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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Find min of a quantum state L2 norm
I have a problem minimizing this norm with respect to $\alpha$:
$\min_{\alpha}||e^{i\alpha}|\psi\rangle-|\phi\rangle ||^2$ (1)
The result is that this achieves min when $\alpha=-\measuredangle \langle\...
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Is there another parameterization of a qutrit 3-level system, besides Gell-Mann?
Question:
Is there a parameterization of a a general qutrit 3-level system similar to:
$$\rho = \begin{bmatrix} p_1 & r_{12}e^{-i\phi_{12}} & r_{13}e^{-i\phi_{13}}\\ \cdot & p_2 & r_{...
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What physical quantity does a density operator represent as an observable?
The density operator is a representation of a state of a quantum system $\rho=|\psi\rangle\langle\psi|$, so it's just an alternative characterisation of a state (or more generally a statistical ...
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Qiskit: density matrix after measurement
I would like to find density matrix after the measurement. The toy code:
...
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How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
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How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert
\sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
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Minimum and maximum of $Tr(\rho\sigma)+\sqrt{1-Tr(\rho^2)}\sqrt{1-Tr(\sigma^2)}$
How do I find the maximum and minimum of the following expression?
$$F_N(\rho,\sigma)=Tr(\rho\sigma)+\sqrt{1-Tr(\rho^2)}\sqrt{1-Tr(\sigma^2)}.$$
I think of using this inequality
$$Tr(\rho\sigma)\leq ...
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How to get the density matrix from a stabilizer table in qiskit
I am new to qiskit and quantum computing in general, so bear with me please. For my bachelor's thesis, I am programming qiskit to first generate a random Clifford circuit (qc) and to then measure the ...
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Reduced density matrix accuracy in amplitude estimation
I am implementing QAE (Quantum Amplitude Estimation), which is very similar to QPE (Quantum Phase Estimation) with a Grover Operator as the U matrix of QPE.
I want to check my results, in the outputs ...
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How to recover the original density matrix from the output of amplitude damping channel?
For amplitude damping, we have the below expression
$$\xi(\rho)=E_0\rho{E_0}^\dagger + E_1\rho{E_1}^\dagger.$$
How can I perform a matrix inverse operation on $\xi(\rho)$ at the receiver to get back ...
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What is the adjoint of the depolarizing channel?
Consider the single qubit depolarizing noise channel given by
$$\Phi(\rho) = \frac{\lambda}{d} \mathbb{I} + (1- \lambda) \rho.$$
What might be the adjoint $\Phi^{*}(\cdot)$ of this channel? In ...
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Averaging over a single Haar-random unitary applied $t$ times
I'm trying to compute the following integral:
$$\int U^{\otimes t}\left|x_1,\cdots,x_t\middle\rangle\middle\langle x'_1,\cdots,x_n'\right|\left(U^\dagger\right)^{\otimes t}\,\mathrm{d}\mu(U)$$
Where $\...
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How to square a density matrix?
Supposing $x$ is a density matrix. I know purity formula is
$$P = \mathrm{tr}(x^2)$$
But I have doubt about calculating $x^2$.
Is it $x\cdot x$ or $x\cdot x^\dagger$?
Can you give me a reference for ...
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How to translate a 4-qubit Grover's algorithm circuit into a state Matrix?
Grover's algorithm circuit may be implemented as follows:
(from here)
It is shown very elegantly by @MartinVesely (How to interpret a 4 qubit quantum circuit as a matrix?) how to translate a 4 qubit ...
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Distribution of density operators under Stochastic Master Equation
Stochastic master equations (SME) are used in studies of open quantum systems. The general form of an SME is:
\begin{align}
\tag{1} d\tilde{\sigma}(t) = - i [H, \tilde{\sigma}(t) ]dt + \frac{1}{2}\...
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De-coherence and Wigners friend?
So in the thought experiment Wigners friend the paradox is ultimately due to a difference of descriptions of density matrices.
