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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How to write the Kraus representation for many-qubit states?

The most general formula of Kraus operator on density matrix is: $$\rho\to \sum_k A_k^\dagger\rho A_k.$$ If I want to write this equation for one qubit, the most general way will be: $\rho_f = (a^*I+b^...
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Helstrom Measurement when two quantum states are close

I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true. To summarize the description, suppose we have two ...
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Why does $\rho^A=\mathrm{tr}_B(\rho^{AB})$ guarantee that $\mathrm{tr}(M\rho^A)=\mathrm{tr}((M\otimes I_B)\rho^{AB})$?

Niesen and Chuang, 2nd edition, page 107, Box 2.6, in its motivation for partial trace, says that if M is an observable on system A and $\tilde{M}$ is the corresponding observable on system AB, then ...
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Does ${\rm tr}(\Pi \rho) = 1$ imply $\Pi\rho\Pi=\rho$?

Suppose I have a density matrix $\rho$ and an orthogonal projector $\Pi$. Is it true that, if $tr(\Pi \rho) = 1$ then it must hold that $$\Pi \rho \Pi = \rho$$? If yes, how can I prove it?
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Mathematics of Measurement then Partial Trace

Say we have the following quantum state: $$ |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|10\rangle)$$ To measure the first qubit and then further trace out the first qubit, my notes have the ...
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Density Matrix of 3 Qubit Quantum Circuit (Qiskit Density Matrix)

If I have a one gate circuit like in the following image: I can calculate the density matrix by following steps: However, I want to do it for the Toffoli gate circuit: and I am confused with the ...
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Data processing inequality for relative entropy in the presence of an amplitude damping channel

Consider the single qubit quantum depolarizing channel, given by $$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}. $$ For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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Are quantum gates superoperators? How to write a quantum circuit as superoperator?

I have some questions related to superoperators: What is the differences between quantum operators and superoperators? For instance quantum gates are also unitary operators but can we say quantum ...
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Are the states in the convex decomposition of a density matrix necessarily orthogonal?

In Nielsen and Chuang's QC&QI, I do not see a statement one way or another. In Steeb and Hardy's Problems and Solutions, orthogonality is asserted. If the $p_i$ in $\sum_i p_i |\psi_i\rangle\...
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Given a three-qubit state, how do you obtain the density matrix for the third qubit

I have a quantum simulator that yields a three-qubit final state. However, I need to measure the first two qubits and apply a one-qubit gate (x,y or z) to the third qubit. How do you reduce a three-...
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How to compute the unitary from the $\chi$ matrix obtained from QPT

I am trying to do quantum process tomography for one qubit and obtain the unitary for the gates that are applied on the qubit. I have studied the theory on process tomography from mike and ike and the ...
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Simulating a quantum circuit with decoherence and noise

Based on the answers given here and here, it is pretty clear that an arbitrary quantum circuit can be simulated with matrix algebra. The difficulty is that this assume perfect fidelity. I am unsure ...
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Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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QuTiP VS RK45: Which one gives the correct results for time-dependent systems?

I am writing a code for a quantum thermal machine which includes both coherent and dissipative time evolutions in its different stages of operation. However, evolving the system with "mesolve&...
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Evaluation of Wigner function representation of a Bloch Sphere

Consider Wigner function representation of a qubit in the basis labeled by $\sigma_z$ and $\sigma_x$ eigenvalues. A general single qubit mixed state has the Bloch representation,$\rho = 1/2 (I + r.\...
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Show that if the Lindblad satisfy $\sum_\mu L_\mu L_\mu^\dagger=\sum_\mu L_\mu^\dagger L_\mu$ then $\rho\propto I$ is a fixed point of an evolution

How can we show that the Lindblad condition: $$\sum_{\mu}L_{\mu} L_{\mu}^{\dagger} = \sum_{\mu} L_{\mu}^{\dagger}L_{\mu},\tag{1}$$ implies that $\rho \propto I$ is the fixed point of the evolution ...
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Show that if the Lindblad satisfy $\sum_\mu L_\mu L_\mu^\dagger=\sum_\mu L_\mu^\dagger L_\mu$ then the von Neumann entropy increases monotonically

How can we show that when the Lindblad operators satisfy the condition: $$\sum_{\mu}L_{\mu} L_{\mu}^{\dagger} = \sum_{\mu} L_{\mu}^{\dagger}L_{\mu},\tag{1}$$ the master equation evolution ...
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Closeness of $\rho$ such that $\text{Tr}(|\psi\rangle\langle\psi|\rho)\le1/2^n+{\cal O}(2^{-2n} )$ for all $|\psi\rangle$ to the maximally mixed state

Consider an $n$ qubit density matrix $\rho$ such that $$\text{Tr}(|\psi\rangle\langle \psi| ~\rho) \leq \frac{1}{2^{n}} + \mathcal{O}\left(\frac{1}{2^{2n}} \right), $$ for every $n$ qubit pure state $|...
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Schmidt vectors for random quantum states

Consider a random quantum circuit $U$ over $n$ qubits, drawn from the Haar measure. Consider the quantum state $$|\psi\rangle = U |0^{n}\rangle.$$ Now, partition $n$ into two and consider the Schmidt ...
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Show that the trace of squared density matrix gives ${\rm tr}(\rho^2)=\frac12(1+\|\mathbf n\|^2)$ [duplicate]

Equation 7.7 is given below: $$\hat\rho = \frac12(I +n_x(\hat X)+n_y(\hat Y)+n_z(\hat Z)) $$ Where $I$ is the identity matrix and $\hat X,\hat Y,\hat Z$ are Pauli matrices. Now my attempt of this was ...
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Is it possible to find a 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio? [closed]

