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Questions tagged [information-theory]

The tag is used for questions connected with information theory in classical and/or quantum sense.

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Are there approximability schemes for NP-Hard problems using quantum algorithms?

I'm looking into some parts of Quantum Complexity Theory and was wondering if we have any approximability schemes for NP-Hard problems using quantum algorithms. I was unable to find any literature on ...
LukasM's user avatar
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What are examples of channels whose Holevo capacity can be computed explicitly?

Given a channel $\Phi:\operatorname{Lin}(\mathbb{C}^n)\to\operatorname{Lin}(\mathbb{C}^m)$, we define its Holevo capacity as $$\chi(\Phi) = \sup_\eta \chi(\Phi(\eta)),$$ with the sup taken with ...
glS's user avatar
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Prove that the coherent information of an antidegradable channel is equal to zero

I want to show that antidegradable channels have zero coherent information, based on Exercise 13.5.6 in [1]. So the solution should use the following relationship: For Hilbert spaces $R, B, E$, a pure ...
forky40's user avatar
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How to systematically find the kernel of a channel from its Kraus operators?

A quantum channel is a completely positive trace-preserving map. Given a quantum state $\rho$ and channel $N$, let the output be $N(\rho)$. Given the Kraus operators of the channel, how can one find ...
user1936752's user avatar
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Noise modelling with Jordan's lemma

In many self-testing protocols / quantum verification protocols, one encounters the so-called "Jordan Lemma": Given two dichotomic Hermitian finite or countably infinite dimensional matrices ...
relativeentropy's user avatar
5 votes
1 answer
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What is the meaning of $\sum_i K_iK_i^\dagger$ for a quantum channel with Kraus operators $K_i$?

Let a channel $N$ be given in terms of its Kraus operators $K_i$ as $$N(\rho) = \sum_i K_i\rho K_i^\dagger.$$ Is the sum $\sum_i K_iK_i^\dagger$ a meaningful quantity? I know that $\sum_i K_i K_i^\...
Pluto's user avatar
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2 votes
1 answer
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Quantum Error Correction Difficulties: "Measurements destroy quantum information" Intuition

Context: In section 10.1.1, Nielsen and Chuang describe the difficulties QEC faces compared to classical error correction. Particularly that Measurements destroy quantum information. Shor states in ...
vollautomatthi's user avatar
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Proof that the relative entropy satisfies $S(\rho\|\sigma)=S(T\rho\|T\sigma)$ iff $\hat TT\rho=\rho$, $\hat TT\sigma=\sigma$ for some $\hat T$

To prove the saturation condition for the strong subadditivity of the von Neumann entropy, the authors of [HJPW2004] make use of the following characterisation of when the monotonicity of the ...
glS's user avatar
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3 votes
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What are examples of states saturating the strong subadditivity of the von Neumann entropy?

A well-known property of classical distribution is that they satisfy strong subadditivity, meaning that for any tripartite joint probability distribution $p(x,y,z)$, we have the inequality $$H(AB)+H(...
glS's user avatar
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19 votes
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Counterexamples in quantum information theory

As was already asked about in this phys.SE question many years ago—which, sadly, got closed and never received an answer—is there a collection of counterexamples in quantum information theory, "...
Frederik vom Ende's user avatar
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Generalizing error propagation formula to multi-parameters

For single parameter phase estimation we have the Cramer-Rao bound $$(\Delta \theta)^2 \geq \frac{1}{F_{Q}[\rho, \hat{A}]},$$where $F_{Q}$ is the quantum Fisher information and where instead of an ...
John Doe's user avatar
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7 votes
2 answers
317 views

How can quantum error correction correct small rotations/continuous errors?

I'm having trouble understanding the so-called "digitization of errors" argument in QEC. Suppose I have to encode my logical qubit into $n$ physical qubits to do error correction. I will use ...
Joan's user avatar
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1 answer
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Does measurement in different bases allow for FTL communication?

Imagine for a moment that we could distinguish between arbitrary quantum states. We’ll show that this implies the ability to communicate faster than light, using entan- glement. Suppose Alice and Bob ...
Nir Sharma's user avatar
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1 answer
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How to get the Kraus decomposition of the amplitude damping channel from its Choi?

I found going from the Choi-matrix of a quantum channel to the Choi-Kraus decomposition a bit difficult. I know that it follows from the eigen-decomposition of the Choi-matrix. But I struggle with ...
Pink Elephants's user avatar
2 votes
2 answers
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Are all extremal points of the feasible set of an arbitrary affine equation pure states?

Suppose I have one constraint on quantum states, i.e., $\Lambda(\rho)=Y$ where $Y$ is a Hermitian matrix and $\Lambda$ is a linear and Hermitian preserving map. Note that $\rho$ and $Y$ can in general ...
narip's user avatar
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Multimode unitary channel in terms of action on characteristic function

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
Phil K.'s user avatar
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1 answer
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How to prove there's no quantum channel that clones all classical states?

Considering a qubit $\scr H =\Bbb C^2$ I have seen a proof of the no-cloning theorem for pure states. I wonder how do you prove it for a classical state? 1)That is, how do I prove that there is no ...
darkside's user avatar
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Mutual information between Alice and Eve in a BB84 intercept resend attack

I'm new to information theory and i need to calculate $I(A,E)$. To calculate it I need conditional entropy $H(A|E)$. I assume the BB84 protocol standard states $\{ |0\rangle,|1\rangle \},\{|+\rangle,|-...
forgetfuled's user avatar
1 vote
1 answer
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If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

Def: The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^...
some_math_guy's user avatar
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1 answer
59 views

Fisher information from likelihood function for discrete quantum case

In the context of a single phase estimation problem of a quantum photonics experiment. For example consider a 3-photon quantum circuit (such as the Mach-Zehnder which depends on some phase shift ...
John Doe's user avatar
  • 911
4 votes
1 answer
103 views

Can post-measurement states have entropy larger than the original state?

