Questions tagged [density-matrix]

For questions about density matrices and related concepts and ideas, e.g. procedures for computing properties of quantum states from their density matrices.

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Question about the probability of failure of the bit flip code

For the bit flip superoperator is $$\mathcal{E}_{BF}(\rho) = (1-p)\rho + p X \rho X$$ where the first term refers to no bit flip and the second term refers to the bit flipping. A single qubit pure ...
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How to find initial quantum states from the density matrix?

Recently, I came across density matrix. Given a qubit $|\psi \rangle = \alpha |0\rangle + \beta |1\rangle$, we can find its density matrix by computing $\rho \equiv |\psi \rangle \langle \psi |$. My ...
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Simon Algorithm without measurement

I want to use the reduced density matrix to compute the measurement probability: $prob(y)=\langle y|\rho|y\rangle$ for a 3-qubit simon algorithm, instead of measuring the ancilla qubits. Here $\rho=...
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Show that while calculating partial traces the probability is independent of the basis of one of the measurements

Consider calculating the probability of the outcome m alone of some composite system $AB$. $p_A(m) = \sum_{n=0}^{d_B-1} p_{AB}(m,n) $ $= \sum_{n=0}^{d_B-1}(⟨α_m|⊗⟨β_n|)\rho_{AB}(|α_m⟩⊗|β_n⟩)$ I'm ...
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Figuring out which experiment is being performed from the results of the experiment

Consider two different experiments involving qubits. In Experiment 1, a qubit is prepared in the mixed state $I/2$, where $I$ is the $2 × 2$ identity matrix. Alice then chooses an orthonormal basis $B$...
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Are projections determined by their action on a full-rank density matrix?

Consider (self-adjoint) projections $P$ and $Q$ defined on a finite-dimensional Hilbert space. If $\rho$ is the maximally-mixed state, then we have that $P \rho P = Q \rho Q$ implies $P = Q$, since $\...
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Review on quantum resources required in mixed-state quantum computing

I am trying to see which features we know are necessary for mixed-state quantum computing to avoid the algorithms being efficiently simulated on classical computers. In the case of pure-state quantum ...
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Schmidt decomposition manages to write a pure state using just d terms

Suppose $|\psi\rangle$ $\in \mathrm{H_A}\otimes\mathrm{H_B}$ is a pure state and we can write a representation of $|\psi\rangle$ like $|\psi\rangle = \sum_j |\alpha_j\rangle|\beta_j\rangle$, where $|\...
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Sampling Haar over two systems

Say $M$ is a matrix acting on $C^r \otimes C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$. With $|\psi\rangle$ sampled from Haar, how can we show that $$ \int (...
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How to show that a density matrix $\rho$ is extreme iff $\rho=|\psi\rangle\!\langle\psi|$?

A density matrix $ρ$ is called extreme if the only way to write $ρ$ as $ρ = p σ + (1 − p) τ$ , with $σ ∈ S_d$, $τ ∈ S_d$, and $p ∈ (0, 1)$ is to have $ρ = σ = τ$ . I want to show that a density matrix ...
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What is an "x-type density matrix"?

i want to know what is a x-type density matrix structure? i want to know the general case of it. is this something like this? can one of matrix elements be 0? unfortunately there is no info about it ...
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Explain the comparison between a state and density matrix

The density matrix $\rho = \frac{1}{2}(|0\rangle \langle0|+|1\rangle \langle1|)$ describes a system which is in state $|0\rangle$ and in $|1\rangle$ with equal probability. Also the state $|\psi\...
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Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?

Suppose I have an arbitrary orthogonal projector $\Pi$ and two density operators $\rho, \sigma$. Is it true that: $$ ||\Pi (\sigma - \rho) \Pi||_1 \le || \sigma - \rho ||_1 $$ where $||\cdot||_1$ ...
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What does $I$ represent in the expression $A\otimes I$?

This paper states: Suppose a two qubit system is in the state $|\psi\rangle=a|00\rangle+b|11\rangle$, and consider the expectation value of any observable $A \otimes I$ that is nontrivial only on the ...
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Is $\rho | \psi \rangle$ invariant in the Wigners friend thought experiment?

Background Let's say I have a gas of $N$ particles where I cannot distinguish between the particles at a temperature $T$. Its density matrix is given by $\rho$. Note, if my friend happens to measure ...
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Why can't the purity of a single qubit be less than $1/2$?

The density matrix of a single qubit system can be defined as, $$ \rho= \frac{1}{2}(\hat I+ \vec r.\hat{\vec \sigma}) $$ From here we can derive, $$ Tr(\rho^2)= \frac{1}{2}(1+r^2) $$ Since $ 0\leq r^...
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Qiskit DensityMatrix.from_instruction when snapshots are present

I have a quantum circuit in which I apply snapshots like this during setup: ...
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What is a "maximally mixed state"?

What is meant by maximally mixed states? Does this mean that there are partially mixed states? For example, consider $\rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right|$ and $\rho_W =...
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Relating upper bound on the total variation distance with a bound on a Pauli observable

Consider an $n$ qubit state $|\psi\rangle$. Let us say we have the following upper bound on the total variation distance between the distribution generated by measuring $|\psi\rangle$ in the standard ...
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How to find the Kraus operators from the process matrix?

I am trying to find the Kraus operator from process matrix. For instance, suppose that for single qubit identity gate, I have the following process matrix: ...
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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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2 answers
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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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1 answer
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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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2 votes
2 answers
135 views

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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7 votes
4 answers
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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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1 answer
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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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3 votes
2 answers
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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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3 votes
1 answer
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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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1 vote
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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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6 votes
0 answers
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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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3 votes
1 answer
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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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2 votes
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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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1 vote
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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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5 votes
2 answers
281 views

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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1 vote
2 answers
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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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