If the physical variable that is measured of the spin system is denoted ...
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How to calculate the log of a density matrix?
In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a ...
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Why my density matrix trace is over 1?
Suppose this operator
$$
\rho=\frac{a^2}{\cosh^2(r)}\sum_{n=0}^{\infty}\tanh^{2n}(r)|0,n\rangle\langle 0,n|+\frac{b^2}{\cosh^4(r)}\sum_{n=0}^{\infty}(n+1)\tanh^{2n}(r)|1,n+1\rangle\langle 1,n+1|
$$
...
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Haar measure : trace of an operator squared and square of the trace of an operator
From doing numerical simulations, I seem to have the following results :
$$ \int d \rho \,\, \text{Tr}(\rho M^\dagger M) = \frac{1}{d} \text{Tr}(M^\dagger M) $$
and
$$ \int d \rho \,\, \left|\text{Tr}(...
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Density matrices of multiples copies of a single Haar-Random state
In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that:
Let $\rho_\mu^m$ be the density matrix of a random $|\psi\rangle^{\otimes m}$ for $|\psi\rangle$ chosen from ...
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In what sense is $\langle\psi|\rho|\psi\rangle$ the fidelity between the pure state $|\psi\rangle$ and the mixed state $\rho$?
The fidelity between a pure state $|\psi\rangle$ and an arbitrary mixed state $\rho$ is given by, $F(|\psi\rangle,\rho)=\sqrt{\langle\psi|\rho|\psi\rangle}$, which is stated to be equal to the square ...
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Cauchy-Schwarz inequality of expectation values of operators
For two operators $A, B$ defined on Hibert space $H_n$, the state is $\rho$, then there is
$$\langle AB \rangle +\sqrt{\langle A \rangle - (\langle A \rangle)^2}\sqrt{\langle B \rangle - (\langle B \...
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Joint probability distribution for variables X and Y that exist at different times
On page $402$ of Nielsen and Chuang, there is the statement:
"there is no notion in quantum mechanics analogous to the joint probability distribution for variables X and Y that exist at different ...
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How to prove the strong convexity of the trace distance?
On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\...
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Density matrix derivation of an entangled state in amplitude dampling channel
The density matrix for an entangled state when both the qubits decohere with probability D1 and D2 in amplitude damping channel is given as $\rho_d$ in \href{https://www.nature.com/articles/nphys2178}...
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How are $\theta, \phi$ and $\lambda$ for the U3 gate derived in Abhijith et al. 2018?
I am looking to implement Quantum PCA from this paper (page 62). They have their code on Github.
I have gone through the paper multiple times but failing to understand how they got numbers (for theta, ...
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How does quantum teleportation work with mixed shared states?
I am given the scenario that instead of the two parties (A & B) sharing the bell state $|\phi_+\rangle$ they share the mixture $\rho_\lambda = \lambda|\phi_+\rangle\langle\phi_+|+(1-\lambda)\frac{\...
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Expansion of multi-qubit density matrix in the Pauli matrix basis
The single qubit density matrix can be expanded as
$$
\rho=\frac{tr(\rho)I+tr(X\rho)X+tr(Y\rho)Y+tr(Z\rho)Z}{2}
$$
which can be shown as,
$\rho$ is a positive operator with $tr(\rho)=1$, ie.,
$\rho=\...
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How to "separate" a separable density matrix?
If I have a bipartite system of two qubits $A$ and $B$, and the density matrix $\rho$ is separable, how do I decompose it into its separable parts?
That is, give $\rho$, expand it as follows:
$$\rho = ...
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Matrix representation for biproduct mixed states
Nielsen and Chuang [10e, p. 74] introduce the Kronecker product $A\otimes_K B$ as a matrix representation of the tensor product $A\otimes B$ of the operators $A$ and $B$ (for clarity I use a subscript ...
