Questions tagged [quantum-information]

This tag is related to Quantum information Theory.

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29 views

Writing Probability distribution in terms of a trace over a density matrix

I have been given and expression for a probability distribution precisely, $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ and I have been asked to show that the above expression ...
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26 views

Query about Partial Transpose

Suppose $A$ is a partitioned matrix whose $ij^{th}$ block is $A_{ij}$ and each block is a square matrix of same size. The partial transpose of matrix $A$ is denoted by $A^{\tau}$. The $ij^{th}$ block ...
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1answer
35 views

How to prove that frame functions on Hilbert spaces are additive?

Let $\newcommand{\calH}{\mathcal{H}}\newcommand{\eff}{\operatorname{Eff}(\calH)}\calH$ be some separable Hilbert space, and denote with $\eff$ the set of effects on $\calH$, that is, the set of ...
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1answer
48 views

Can't understand how the authors got this equation?

Let us consider a two-qubit system (A and R) initially entangled as given by: $$ \left| {{\Psi _{AR}}} \right\rangle = \alpha \left| {{0_A}} \right\rangle \otimes \left| {{1_R}} \right\rangle + \...
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1answer
21 views

Is the convex hull of product channels strictly contained in the set of all channels

For two finite dimensional Hilbert spaces $\mathcal{H},\mathcal{H'}$ let $T(\mathcal{H},\mathcal{H'})$ be the set of all CPTP maps (quantum channels) $L(\mathcal{H})\to L(\mathcal{H}')$. Let $\mathcal{...
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1answer
56 views

Is the support of $\rho_{AB}$ always contained in the support of $\rho_A\otimes\rho_B$?

Given a positive unit trace Hermitian matrix, i.e. a density operator, $\rho_{AB}$ on Hilbert spaces $\mathcal{H}_A\otimes\mathcal{H}_B$. Consider its marginals $\rho_A$, $\rho_B$. Do we have the ...
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47 views

Sums of largest arithmetic and geometric means of a probability vector

The Question Let $p\in\mathbb{R}^d_{\geq 0}$ with $\sum\limits_{i=1}^d p_i = 1$ be a probability vector. Define the matrices $a=(a_{ij})_{i,j}$, $g=(g_{ij})_{ij}\in\mathbb{R}^{d\times d}$ via \begin{...
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1answer
69 views

Bipartite Version of Schur's Lemma

Let $V$ be a finite-dimensional vector space, let $\pi$ be a representation of some group $G$ (I'm really interested in $G=SO(n)$) on $V$, and suppose that a linear operator $S$ on $V\otimes V$ ...
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45 views

Quantum Typicality

Let $\rho, \sigma \in \mathcal{D}(\mathcal{A})$ with $\text{supp}(\rho) \subseteq \text{supp}(\sigma)$, and spectral decomposition \begin{align*} \rho = \sum_{x}p_x |\psi_x\rangle\langle\psi_x| ~~\...
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1answer
57 views

$\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{\operatorname{Tr}[\rho M]}{\operatorname{Tr}[\sigma M]}$ satisfies Data Processing Inequality

$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{H}_1, \mathcal{H}_2$ be a Hilbert spaces and $\rho, \sigma$ be density matrices on $\mathcal{H}_1$. Define $$D(\rho\parallel\sigma) := \sup_{0 \leq M ...
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15 views

$\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{Tr[\rho M]}{Tr[\sigma M]}$ satisfies DPI [duplicate]

Let $\mathcal{H}_1, \mathcal{H}_2$ be a Hilbert spaces and $\rho, \sigma$ be density matrices on $\mathcal{H}_1$. Define $D(\rho||\sigma) := \sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{Tr[\rho M]}{Tr[...
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1answer
26 views

Proof that the adjoint operator of $\Phi$ is $\Phi(\sigma)=\sum_{i}A_{i}^{*}\sigma B_{i}$ in terms of Kraus operators.

