Questions tagged [density-matrix]

A density matrix is a matrix that can be used to describe a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state.

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87 views

How can we upper bound the norm of a partial trace?

Suppose we have the normalised states $|\phi_{1}\rangle,|\phi_{2}\rangle \in A \otimes B$ where $A$ and $B$ are $d$-dimensional complex vector spaces. Suppose $|\langle\phi_{2}|\phi_{1}\rangle| < ...
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Produce a quantum state with its density matrix an identity matrix up to an constant

For a n-qubit quantum state $|\psi\rangle=\displaystyle\sum_{i=0}^{2^N-1}|i\rangle$, by definition it's density matrix is $|\psi\rangle\langle\psi|=\displaystyle\sum_{i,j=0}^{2^N-1}|j\rangle\langle i|$...
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Is the set of two-qubit absolutely separable states convex?

Companion question on MathOverflow Let us order the four nonnegative eigenvalues, summing to 1, of a two-qubit density matrix ($\rho$) as \begin{equation} 1 \geq x \geq y \geq z \geq (1-x-y-z) \geq 0. ...
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93 views

Depolarization of density operator with zeros in diagonal

I suppose a quantum state with density matrix like the following is not valid. $$ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. $$ Now, let's say I have a valid density operator representing ...
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91 views

Trace distance of two classical-quantum state with hashing

Let's say I have a classical-quantum(cq) state $\rho_{XE}$, where the classical part $(X)$ is orthogonal. It's trace distance from another uniform density operator is defined to be: $$ \frac{1}{2}||\...
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Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?

The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
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Proof of the no-communication theorem

Let $A, B$ be (finite-dimensional) Hilbert spaces, and $\rho$ some mixed state of $A \otimes B$. I am trying to show that a measurement performed on the '$A$-subsystem' does not affect $\rho^B = \text{...
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What is the relation between density matrices and phase-space probability distributions?

According to its tag description, a density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical ...
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Computation of qubits with quantum gates using density matrix form

I'm making a quantum circuit with qubits and quantum gates. While I'm doing it, I have some problem with it. My calculation process is below. As you can see, start qubit is $|0 \rangle$ and after 'X' ...
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How to represent the state vector form of a qubit in density matrix representation? [duplicate]

While I'm studying state vector and density matrix. I wonder how to write qubit state as density matrix. qubit state can be represented with state vector form. But how about density matrix?
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Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?

Below is a question and an answer. How does quantum information relate to, diverge from or reduce to Shannon information, which used log probabilities? What people are more often interested in are ...
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129 views

Trace distance bound after partial trace

Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.} I know that the trace distance ...
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Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
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What is the intuition behind "states with support on orthogonal subspaces"?

I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia: "In mathematics, ...
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How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?

Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate: $$\newcommand{\tr}{\operatorname{trace}} \tr(A \sqrt{B} A \sqrt{B}). $$ I saw that this trace can be ...
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Are perfectly LOCC-indistinguishable states necessarily identical?

Let $\rho,\sigma\in\text{L}(\mathcal{H}_{XAB})$ be given by $$ \rho = \sum_x |x\rangle\langle x|\otimes p_x\rho_x, \quad \sigma = \sum_x |x\rangle\langle x|\otimes q_x\sigma_x, $$ and consider ...
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Initialising impure density matrices

I wish to initalise the state $\rho=(1-\frac{p}{2})|0\rangle \langle0|+\frac{p}{2}|1\rangle\langle1|$, where p is some measure of decoherence. This is a mixed state. There are some suggestions on here ...
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Information about two algorithms of Matrix product state

In qiskit backends, there is Matrix_product_state. With this backend, I can simulate circuit for several qubits. And I found some mysterious problem about MPS. With 25,26,27 qubits, the simulating ...
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56 views

Semi-definite program for conditional smooth max-entropy

I am aware of a SDP formulation for smooth min-entropy: question link. That program for smooth min-entropy was found in this book by Tomachiel: page 91. However, I am yet to come across a semi-...
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Give an explicit derivation of the exact formula for the two-qubit absolute separability Hilbert-Schmidt probability $\approx 0.00365826$

The two-qubit eigenvalue ($\lambda_i$ >= 0, $i=1,\ldots,4$, $\lambda_4=1-\lambda_1-\lambda_2-\lambda_3$) condition of Verstraete, Audenaert, de Bie and de Moor AbsoluteSeparability (p. 6) for ...
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Can Alice and Bob distinguish entangled state coefficients?

