# Questions tagged [quantum-information]

This tag is related to Quantum information Theory.

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### Inequality for square root of partial trace

Some non-trivial problem involving partial trace, any help is appreciated! Consider a system consisting of subsystems 1 and 2, and ${\rm Tr}_1$ (${\rm Tr}_2$) is a partial trace over subsystem 1 (2), ...
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### The trace distance between two gaussian states [closed]

If I know two gaussian states' means and covariance matrixes (both of them are two-mode gaussian states), how can I compute the trace distance between them?
1 vote
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### Why does block coding via typical strings give messages longer than $nH(p)$?

This semester, I am taking a course on quantum information and quantum computing. Since I am rather new to information theory I have a problem with understanding a paragraph in my lecture notes. The ...
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### Queues wait for other queues: A communication problem

I am working on a problem which involves a single server that requires multiple inputs to do a computation. Each of these inputs arrive as a Poisson process with rate $\lambda$. Hence, a situation ...
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### Integration by substitution in Selberg's Integral

I am reading the article "Hilbert--Schmidt volume of the set of mixed quantum states" (https://arxiv.org/abs/quant-ph/0302197). I do not understand the step in which we start from (4.2) and ...
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### Isometric maps between spaces with different dimensions

I have the following problem: Consider complex Euclidean spaces H and H′ such that $dim(H) ⩽ dim(H′)$. Let ${|v_i⟩}^{k}_{i=1} ⊂ H$ and ${|w_i⟩}^{k}_{i=1} ⊂ H′$ denote orthonormal sets of vectors. Show ...
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### Sum of entangled operators is entangled

Let $\mathcal{H},\mathcal{K}$ be Hilbert spaces ($\cong \mathbb{C}^2$ if necessary) and let $h_1,h_2$ denote semi-positive $\ge 0$ operators on $\mathcal{H}\otimes \mathcal{K}$. If $h_1,h_2$ are ...
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1 vote
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### Range is product then so must the operator

Consider an operator $\eta$ on the (tensor) product of Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_2$ such that its range is of the form $V_1\otimes V_2$. Does this imply that $\eta$ must be a ...
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### Different ways of defining Entanglement

Consider (finite-dim) Hilbert spaces $\mathcal{H},\mathcal{K}$ and let $A$ denote an operator on $\mathcal{H}\otimes \mathcal{K}$. There are then two ways of defining the entanglement" of ...
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### A proof of no-cloning theorem in the case of pures states in a qubit system

I am working on this problem Consider a qubit $\scr H =\Bbb C^2$ and pure states. Prove the no-clonning theorem ( hint: Use the linearity of the channel to arrive to a contradiction) I wonder if the ...
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1 vote
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1 vote
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### Schmidt decomposition in larger Hilbert spaces

Consider a bipartite quantum system described by the density operator, $\hat{\rho}$, an operator acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. This matrix will be a ...
1 vote
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### Outer product as an operator in an infinite Hilbert space

The outer product between a bra-ket $|a\rangle\langle a|$ where if $|a\rangle\in\mathcal{H}$ and $\langle a|\in\mathcal{H}_{dual}$ is a vector in the tensor vector space formed by the Hilbert space ...
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### Going between two sets of vectors in $\mathbb{C}^d$

Let $|\psi_1 \rangle, \dots, |\psi_n \rangle, |\phi_1 \rangle, \dots, |\phi_n\rangle \in \mathbb{C}^m$. Show that if $\langle \psi_i | \psi_j \rangle = \langle \phi_i | \phi_j \rangle$, then there ...
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### Concentration around $L_2$ average

An exercise from Aubrun-Szarek book: Alice and Bob Meet Banach The Interface of Asymptotice Geometric Analysis and Quantum Information Theory: I want to know how to prove this: Exercise 5.46 - Let $f$ ...
289 views

### How to find expectation value $p_y$ from the Bloch sphere?

Consider an arbitrary state: $$|\psi\rangle = a|0\rangle+b|1\rangle,$$ where $a=cos\left(\frac{\theta}{2}\right), b=sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is the ...
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1 vote
35 views

### Example of statistical data with the property of being contextual that is generated by quantum mechanics

Definition for the specifics of the question as well as an example of contextuality in quantum mechanics I have a set of measurements acting on a 2 qubit state for whom the statistics of the ...
1 vote
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### How to diagonalize an infinte dimensional operator

I want to take logarithm of an infinite dimensional operator given by $\rho = \int\int dx_1 dx_1'C(x_1,x_2)C^*(x_1',x_2)|x_1\rangle\langle x_1' |$, where $C(x_1,x_2)$ is a gaussian function in $x_1$ ...
56 views

### Predual of von Neumann algebra M is closed in the dual $M*$ [closed]

I am reading Stephane Attals Book Open Quantum Systems 1, chapter Elements of Operator Algebras and Modular Theory. And have issues understanding the proof given here. First of all in the "...
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1 vote
83 views

Consider the following circuit : where $|\psi\rangle$ is a qubit in $\mathbb{C}^2$, $|0\rangle= \begin{pmatrix}1 \\ 0 \end{pmatrix}$, $T= \begin{pmatrix}1 & 0\\ 0 & e^{i\pi/4} \end{pmatrix}$ ...
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### quantum guesswork of an ensemble

I am reading an article about quantum guesswork [ Guesswork of a quantum ensemble by michele Dall'Arno ] and i am wondering why he considers ensembles whose traces sum up to 1 in the introduction. I ...
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