I am studying basics of quantum information theory and I am trying to prove the fact that the adjoint operator of $\Phi\in\mathcal{L}(\mathcal{L(\mathcal{H})},\mathcal{L(\mathcal{K})})$ is $\Phi^{*}(\...
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63 views

Bimodule Maps and Opposite Algebras

Let $B$ be a finite dimensional $C^\star$ algebra, and $B\subseteq \mathcal{B(H)}$ for some finite-dimensional Hilbert space $\mathcal{H}$. By $B'$, let us denote the commutant of $B$ in $\mathcal{B(H)...
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14 views

Maximum number of independent commuting Paulis

Let $P_n$ be the Pauli group on $n$ qubits. Let $H \subseteq P_n$ s.t. $H$ is commuting and consist of independent elements. it is well known that $|H| \leq n$. However I have not been able to find a ...
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46 views

Computing a certain functional derivative.

I'm trying to see what happens when we extended the notion of a convex roof definition to infinite dimensions. In finite dimensions if we can define an entanglement measure $e$ on pure states of $\...
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1answer
46 views

Proof Rényi entropy is non-negative

The Rényi entropy is defined as: \begin{equation} S_\alpha = \dfrac{1}{1-\alpha}\log(\text{Tr}(\rho^\alpha)) \end{equation} for $\alpha \geq 0$. This can be rewrited in terms of $\rho$ eigenvalues, $\...
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1answer
52 views

Can a nonzero win-win situation appear in a purely competitive 2 players game?

I thought using quantum theory to achieve that goal, namely that the distinction between players is artificial and in fact each player is a superposition of players given in the data, since they ...
1
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1answer
38 views

Density operators and spectral decomposition

There are two questions I'm curious about, and both of them may be incredibly silly. I just haven't been able to convince myself otherwise. We know that a density operator $\rho_E$ has the form $\...
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9 views

Existence of LOCC orthogonal projection measurement.

Let $M = \{M_x\}_{x\in X}$ be a POVM on $\text{End}(H_{AB})$. By Naimark's dilation theorem there exists an orthogonal projection measurement $P = \{P_x\}_{x\in X}$ on some larger space yielding the ...
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43 views

Defining operator norm as the spectral norm of its matrix

Suppose $T$ maps between symmetric positive definite $d$-by-$d$ matrices $X,Y$ as follows: $$\operatorname{vec}(Y)=M\operatorname{vec}(X)$$ $M$ is a symmetric positive definite $d^2$-by-$d^2$ matrix. ...
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52 views

Classical Information Theory or Quantum Information Theory - which one has more applications and will be more important for the future?

Please do forgive me as I have no previous background in Information Theory whatsoever, but I‘m essentially wondering which one out of CIT or QIT will be more important in the future for research and ...
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1answer
44 views

quantum tensor equation simplification

I've started learning quantum computing of late and got interested in some secret sharing. In a recent post on the site https://quantumcomputing.stackexchange.com/questions/13195/grover-search-with-...
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41 views

Proving $S(A:B)+S(A:C) \le 2S(A)$ using subadditivity

I am fairly new to this, but my attempt below involves using a fourth system R to purify. $$S(A,B,C)+S(B) \le S(A,B)+S(B,C),$$ introducing a system R which purifies ABC, we get $$S(A,B,R)+S(B) \le S(A,...
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1answer
57 views

Confused regarding norm and why is $\langle a|b\rangle\langle b|a\rangle = ||\langle a|b\rangle||^2$

Given $\langle v|v\rangle=\sum_{i}a_{i}^2$ then $|||v\rangle||=\sqrt{\langle v|v\rangle}=\sqrt{\sum_{i}|a_{i}|^2}$. Also, $\langle a|b\rangle=\langle b|a\rangle^{*}$. So how then is $\langle a|b\...
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1answer
33 views

When getting the outer product representation w.r.t an input and output basis, what matrix is $\langle w_{j}|A|v_{i}\rangle$ actually in?