Suppose Alice and Bob share the quantum state $\frac{1}{\sqrt 2}(|x\rangle + (-1)^b |y\rangle)$ for some $x\neq y \in \{0,1\}^2$ and $b \in \{0,1\}$. They both do not know $x,y$, and use some ...
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Is there a proof or example to show that a noiseless subsystem is not necessarily closed under addition?

In a text (section 3.6 page 92) about noiseless subsystems by D. Lidar, it is mentioned:'A subsystem is a tensor factor in a tensor product, and this does not have to be a subspace (e.g., in general ...
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Is the partial trace $\mathrm{Tr}_B(\rho)$ equal to $\sum_k \mathrm{Tr}[(\sigma_k\otimes I)^\dagger \rho]\sigma_k$?

Assume a composite quantum systes with state $|\psi_{AB}\rangle$ or better $\rho=|\psi_{AB}\rangle\langle\psi_{AB}|$. I want to know the state of system $A$ only, i.e. $\rho_A$. Is there any ...
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Reduced Density Matrix Equation of Motion to describe an Ellipse

Given a pure two qubit state $|\psi_{AB}\rangle$. If we trace out system $B$, the remaining density matrix $\rho_A = Tr_B|\psi_{AB}\rangle\langle\psi_{AB}|$, can be represented as a point lying ...
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How to get the state of an individual qubit in a composite system?

Given a composite system with $N$ qubits represented by some $2^N$-dimensional vector, how would I get the quantum state of an individual qubit? Note that I understand some states are not separable ...
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Can we compute a full density operator $\rho_{AB}$ from its reduced density operators $\rho_A$ and $\rho_B$?

Given density operator of a composite system, say $\rho_{AB}$, we can always calculate reduced density operators of individual system i.e. $\rho_{A}=Tr_{B}(\rho_{AB})$ and $\rho_{B}=Tr_{A}(\rho_{AB})$....
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Calculate probability of a state after depolarization

Let's say I have a particle in the quantum state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd ...
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Relation between Wigner quasi-probabability distribution and statistical second-moment

Is there any relation between the Wigner quasi-probability distribution function $W$ and the statistical second-moment (also known as covariance matrix) of a density matrix of a continuous variable ...
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What can be inferred about the closeness of reduced qubit states from the closeness of the bipartite quantum state?

Given a qubit state $|\psi\rangle \in \mathcal{H}$, and two bipartite general mixed states $\rho$ and $\sigma$, such that, $$\langle \psi|\otimes \langle \psi|\rho - \sigma |\psi\rangle \otimes |\psi \...
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Is the Hilbert-Schmidt probability simply zero that a generic rank-2 two-qubit ("pseudo-pure") density matrix is separable?

The multifacted evidence is very compelling--although not yet presented in a formal proof--that the Hilbert-Schmidt probability that a generic (full rank/rank-4) two-qubit density matrix is separable ...
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Semi-definite program for smooth min-entropy

The conditional min-entropy is defined as (wiki): $$ H_{\min}(A|B)_{\rho} \equiv -\inf_{\sigma_B}\inf_{\lambda}\{\lambda \in \mathbb{R}:\rho_{AB} \leq 2^{\lambda} \mathbb{I} \otimes \sigma_B\} $$ And ...
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Mixed state vs superposition , experiment test

To distinguish between a coherent and de-cohered stage of the same system what experiments can provide the answer? The term Experiment is used here in the Bohr-Einstein-debate sense, a realizable ...
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Joint system of RAB after purification of A into R