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Question regarding the trace-preserving quantum operator trace distance
In Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, the proof of Theorem 9.2 (Trace-preserving quantum operations are contractive) on ...
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How to compute the maximum possible coherence of a two-particle Bell state?
I am reading through some notes and am stuck on a bit of math that shows the max possible coherence. Our wave function is $|\psi\rangle =\frac{|01\rangle+|10\rangle}{2}$ and doing $|\psi\rangle \...
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Does closeness in trace distance imply close measurement outcomes?
Suppose we have two density matrices $\rho$ and $\rho'$ such that $\|\rho - \rho'\|_1 \leq \varepsilon$. Let $\{\Lambda, I - \Lambda\}$ be elements of some POVM. If it holds that
$$Tr(\Lambda\rho) \...
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Are there quantum algorithms which initial state is a mixed one?
In this answer, it is stated that it is not yet known how to efficiently classically simulate separable mixed states, a statement supported in a comment to this answer.
However, I can't imagine an ...
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Does applying a random Pauli matrix to a density matrix result in the identity?
Nielsen and Chuang's textbook, Equation 8.101 (section 8.3.4 'Depolarizing Channel') shows that applying a random Pauli to a density matrix representing one qubit equals the identity (times one half):
...
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State tomography with Pauli basis measurements for a high number of qubits
My end goal is to recover the quantum state in its computational basis or reduced density matrix of a high number qubit circuit in a real QPU. Taking into account that the number of qubits will be ...
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Is there an expression for the partial trace of a vectorized density matrix?
Is there an expression for the partial trace of vectorized density matrix? I did some literature review but didn't find not much relevant information.
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Action of a channel on an "unphysical" state
Suppose we are given a rule $\Phi$ which is completely positive and trace preserving operation takes an input qubit state $\rho$ to an output qubit state $\rho^\prime$ (as an example of such rule see ...
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How simulation of noisy quantum circuits is done in Qiskit using the statevector method
While performing VQE calculations of medium-sized molecules like H2O, using Qiskit AerSimulator with noise, I noticed that even for a large number of shots, the speed of simulation using statevector ...
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CNOT gate an elementary example of a single qubit quantum operation
A natural way to describe the dynamics of an open quantum system is to regard it as arising from an interaction between the system of interest and an environment, which together form a closed quantum ...
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Writing a Density matrix in terms of the magnitude of the Bloch Vector
Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
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measurement probability from density operator?
I've been through this before but I can't fully get my head round this upon review.
So the density operator $\hat{\rho}=\sum_j p_j|\psi_j\rangle\!\langle \psi_{j}|$ for pure states $|\psi_{j}>$ at ...
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Qutip: Mesolve gives different and weird results with different fock state numbers
I have been trying to simulate the average number of particles at 3 sites of coupled harmonic oscillators.
I have used the code from the below tutorial:
https://notebook.community/ajgpitch/qutip-...
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Simulating density matrices in quantum simulators
I would like to load random quantum states sampled from a given density matrix based on its classical probabilities ie based on the definition of the given density matrix: $\rho = \sum_i p_i |\psi _i \...
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(Proof verification) Kaye Exercise 3.5.5, partial trace in larger system
This is the exercise as stated in Kaye's book, Introduction An Introduction to Quantum Computing:
Show that for any density operator $\rho$ on a system $A$, there exists a pure state $|\psi\rangle$ ...
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2
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Why density matrix representation usually written in the following form?
The pure state
$|\psi\rangle = \sum \alpha_i |i\rangle$,
$\rho = \sum_{i,j}\alpha_i^*\alpha_j |j\rangle\langle i|$,
why is the form not be
$\rho = \sum_{i}\alpha_i^*\alpha_i |i\rangle\langle i|$ ???
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Is Density Matrix simulation same as Tensor Network simulation?
I read that there are two major ways of simulating quantum circuits - State Vector Simulation and Tensor Network Simulation. But Qiskti provides a backend to simulate state-vectors and a backend for ...
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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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