Given a set of measurement operators $\{M_i\}$ that sum to unity, consider the post-measurement states on some $\rho$ as $\rho_i:=(\sqrt{M_i}\rho\sqrt{M_i})/p_i$ and $p_i:=\mathrm{Tr}(M_i\rho)$. It's ...
Shadumu's user avatar
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2 votes
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What does it mean to take the maximum over all POVMs?

What does it mean to take the maximum over all POVMs (in the definition of accessible information for example) ? The set of POVMs is infinite, how can we be sure that the maximum is attained by one of ...
yosh's user avatar
  • 127
2 votes
2 answers
149 views

Does independence from the input state imply a tensor product structure for the unitary?

Let $U$ be a unitary acting on Hilbert space $\mathcal H = \mathcal H_A \otimes \mathcal H_B$ such that $$\mathrm{Tr}_A(U \vert \psi \rangle \langle \psi \vert_A \otimes \vert 0 \rangle \langle 0 \...
SescoMath's user avatar
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0 answers
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Partial decoherence of a general one-photon state

Let $\rho_1$ be the pure one-photon state described by the ket $$|\psi_1\rangle = \int dk\ A(k)a^\dagger(k)|0\rangle$$ for a complex amplitude function $A(x)$ and an empty ket $|0\rangle$. This state ...
Bentanglement's user avatar
3 votes
2 answers
183 views

Equal partial traces

Given an arbitrary state $\rho_{AB}$, is it always possible to construct an extension $\rho_{ABC}$ such that $$Tr_B(\rho_{ABC}) := \rho_{AC} = \rho_{AB} := Tr_C(\rho_{ABC})$$ If yes, does there exist ...
user1936752's user avatar
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1 vote
1 answer
246 views

How is Quantum Computing expressed in the language of abstract algebra?

I've lately been taking further coursework in abstract algebra, and it has struck me as fairly reminiscent of quantum computing. Of course, Pauli matrices, etc. have relevant roots within abstract ...
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Entropic uncertainty relations with measurements and memory [duplicate]

In entropic uncertainty relations involving measurements and memory, one has a quantum state $\rho_{AB}$. Alice holds register $A$ and performs one of two measurements denoted by observables $R$ and $...
user1936752's user avatar
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3 votes
0 answers
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Fisher information of parametric channel

Suppose $\Phi_\theta$ is a quantum channel whose action can be written for any state $\rho\in \mathcal S(\mathcal H_S)$ in the Stinespring representation as $\Phi_\theta(\rho)= \text{Tr}_E(U_\theta (\...
Quantastic's user avatar
1 vote
0 answers
65 views

Distinguish two states with their priors probability

EDIT: This is a computer programming / coding exercise The states $\left|\psi\right>$ and $\left|\phi\right>$ are defined as $∣ϕ⟩=\cos(θ_ϕ)\left|0\right>+\sin(θ_ϕ) \left|1\right>$ with ...
Minh Triet's user avatar
1 vote
0 answers
38 views

Composing beam splitters

Let $a, b$ and $c$ be independent modes in a system $S$ and in environments $E_1$, $E_2$ respectively. Suppose $a$ goes through a beam-splitter characterized by a parameter $\theta$ coupling it to ...
Quantastic's user avatar
3 votes
1 answer
314 views

Erasure errors in quantum error correction

Consider an $[n, k, d]$ classical code. This code can correct up to $d-1$ erasures. For example, if we have the code that maps $0\rightarrow 00$ and $1\rightarrow 11$, this code has distance $2$. ...
user1936752's user avatar
  • 2,997
3 votes
1 answer
97 views

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 ...
IamKnull's user avatar
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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 ...
user1936752's user avatar
  • 2,997
2 votes
1 answer
46 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 ...
yosh's user avatar
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3 votes
1 answer
83 views

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 ...
IamKnull's user avatar
  • 433
3 votes
1 answer
281 views

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 ...
Noobgrammer's user avatar
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1 answer
136 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: $\...
Rex's user avatar
  • 25
0 votes
1 answer
64 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 ...
Song of Physics's user avatar
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0 answers
60 views

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, ...
MonteNero's user avatar
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2 votes
1 answer
73 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 ...
Sowmitra Das's user avatar
2 votes
2 answers
115 views

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 ...
ZE1's user avatar
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0 votes
0 answers
46 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 ...
madeel's user avatar
  • 321
1 vote
1 answer
63 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 $\rho$ ...
quantum_theo's user avatar
2 votes
1 answer
174 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 ...
echinodermata's user avatar
1 vote
1 answer
156 views

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 ...
truebaran's user avatar
  • 153
2 votes
0 answers
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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 ...
Abir's user avatar
  • 135
2 votes
2 answers
79 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) +...
JRT's user avatar
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1 vote
1 answer
79 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 ...
Abir's user avatar
  • 135
1 vote
1 answer
95 views

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 ...
epelaez's user avatar
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1 vote
0 answers
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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....
user1936752's user avatar
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