If $A:V\to W$, where $|v_{i}\rangle$ is an orthonormal basis for $V$ and $|w_{j}\rangle$ is an orthonormal basis for $W$, then $A=I_{W}AI_{V} = \sum_{ij}|w_{j}\rangle\langle w_{j}|A|v_{i}\rangle\...
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18 views

Can logical system rules be transformed to some kind of quantum logical circuits?

I have read some of the basic theory involving quantum logic circuit theory. Now, if I understood correctly, every quantum circuit: Has the same number of input as outputs Is reversible (we can ...
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63 views

When do stochastic processes increase mutual information?

Suppose we have a composite system that is made up of two random variables $A,B$. And suppose also that the mutual information of the system is non-zero, $I(A;B) > 0$. We then apply a stochastic ...
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2answers
16 views

Proving $\frac{1}{2^n}\sum_{z\in\{0,1\}^n} (-1)^{z\cdot (x\oplus y)}=\delta_{xy}$, where $x\oplus y$ is the bitwise sum

In quantum algorithms I often find this identity: $$\frac{1}{2^n}\sum_{z\in\{0,1\}^n} (-1)^{z\cdot (x\oplus y)}=\delta_{xy}$$ where $x\oplus y$ is the bitwise sum. I am not able to prove in general ...
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0answers
109 views

Finding a class $C$ of bipartite PPT states such that entanglement of $\rho \in C$ implies entanglement of $\rho + \rho^{\Gamma}$.

Consider an entangled bipartite quantum state $\rho \in \mathcal{M}_d(\mathbb{C}) \otimes \mathcal{M}_{d'}(\mathbb{C})$ which is positive under partial transposition, i.e., $\rho^\Gamma \geq 0$. As ...
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26 views

Coherent Information of Entanglement Breaking channels

The book by John Watrous, "The Theory of Quantum Information" is an exciting read for anyone wanting to research quantum information theory. The following question presumes some background ...
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38 views

if $(|a\rangle - |b\rangle)\langle b| + |b\rangle(\langle a| - \langle b|) = 0$ then $|a\rangle = |b\rangle$, where $|a\rangle, |b\rangle$ unit

Essentially the problem above. I've tried approaching it by arguing that $|b\rangle(\langle a| - \langle b|)$ is the adjoint of $(|a\rangle - |b\rangle)\langle b|$, and since $|b\rangle$ nonzero, ...
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2answers
134 views

How are spin network edges related to anti-symmetric projectors on the Hilbert space of the fundamental rep of SU(2)?

In the paper here https://arxiv.org/pdf/gr-qc/9905020.pdf we see an introduction to Spin-networks of the original Penrose type i.e an undirected open graph whose edges have labels that are irreducible ...
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1answer
49 views

Showing if scalar product of vectors is 1 then they are the same quantum state

How do you show that if $|\langle \psi|\phi\rangle| = 1$, then $\phi$ and $\psi$, both of dimension $d$, represent the same quantum state? (Same quantum state iff there exists a $\theta$ s.t. $|\psi\...
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0answers
36 views

What kind of terms dominate this summation? : Specific question

Here is my summation term. $\Sigma_{l=0}^{np}\langle e_{1}|f_{1}\rangle^{l}\langle e_{1}|f_{2}\rangle^{np-l} \langle e_{2}|f_{1}\rangle^{nq-l} \langle e_{2}|f_{1}\rangle^{n(1-q)-np+l} {nq\choose l} {n(...
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1answer
60 views

Is partial trace additive in sense of direct sum?

I have an intuition, but not sure exactly, whether the partial trace is additive in the sense of direct sum. My intuition is that partial trace is additive and direct sum acts as a sum but with ...
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0answers
55 views

A precise formula for the summation of an inner product

We have 2 strings $|v\rangle$ and $\langle u|$, $|v\rangle=|e_{1}\rangle^{np}|e_{2}\rangle^{n(1-p)}$ where $e_{1}$ occurs $np$ times and $e_{2}$ occurs $n(1-p)$ times and $\langle u|=\langle f_{1}|^{...
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1answer
122 views

Why can we disentangle the squeezing operator $\exp[\frac12(\xi a^{\dagger 2}-\xi^* a^2)]$ via the $\mathfrak{su}(1,1)$ algebra?

