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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Classical versus quantum correlations and partial traces

Given a bipartite state $\rho_{AB}$ living in the Hilbert space $\mathcal H(A\otimes B)$ we can always define two local states on $A$ and $B$ respectively by taking the appropriate partial traces: $$\...
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Can one always find purifications which preserve equality of statistical mixtures?

When pure states $|\psi_1⟩$, $|\psi_2⟩$ and $|\phi_1⟩$, $|\phi_2⟩$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ have identical statistical mixtures $$\frac{1}{2}(|\psi_1⟩⟨\psi_1| + |\psi_2⟩⟨\psi_2|) = \...
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How do get $\rho_{BA}$ if I have $\rho_{AB}$

If Alice and Bob share the state: $$\left| {{\psi _{AB}}} \right\rangle = \sin \theta \left| {10} \right\rangle + \cos \theta \left| {01} \right\rangle $$ then $\rho_{AB}$ can be obtained as: $${\...
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Is there an algorithm that can decide if a state is a mixed state or a pure state?

Given we are using the computational basis,is there a quantum algorithm that can decide if an arbitrary input state $\vert A\rangle$ ( using $N$ qubits) is a pure state or a mixed state? $\vert A\...
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What are the conditions ensuring a two-qubit density matrix is positive semidefinite?

I've seen some papers writing $$\rho=\frac{1}{4}\left(\mathbb{I} \otimes \mathbb{I}+\sum_{k=1}^{3} a_{k} \sigma_{k} \otimes \mathbb{I}+\sum_{l=1}^{3} b_{l} \mathbb{I} \otimes \sigma_{l}+\sum_{k, l=1}^{...
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On existence of orthonormal basis for each subsystem in Separable state [closed]

A separable state in $\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is given by $$\rho_{s}=\sum_{\alpha,\beta}p(\alpha,\beta)|\alpha\rangle\!\langle\alpha|\otimes|\beta\rangle\!\langle\beta|.$$ Now, my ...
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Write the difference of 2 density operators in terms of a spectral decomposition

An exercise question (9.7) from Quantum computation and Quantum Information by Michael E. Nielson and Isaac L. Chuang says that I can write the difference of any 2 arbitrary density operators $\rho,\...
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How to describe the evolution of a density matrix using the Choi matrix?

How do I apply the Choi matrix on a Density matrix. Say my process is a Hadamard gate, and my input state is the ground state on 1 qubit (qubit id 0). $U = H = \dfrac{1}{\sqrt{2}} \begin{bmatrix}1&...
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Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
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Can one quantify entanglement between different parts of a system?

Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
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How can a density matrix be prepared on a quantum register?

I am currently trying to implement the VQSE algorithm. There the biggest eigenvalues and their corresponding eigenvectors of a density matrix $\rho$ are computed. In contrast to VQE, the matrix $\rho$ ...
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Properties of frames in quasiprobability representation

Let $\mathbb{C}^{d}$ be a complex Euclidean space. Let $\mathsf{H}(\mathbb{C}^{d})$ be the set of all Hermitian operators, mapping vectors from $\mathbb{C}^{d}$ to $\mathbb{C}^{d}$. I had some ...
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Does the definition of separability of pure states require the components of the summands to be pure?

Does the definition of separability of pure states require the components of the summands to be pure? More precisely, let $\rho$ be a pure state (i.e., $\rho=|\phi\rangle\langle\phi|$) on the space $...
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When does Hermitian operator with unit trace become a density operator?

The definition of density operators is that (i) positive semidefinite; and (ii) unit trace. Given a Hermitian matrix $\rho$ (say, the size is $2\times 2$) with unit trace, I know that such matrix may ...
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How to find the $A_i$ in the matrix product state representation?

From what I understand, MPS is just a simpler way to write out a state, compared to the density matrix. But how do I get those $A_i$ matrices? From all the examples I read, people just somehow "...
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Why are commuting density operators said to be “classical states”?

In quantum information it is commonly said that a set of states $S=\{ \rho_i \}_i$ is classical if $[\rho_m, \rho_n] = 0, \,\forall \rho_m,\rho_n \in S$. This is meant in the sense that all observed ...
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How to trace out qubits from a multipartite density matrix [duplicate]

I have a density made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to trace out 2nd and 3rd qubits. Is there any reference ...
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How to find the distance between a given $\rho$ and the nearest pure state(s)?

I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state: $$ \min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
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Quantum teleportation of a mixed state through a pure state?

Let's assume we have a register of qubits present in a mixed state $$\rho = \sum_i^n p_i|\psi_i\rangle \langle \psi_i|$$ and we want to teleport $\rho$ through a random pure state $|\phi\rangle$. What ...
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Modeling building blocks for quantum computation

If I would design library for quantum computation I would naively consider a sequences of entangled qudits with unit length as a building blocks. I.e., unit length elements from $$\mathbb{C}^{d_{1}}\...
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What is the superop simulator in Qiskit for?

I'm trying to understand what the use case of a superop simulator would be. My understanding is that density matrix is generally more resource intensive than state vector, but it has additional ...
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Given $|\psi\rangle=(U_A\otimes U_B)|0,0\rangle$, is $|\psi\rangle\!\langle\psi|$ always a product state?

say I have some state in the combined space $\psi$ ∈ $H_A\otimes H_B$, where $\psi=U_A \otimes U_B|0,0\rangle$ (operators from respective spaces), and $\rho_A, \rho_B$ the respective density matrices. ...
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How to get Bloch sphere Cartesian coordinates from density matrix

I am vexed by a particular derivation. Given a state $\psi$ and corresponding density matrix $\rho = |\psi\rangle \langle \psi|$, or $\rho = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$, I ...
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Does $\mathrm{Tr}(\rho\sigma) > 0$ prove that a state $\sigma$ is separable?

