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. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

Filter by
Sorted by
Tagged with
6
votes
1answer
128 views

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 ...
4
votes
1answer
52 views

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, ...
2
votes
1answer
51 views

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 ...
5
votes
1answer
147 views

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 ...
1
vote
1answer
53 views

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 ...
0
votes
1answer
29 views

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 ...
1
vote
1answer
31 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-...
2
votes
1answer
82 views

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 ...
3
votes
2answers
54 views

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 ...
4
votes
1answer
131 views

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 ...
3
votes
1answer
53 views

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 ...
2
votes
0answers
64 views

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 ...
2
votes
1answer
59 views

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 ...
2
votes
2answers
48 views

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 $\rho_{B}=Tr_{A}(\rho_{AB})$....
2
votes
2answers
56 views

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 ...
3
votes
1answer
38 views

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 ...
1
vote
1answer
51 views

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 \...
3
votes
1answer
57 views

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 ...
4
votes
1answer
121 views

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 ...
0
votes
2answers
76 views

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 ...
1
vote
2answers
45 views

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 ...
1
vote
0answers
18 views

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 ...
0
votes
0answers
38 views

Convert a two-ququart (16 x 16) density matrix into normal form--so that the components of the Bloch vectors of the two reduced systems are all zero

The two-ququart ($16 \times 16$) "Hiesmayr-Loffler" density matrix https://iopscience.iop.org/article/10.1088/1367-2630/15/8/083036/meta, (https://arxiv.org/abs/2004.06745 eq. (13)), What ...
1
vote
1answer
31 views

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 $\...
2
votes
7answers
193 views

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,...
-1
votes
1answer
25 views

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 ...
3
votes
1answer
38 views

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 ...
1
vote
0answers
96 views

Increasing the von Neumann entropy despite the measurment?

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 ...
3
votes
2answers
61 views

$M(\rho)=\operatorname{Tr}_2\left(\ U\ \rho\otimes\rho_2\ U^{\dagger}\right)$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, ...
2
votes
2answers
206 views

Is my $|0\rangle$ state mixed or pure?

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$...
1
vote
1answer
57 views

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\...
2
votes
1answer
166 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 ...
1
vote
1answer
66 views

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 $$\rho = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}.$$
4
votes
1answer
83 views

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) := \...
0
votes
1answer
36 views

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 ...
1
vote
1answer
49 views

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 &...
2
votes
1answer
59 views

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 ...
1
vote
0answers
24 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 ...
1
vote
0answers
42 views

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. ...
2
votes
1answer
178 views

Understanding the outer products in density matrices

I don't understand a simple property of the outer product when doing density matrices. I am studying nielsen and chuang's book. At equation 2.197 they do show the density matrix of the state of ...
5
votes
3answers
209 views

How to prepare mixed states on a quantum computer?

I am a little bit confused by density matrix notation in quantum algorithms. While I am pretty confident with working with pure states, I never had the need to work with algorithm using density ...
3
votes
2answers
66 views

Applying density matrix based criterion to verify separability

In order to figure out if a given pure 2-qubit state is entangled or separable, I am trying to compute: the density matrix, then the reduced density matrix by tracing out with respect to one of the ...
3
votes
2answers
183 views

Implementation of tomography on IBM Q

I wanted to ask how do you implement a circuit that finds the non-diagonal values of the density matrix of a quantum state on IBM Q?
1
vote
0answers
34 views

Are there different orderings of the fifteen SU(4) generators in common use?

I've recently performed certain analyses (Archipelagos of Total Bound and Free Entanglement) pertaining to eq. (50) in Separable Decompositions of Bipartite Mixed States , that is \begin{equation} ...
4
votes
3answers
123 views

Why do purifications only differ by a local unitary?

Let's consider $\rho_A$ a density matrix. I introduce a space $B$ and an entangled state $|\Psi\rangle$ (the purification) so that: $$\newcommand{\tr}{\operatorname{Tr}}\rho_A = \tr_B(|\Psi\rangle \...
3
votes
0answers
33 views

Find the qutrit analogue of certain qubit and ququart formulas of Li and Qiao for testing separability

In eqs. (33), (43)-(46), (56) of their paper, "Separable Decompositions of Bipartite Mixed States" https://arxiv.org/abs/1708.05336, Li and Qiao present a number of formulas pertinent to testing the ...
3
votes
2answers
60 views

Simulating Classical Probabilistic Transitions with superoperators

I'm working on the following exercise: "Show how a classical probabilistic transition on an M -state system can be simulated by a quantum algorithm by adding an additional M -state ‘ancilla’ ...
4
votes
1answer
301 views

Why are $d^2$ dimensions required to describe a density matrix?

A density matrix is defined as: $$\sum p_i |\psi_i \rangle \langle \psi_i|$$ If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
1
vote
1answer
100 views

Density matrix for a diagonally polarized photon

I am struggling with the density matrix for diagonally polarized photons. Can I think of diagonally polarized photons as a mixture of vertically and horizontally polarized photons?
3
votes
0answers
152 views

What is “Lindblad Superoperator” in Stochastic Master Equation

I was reading a paper titled "Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations" and was confused about a stochastic master ...