In the context of quantum mechanics, we define the squeezing operator $S(\xi)$ as: $$S(\xi)\equiv \exp[\frac12(\xi a^{\dagger 2}-\xi^* a^2)],$$ where $a^\dagger$ and $a$ are the so-called creation and ...
4
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1answer
114 views

Quantum information theory and nonlocal games

We are running a seminar among PhDs in math and try to learn some quantum information theory and quantum computation. Being an analyst I'd like to make a contribution with a topic in that direction. I ...
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0answers
22 views

Definition of indistinguishable states in classical statistics and separability

The problem of indistinguishable particles motivated new statistics such as Bose-Einstein or Fermi-Dirac that were later formalized by von Neumann as Quantum Statistics. In quantum statistics, states ...
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28 views

What does a private subspace mean?

I am referring to the following definition: "Given a channel $\Phi$ on $\mathcal{H}$ and a subspace $\mathcal{C}$, we say $\mathcal{C}$ is private for $\Phi$ if there is a density operator $\rho_0$ ...
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1answer
56 views

Understanding how to calculate orthogonal projection operator

Let $\mathcal{H} =\mathbb{C}^2, \mathcal{M}_1 = \mathbb{C}|0\rangle$ with $|\psi\rangle = \alpha |0\rangle + \beta|1\rangle$. Show $Pr(\mathcal{M_1}) = |\alpha|^2.$ We know that $\mathcal{M_1}$ is a ...
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2answers
47 views

All states are simple averages of pure states

I was going through the following book "Alice and Bob meet Banach: The interface of Asymptotic Geometric Analysis and Quantum information theory" by Aubrun and Szarek. I found a problem on basic ...
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1answer
30 views

Entries of a unitary matrix

In the solution a a problem in quantum computation I saw this line: $$U_{ij}=\langle\psi_i|\left(\sum_k|\phi_k\rangle\!\langle\psi_k|\right) |\psi_j\rangle.$$ Where $U_{ij}$ are the entries of a ...
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23 views

Quaternionic quantum channels.

What is the proper definition of quaternionic quantum channel and do we have the choi kraus representation theorem for quaternionic case as well?
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1answer
59 views

Prove that the size of generating set of a group is at most log(|G|)

I am studying Nielsen & Chuang's book. In the appendix, they prove a little lemma that if a set $\langle g_1, g_2,...,g_l \rangle$ generates a group $G$, then the size of this set would at most $\...
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1answer
101 views

Trace Differentiation with Pauli operators, finding $\frac{d x}{d t}$ and $\frac{d z}{d t}$ from the master equation [closed]

I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $\theta$. For context this is the setup and derivation I have for continuous measurement along ...
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1answer
31 views

Calculation with finite sums

I'd like to know whether the following calculations are correct, as I obtain an unexpected result. Start from $$\frac{1}{N}\sum_{n=1}^NE_n^2\left(1+\sum_{j\ne k}^N\rho_{jk}\exp\left(i\frac{2\pi n}{N}(...
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0answers
34 views

Completely positive, unital and primitive maps acting on unitaries

Consider a completely positive, unital and primitive map $E$ on the operators on a finite-dimensional Hilbert-space and suppose that there exists a unitary and hermitian operator $U$ (hence $U^2=1$) ...
3
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1answer
56 views

Does $S(\rho_{AB})=S(\rho_A)+S(\rho_B)$ imply that $\rho_{AB}=\rho_A\otimes\rho_B$?

Let $\rho_{AB}$ $\in$ $\mathbb{H_A} \otimes \mathbb{H_B}$ be a density operator with reduced density operators $\rho_A = tr_B[\rho_{AB}]$ and $\rho_B = tr_A[\rho_{AB}]$. I assume that the von Neumann ...
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0answers
29 views

Interest of superdense coding

When you enter the field of quantum communications, you run at some point into the concept of superdense coding. Basically it is a way to encode classical bits on the qubits of a quantum channel, ...