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.
293
questions
1
vote
1
answer
26
views
Quantum Relative entropy- the math and intuition
I am new to quantum information theory and have been reading Mark Wilde's notes on quantum relative entropy.
http://www.markwilde.com/teaching/2015-fall-qit/lectures/lecture-19.pdf
I have three basic ...
- 11
0
votes
0
answers
11
views
How can i get the superposition of 8 bits from 26 bits outputs in qiskit
the only idea i see is get state vector from density_matrix
for example, i have the circult:
...
- 13
1
vote
0
answers
15
views
List of inequalities for purity of a traced out bipartite system
I would like to know if there are inequalities related to the purity of the partial trace of a bipartite system.
The purity $P$ of a density matrix $\rho$ is given by
$$P(\rho) = Tr(\rho^2).$$
The ...
- 2,041
0
votes
1
answer
29
views
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\...
- 13
1
vote
2
answers
67
views
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:
$$\rho = \begin{bmatrix} p_1 & r_{12}e^{-i\phi_{12}} & r_{13}e^{-i\phi_{13}}\\ \cdot & p_2 & r_{...
- 155
2
votes
1
answer
92
views
What physical quantity does a density operator represent as an observable?
The density operator is a representation of a state of a quantum system $\rho=|\psi\rangle\langle\psi|$, so it's just an alternative characterisation of a state (or more generally a statistical ...
- 117
0
votes
1
answer
54
views
Qiskit: density matrix after measurement
I would like to find density matrix after the measurement. The toy code:
...
- 103
4
votes
3
answers
70
views
How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
0
votes
1
answer
65
views
How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\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 ...
- 790
1
vote
1
answer
47
views
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
$$Tr(\rho\sigma)\leq ...
- 11
2
votes
2
answers
112
views
How to get the density matrix from a stabilizer table in qiskit
I am new to qiskit and quantum computing in general, so bear with me please. For my bachelor's thesis, I am programming qiskit to first generate a random Clifford circuit (qc) and to then measure the ...
- 37
1
vote
0
answers
43
views
Reduced density matrix accuracy in amplitude estimation
I am implementing QAE (Quantum Amplitude Estimation), which is very similar to QPE (Quantum Phase Estimation) with a Grover Operator as the U matrix of QPE.
I want to check my results, in the outputs ...
- 1,124
3
votes
1
answer
181
views
How to recover the original density matrix from the output of amplitude damping channel?
For amplitude damping, we have the below expression
$$\xi(\rho)=E_0\rho{E_0}^\dagger + E_1\rho{E_1}^\dagger.$$
How can I perform a matrix inverse operation on $\xi(\rho)$ at the receiver to get back ...
- 31
4
votes
1
answer
199
views
What is the adjoint of the depolarizing channel?
Consider the single qubit depolarizing noise channel given by
$$\Phi(\rho) = \frac{\lambda}{d} \mathbb{I} + (1- \lambda) \rho.$$
What might be the adjoint $\Phi^{*}(\cdot)$ of this channel? In ...
- 315
3
votes
1
answer
112
views
Averaging over a single Haar-random unitary applied $t$ times
I'm trying to compute the following integral:
$$\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)$$
Where $\...
- 4,278
1
vote
1
answer
58
views
How to square a density matrix?
Supposing $x$ is a density matrix. I know purity formula is
$$P = \mathrm{tr}(x^2)$$
But I have doubt about calculating $x^2$.
Is it $x\cdot x$ or $x\cdot x^\dagger$?
Can you give me a reference for ...
- 127
2
votes
2
answers
116
views
How to translate a 4-qubit Grover's algorithm circuit into a state Matrix?
Grover's algorithm circuit may be implemented as follows:
(from here)
It is shown very elegantly by @MartinVesely (How to interpret a 4 qubit quantum circuit as a matrix?) how to translate a 4 qubit ...
- 191
4
votes
0
answers
40
views
Distribution of density operators under Stochastic Master Equation
Stochastic master equations (SME) are used in studies of open quantum systems. The general form of an SME is:
\begin{align}
\tag{1} d\tilde{\sigma}(t) = - i [H, \tilde{\sigma}(t) ]dt + \frac{1}{2}\...