Given a pure state $|\psi\rangle_{AB}$ on a joint system $AB$, we can consider the reduced density operator $\sigma_A = Tr_B(|\psi \rangle \langle \psi|)$ on $A$ and subsequently purify this state ...
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Density matrix of spatially (i.e., at the same time instant) vs. causally ( i.e., one evolves into the other) correlated quantum systems

In the classical case, if Y is the output of a classical channel whose input is X, it makes sense to speak of a joint distribution $P_{XY}$. In the quantum case, if a state $\rho_A$ is input to a ...
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Quantum operation to get rid of small but nonzero eigenvalues

Updated and edited question: Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\...
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Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$

I am trying to show that for any qubit state p, the following holds: $$I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$$ I have tried different manipulations,...
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Can an ensemble of pure states give probability less than 1?

I am calculating the reduced density matrix of a bipartite system, I ended up getting that it was the sum of two density matrices of pure states each with a probability $1/3$. My understanding was ...
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What does it mean geometrically (in terms of vectors in the Bloch sphere) if the commutator of two density matrices $ρ_1$ and $ρ_2$ vanishes?

When the commutator of two operators vanishes then we can measure one without affecting the other. I'm not sure how this translates in the case of density matrices. If the density matrices are ...
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Increasing the von Neumann entropy despite the measurement?

Background Assume we have a density matrix $\rho$ of a sub-ensemble. However, we have an imperfect measuring instrument. While it does perform a measurement, we do not know exactly when it performs ...
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$M(\rho)=\operatorname{Tr}_2[U(\rho\otimes\rho_2)U^{\dagger}]$ is unitary $\iff U=U_1\otimes U_2$, a product of $2$ unitary operators?

Let $\rho : V_1 \to V_1 $ and $\rho_2 : V_2 \to V_2 $, where $V_1$ and $V_2$ are Hilbert spaces. Suppose that $U:V_1\otimes V_2 \to V_1\otimes V_2$ is a unitary operator. Define a map $M : L(V_1, ...
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Is my $|0\rangle$ state mixed or pure?

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$...
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Composition of tensor product

I don't have much confidence with density matrices, and I would like to be sure about a property of composition of tensor products operations. Specifically, $$ \sum_i \sum_j |a_i\rangle|b_i\rangle\...
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534 views

How to perform Quantum Process Tomography for three qubit gates?

I am trying to perform Quantum process tomography (QPT) on three qubit quantum gate. But I cannot find any relevant resource to follow and peform the experiment. I have checked Nielsen and Chuang's ...
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What is the density matrix of $|+\rangle$ with respect to basis $\{|+\rangle, |-\rangle\}$?

Prove that the density matrix of $|+\rangle$ with respect to basis $\{|+\rangle, |-\rangle\}$ is given by $$\rho = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}.$$
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following: $$ F(\rho, \sigma) := \...
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Is the set of density operators invariant under the induced action of the unitary group?

Show that the set of density operators is invariant under the induced action of $U(H)$ on $End(H)$. I know that a density operator must be positive and have a trace equal to one. But I don't know ...
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Need some help with Purity calculation

maybe you could help me a little about my calculation of a quantum pure state with purification. I have this density matrix: \begin{equation} \rho= \begin{pmatrix} 0.4489 & 0.2304 & 0.2162 &...
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Computation of a reduced density matrix

On wikipedia, the article on quantum entanglement gives an example of the computation of a reduced density matrix. I would like to understand precisely what's going on with the computation. First the ...
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108 views

Numerical methods for finding an eigen basis of a degenerate Liouvillian

I'm trying to find the steady-state of a master equation, $$\dot{\rho}(t) = \mathcal{L}\rho(t),\tag{1}\label{1}$$ In the form where we vectorise the density matrix and matrixify (??) the Liouvillian ...
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What does "decoherence attenuates the density matrix" mean?

I'm reading the paper Implementation of the Quantum Fourier Transform. On page 4, they write To a first approximation, decoherence during the course of the QFT attenuates the entire density matrix. ...