Is it possible to find 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio and if it is how is it done?
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Reduced density matrix of a Haar random state and its Schmidt decomposition

Consider a Haar random quantum state $|\psi\rangle$. Note that $$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}.$$ $\mathbb{I}_n$ is the identity operator on $n$ qubits. ...
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Finding Wigner function of four maximal entangled Bell state

How can we find a Wigner function for the four maximally entangled Bell states $(|00\rangle \pm |11\rangle)/\sqrt{2}$, $(|01\rangle \pm |10\rangle)/\sqrt{2}$? I have used the basis where labels for ...
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Tripartite quantum marginal problem

Consider a tripartite quantum system with the three subsystems labeled $A, B,$ and $C$. Now take two states $\rho_{AB}$ on the joint system $AB$ and $\rho_{BC}$ on the joint system $BC$. Under what ...
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Random quantum states and Schur-Weyl duality

Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator: $$ \rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC. $$ Let's ...
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Special properties of a channel whose Kraus decomposition contains Identity

I would like to know if there are any special properties of channels that permit a Kraus representation that includes an identity? That is, if I am given a Kraus representation of a CPTP map $\Phi$ ...
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SWAP test and density matrix distinguishability

Let us either be given the density matrix \begin{equation} |\psi\rangle\langle \psi| \otimes |\psi\rangle\langle \psi| , \end{equation} for an $n$ qubit pure state $|\psi \rangle$ or the maximally ...
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how to know the appropriate time to know the SWAP gate operation in dipole interaction

Consider dipole-dipole interaction between two qubits,$H_{int} = g \boldsymbol\sigma_{1}\cdot\boldsymbol\sigma_{2}=g(X_1X_2+Y_1Y_2+Z_1Z_2)$. How can I show that by turning on this interaction for an ...
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Proof for Cardinality of the Clifford Group

In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
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Compute ${\rm tr}(a_k a_{k'}\rho)$ with $\rho=e^{-\beta H}/Z(\beta)$ Gibbs state and $a_k$ ladder operators

Consider a harmonic oscillator with hamiltonian $H=\sum_k\omega_k a_k^\dagger a_k$ and a state $\rho=\frac{e^{-\beta H}}{Z(\beta)}$ where $Z(\beta)=\text{tr}[{e^{-\beta H}}]$. The quantity $$A:=\sum_{...
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Interpretation of the unitaries involved in the eigenvalue decomposition of a density operator

If $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle \psi_{i}|$, this ensemble doesn't require $\langle \psi_{i}|\psi_{j}\rangle$=0. Given that $\rho$ is positive semi-definite, by the spectral theorem it ...
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Is there a way to write down the eigenstates of this two-qubit density matrix?

I am considering the density matrix which represents an arbitrary state for a pair of qubits. When written out in terms of the Pauli operators, this is as follows (certain terms vanish for another ...
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Relation between symmetric subspaces and $n$-exchangeable density matrices

Let us consider $n$ elements, each taken from the set $\{1, 2, \ldots, d\}$ and let $S_n$ be the set of all permutations on these $n$ elements. Define a permutation operator on the set of $n$ qudits ...
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How to combine/calculate for interference using density matrices?

Let's assume following two density matrices are corresponding to the A and B in the Stern-Gerlach apparatus bellow (I know Stern-Gerlach is a more of a physics experiment but I think it can equally be ...
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Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?

States belonging to some space $\mathcal H$ can be described by density operators $\rho\in L(\mathcal H)$ that are positive and have trace one. Pure states are the ones that can be written as $\rho=|\...
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Average output state of random quantum circuits

Let $|\psi\rangle = C_1 |0^{n}\rangle$ be a quantum state, such that $C_1$ is a Haar random unitary circuit. Consider a density matrix $\rho$ as follows \begin{equation} \rho_1 = \mathbb{E}[|\psi\...
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Finding Kraus operators from the output density matrix

I have a question regarding Kraus operators. Any quantum channel can be written in terms of Kraus operators as $E(\rho)= \sum_{i=0}^n K_i \rho K_i^{\dagger}$ where $\rho$ is the initial density ...
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Concavity of Conditional Quantum Entropy

Let's say I have a bipartite density operator $\gamma_{12} = (1 - \epsilon) \rho_{12} + \epsilon\sigma_{12}$, for $0 \le \epsilon \le 1$, i.e., a convex combination of $\rho_{12}$ and $\sigma_{12}$. I ...
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How to calculate the spectral norm of the density operator used in Molina et al. 2012 paper?

In Molina et al (2012)'s article on quantum money, the proof of security of Wiesner's quantum money scheme depends on the fact that the density operator $$Q = \frac{1}{4}\sum_{k \in \{0, 1, +, -\}}\...
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How is a classical bipartite state written in quantum notation?

As in the title, is a classical bipartite state on $AA'$ given by $$\sum_{ij} p_{ij} \vert i\rangle\langle i\vert_A \otimes \vert j\rangle\langle j\vert_{A'}$$ with $\sum_{ij}p_{ij} = 1$. In ...
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Increasing entropy for projective LCPT mapping

Given a set of projectors $\{P_i\}$ acting on a space $\mathcal H_S$, let $\Phi$ be the LCPT map defined by $$\Phi(\rho)=\sum_i P_i\rho P_i.$$ The goal is to show that $S(\Phi(\rho))\ge S(\rho)$. The ...
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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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200 views

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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1answer
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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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