As an example I have the density matrix: $\rho = \frac{1}{3}(| \phi^+ \rangle\langle\phi^+| + | 00 \rangle\langle 00|+| 11 \rangle\langle11| )$ And the two-qubit state is: $\frac{1}{3}(| \phi^- \...
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Can we write the density operator as a sum of mixed states?

In every resource I find (like Nielsen and Chuang or online courses), the density operator is defined as follows: we consider a sequence of pure states $\left|\psi_i\right\rangle$ with associated ...
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How can we prove that the covariance satisfies $\mathrm{Cov}_\rho(X,Y)=\mathrm{Cov}_\rho(Y,X)$?

While attempting to prove the Cauchy Schwarz Inequality I came across this problem. First of all, if we are given a $\rho$ density matrix and two matrix of obserables $X,Y$, after defining the ...
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Why does the trace of density operators need to be one?

Usually, the textbook starts with a few assumptions of what density operator $\rho$ has. One of them is $Tr(\rho) = 1$. Why is that?
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Purification applied to indistinguishability

In Zhandry's compressed oracle paper, one can read the following: Next, we note that the oracle $h$ being chosen at random is equivalent (from the adversary’s point of view) to $h$ being in uniform ...
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Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

Let us consider two quantum channels $\Phi:M_d\rightarrow M_{d_1}$ and $\Phi_c:M_d\rightarrow M_{d_2}$ that are complementary to each other, i.e., there exists an isometry $V:\mathbb{C}^d\rightarrow \...
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Quantum indistinguishability using density operators

There is something that bugs me concerning the use of density matrices. For instance, to argue that quantum teleportation does not spread an information faster than light, Nielsen and Chuang state the ...
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Setting the Initial State of Density Matrix Simulation in Cirq

I am trying to use the DensityMatrixSimulator in Cirq. I want to append gates to the circuit conditional on measurements. In order to reduce computational resources, I am trying to run these ...
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Does a partial transpose always have real eigenvalues?

I am working with a tripartite system, but when I partially transpose the $8\times 8$ density matrix I get two complex eigenvalues. I know the criteria for the positive and negative eigenvalues, but ...
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How do you mix two pure states to obtain a mixed state?

If we have the following two states \begin{equation} |\psi\rangle_1 = \frac{1}{\sqrt{2}}|0\rangle_A|0\rangle_B + \frac{1}{\sqrt{2}} |1\rangle_A |1\rangle_B \end{equation} \begin{equation} |\psi\...
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Calculate the von Neumann Entropy of a two-qubit entangled state

After working through an exercise I got a confusion answer/solution that either may be because I've made a mistake or I'm not understanding von Neumann Entropy. I have the two qubit system $$ | \psi \...
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Stinespring dilation using ancilla in mixed state?

The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
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What is the conjugate transpose of $|0\rangle_{A}|1\rangle_{B}$?

Suppose a composite state is in $|0\rangle_{A}|1\rangle_{B}$. Then their conjugate transpose would be $\langle 0|_{A}\langle 1|_{B}$? Note: Why this question? Because I was checking MIT's "...
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How would I compute a density matrix of a complex qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state, $$ \frac{1}{9}\begin{bmatrix} 5 & 1 & −i \\ 1 & 2 &...
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How would I compute a density matrix of a 2 qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this? $$ |00> \;with \;probability \; 2/4 \\ ...
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How many real numbers are required to describe density matrix for $n$ qubits?

(All of these coming from the topic of simulation of quantum systems) A density matrix $\rho$ Which describe state of $n$ qubits will have $2^{n} \times 2^{n}$ size. We have couple of conditions like ...
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Derivation of Equation $8.7$ in Nielsen Chuang [duplicate]

Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358) $$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\...
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What does “bipartite system” mean? [duplicate]

I have seen this term appearing multiple times when discussing density matrices. For example here is an excerpt from the lecture notes: We are now ready to introduce the idea of a reduced density ...
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How to express a probability distribution $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ in terms of a trace of a density matrix?

I have been given and expression for a probability distribution $$P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$$ and I have been asked to show that the above expression can be ...
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Is there an easy way to calculate the eigenvalues of the partial transpose of a given matrix? [duplicate]

Consider the state $$|\psi\rangle=(\cos\theta_A|0\rangle+\sin\theta_A|1\rangle)\otimes(\cos\theta_B|0\rangle+e^{i\phi_B}\sin\theta_B|1\rangle).$$ To calculate the $\rho^{T_B}$ I first calculate the $\...
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How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?

How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?
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How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?

How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state $$ \rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
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What is the difference between $|0\rangle+|1\rangle$ and a balanced mixture of $|0\rangle$ and $|1\rangle$? [duplicate]

Suppose I have a quantum state $\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$. Also I have a mixture of two quantum states $S_{1} = |0\rangle$ and $S_{2} = |1\rangle$. In this mixture $50\%$...
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How do I determine if a given pure two-qubit state is separable?

I'm trying to self-study some topics about quantum computing and I came across a topic of state separability. Talking about that, I wanted to determine separability on the following state (from Qiskit ...
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What do the off-diagonal elements of a density matrix physically represent?

For simplicity, let's take a density matrix for a single qubit, written in the $\{|0\rangle,|1\rangle\}$ basis: $$ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10}^* & 1-\rho_{00} \...
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$\rho_{SE}(0)=\rho_S(0)\otimes\rho_E(0)$: No coupling or no entanglement?

We know that the entangled states cannot be expressed like product state, e.g. $|\omega\rangle = |\psi\rangle\otimes|\phi\rangle$. In the density matrix describing the correlations between system $S$ ...
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What's the 'physical consistency' in the partial trace scenario?

I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand: Physical consistency requires that any ...