)_{\rho}{=}2 \log_2 \{\min (d_A, d_B)\}$
The conditional quantum mutual information (CQMI) of a state $\rho^{ABC}$ respects the dimension bound $I(A:B|C)_{\rho}{\leq}2 \log_2 \{\min (d_A, d_B)\}$ (Mark Wilde's book, exercise 11.7.9). One ...
- 135
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vote
0
answers
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views
What is the classical Fisher information of a parametrized coherent state $|\alpha_\theta\rangle$?
Suppose $|\alpha_\theta\rangle$ is a coherent state depending on the real parameter $\theta$. What is the classical Fisher information it carries?
There are explicit formulae for the quantum Fisher ...
2
votes
2
answers
55
views
How to evaluate the mutual information of a classical-quantum state?
Suppose I have the state $\sum_i p_i \vert i\rangle\langle i\vert \otimes\rho_i$ on subsystems $X$ (classical part) and $B$ (quantum part) and I wish to evaluate the mutual information $I(X:B) = S(X) +...
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views
How to justify the conclusion $|E_{sq}(\rho)-E_{sq}(\sigma)|\le f(\epsilon)$, when proving the continuity of the squashed entanglement?
I am following the paper by Christandl and Winter introducing squashed entanglement. My question is particularly on the continuity proof of squshed entanglement mentioned after conjecture 14 and ...
- 135
1
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1
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87
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G-twisted Pauli twirl circuit
Pauli twirls are obtained by taking a unitary $U$, and finding some Pauli gates $P_1, P_2$ such that $P_1 U P_2$. So, for example, one possible twirl of the $S$ gate would be $YSX$.
In the paper ...
- 2,865
1
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0
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19
views
Average-case error and worst-case error in entangment-assisted classical communication
I am reading this paper where one considers a quantum channel $\mathcal{N}$. We are interested in the maximum rate at which we can send classical messages over this channel in the one-shot setting i.e....
- 2,597
2
votes
1
answer
38
views
What are the antisymmetric terms in $\sigma_{mn}$ in the expression for the Fisher information?
Given a parameter-dependent density operator $\hat\rho^\lambda$ and its spectral decomposition $\{\rho_m^\lambda, |\psi_n^\lambda\rangle\}$, Eq. $(17)$ from this review shows that one can compute its ...
1
vote
1
answer
59
views
Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?
In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says:
A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i,
The ...
- 205
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votes
0
answers
29
views
Dense coding for PR boxes
We can think of an entangled state as a resource for accomplishing various tasks. For example, passing a CHSH test, performing a dense coding operation (when enhanced by transmission of a single qubit)...
- 55.7k
3
votes
3
answers
111
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What is the conditional min-entropy of a pure bipartite state?
In this paper, it is stated that the conditional min-entropy $H(A|B)_{\rho_{AB}}$ of $A$ conditioned on $B$ for any $\textbf{pure}$ quantum system $\rho_{AB}=|\psi_{AB} \rangle \langle \psi_{AB} |$ is
...
- 55
4
votes
1
answer
67
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Clarification about inverses in sandwiched Renyi divergence
The sandwiched Renyi divergence is defined as in
$$
\tilde{D}_\alpha(\rho\|\sigma):=\frac{1}{\alpha−1}\log tr[(\sigma^{\frac{1−\alpha}{2\alpha}}\rho \sigma^{\frac{1−\alpha}{2 \alpha
}})^\alpha]
$$
The ...
- 73
1
vote
0
answers
33
views
How can $\chi(\hat{A},\hat{B}:C) \le \chi(\hat{A},B:C)$ be true?
The holevo information of $\rho_{ABC}$ w.r.t to measurements on A and B (for the sake of this we'll assume local measurements suffice), is given by $$\chi(\hat{A},\hat{B}:C)$$ where $\hat{A}$ and $\...
- 1,115
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votes
1
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99
views
Twirling of quantum states: Maximally entangled states
I have been reading the paper "Resource theory of unextendibility and non-asymptotic quantum capacity" (https://arxiv.org/pdf/1803.10710.pdf) by Kaur et.al, I have two questions ...
- 13
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vote
0
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55
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Optimal one-shot input for $\mathcal{N}^{\otimes n}$ and optimal one-shot input for $\mathcal{N}^{\otimes k}$ where $k< n$?
Suppose I have a classical or quantum channel $\mathcal{N}$. I wish to use it for some communication task in the one-shot setting i.e. where I have $n$ i.i.d. copies of $\mathcal{N}$ denoted by $\...
- 2,597
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0
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Which quantum entropies are meaningful with respect to continuous distributions of states?
When using a quantum channel to transmit classical information, we consider an ensemble $\mathcal{E} = \{(\rho_x, p(x))\}$ consisting of states $\rho_x$ labelled with a symbol $x$ from a finite ...
- 6,143
1
vote
1
answer
70
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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 ...
- 13
2
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0
answers
81
views
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! Please skip to the last section if you're familiar with one-shot information theory.
Defintions
The max-...
- 2,597
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0
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36
views
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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2
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2
answers
78
views
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 ...
4
votes
0
answers
39
views
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),$$
for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS♦
- 23.4k
1
vote
1
answer
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Does the max-relative entropy satisfy $0 < D_{\max}(\rho \parallel I_A \otimes \sigma_B) < 1$?
The quantum conditional min-entropy is defined as
$$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 \...
- 365
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1
answer
71
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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
$$D_{\max}(\rho \parallel I_A \...
- 365
1
vote
1
answer
45
views
Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$
I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent.
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B ...
- 365
1
vote
2
answers
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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 ...
- 53
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0
answers
77
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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 $\{ \...
- 6,143
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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 ...
- 791
1
vote
1
answer
155
views
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 ...
- 332
3
votes
1
answer
191
views
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 ...
- 365
1
vote
0
answers
169
views
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)(|\...
- 41
1
vote
0
answers
35
views
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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votes
1
answer
86
views
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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6
votes
1
answer
312
views
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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2
votes
1
answer
112
views
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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2
votes
2
answers
186
views
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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