- 2,041
2
votes
1
answer
240
views
De-coherence and Wigners friend?
So in the thought experiment Wigners friend the paradox is ultimately due to a difference of descriptions of density matrices.
If the physical variable that is measured of the spin system is denoted ...
- 377
2
votes
1
answer
210
views
How to calculate the log of a density matrix?
In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a ...
- 149
0
votes
1
answer
79
views
Why my density matrix trace is over 1?
Suppose this operator
$$
\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|
$$
...
- 127
3
votes
1
answer
83
views
Haar measure : trace of an operator squared and square of the trace of an operator
From doing numerical simulations, I seem to have the following results :
$$ \int d \rho \,\, \text{Tr}(\rho M^\dagger M) = \frac{1}{d} \text{Tr}(M^\dagger M) $$
and
$$ \int d \rho \,\, \left|\text{Tr}(...
- 33
2
votes
1
answer
133
views
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 ...
- 4,278
0
votes
1
answer
80
views
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 ...
- 667
0
votes
1
answer
100
views
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
$$\langle AB \rangle +\sqrt{\langle A \rangle - (\langle A \rangle)^2}\sqrt{\langle B \rangle - (\langle B \...
- 113
0
votes
1
answer
34
views
Joint probability distribution for variables X and Y that exist at different times
On page $402$ of Nielsen and Chuang, there is the statement:
"there is no notion in quantum mechanics analogous to the joint probability distribution for variables X and Y that exist at different ...
- 167
2
votes
1
answer
68
views
How to prove the strong convexity of the trace distance?
On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\...
- 167
0
votes
0
answers
35
views
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}...
1
vote
0
answers
37
views
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, ...
- 135
6
votes
2
answers
348
views
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{\...
- 63
3
votes
1
answer
125
views
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=\...
- 667
2
votes
0
answers
70
views
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:
$$\rho = ...
- 280
1
vote
1
answer
36
views
Matrix representation for biproduct mixed states
Nielsen and Chuang [10e, p. 74] introduce the Kronecker product $A\otimes_K B$ as a matrix representation of the tensor product $A\otimes B$ of the operators $A$ and $B$ (for clarity I use a subscript ...
- 113
0
votes
0
answers
51
views
Question regarding the trace-preserving quantum operator trace distance
In Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, the proof of Theorem 9.2 (Trace-preserving quantum operations are contractive) on ...
- 217
2
votes
1
answer
96
views
How to compute the maximum possible coherence of a two-particle Bell state?
I am reading through some notes and am stuck on a bit of math that shows the max possible coherence. Our wave function is $|\psi\rangle =\frac{|01\rangle+|10\rangle}{2}$ and doing $|\psi\rangle \...
- 23
4
votes
1
answer
120
views
Does closeness in trace distance imply close measurement outcomes?
Suppose we have two density matrices $\rho$ and $\rho'$ such that $\|\rho - \rho'\|_1 \leq \varepsilon$. Let $\{\Lambda, I - \Lambda\}$ be elements of some POVM. If it holds that
$$Tr(\Lambda\rho) \...
- 437
3
votes
1
answer
91
views
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 ...
- 4,278
1
vote
2
answers
109
views
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):
...
- 1,311
1
vote
1
answer
103
views
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 ...
2
votes
1
answer
87
views
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.
- 155
5
votes
1
answer
62
views
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 ...
- 578
0
votes
1
answer
58
views
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 ...
- 214
0
votes
2
answers
210
views
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 ...
- 667
2
votes
1
answer
92
views
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 ...
- 450
0
votes
1
answer
51
views
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 ...
- 155
1
vote
0
answers
98
views
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-...
- 11
2
votes
0
answers
74
views
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 \...
- 61
2
votes
1
answer
48
views
(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 $|\psi\rangle$ ...
2
votes
2
answers
91
views
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
$\rho = \sum_{i}\alpha_i^*\alpha_i |i\rangle\langle i|$ ???
- 437
4
votes
1
answer
128
views
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 ...
