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dynamically updating the states of individual qubits in a composite \( n \)-qubit system

Here’s the scenario: I am working with a 6-qubit composite state, where the first three qubits $( e_1, e_2, e_3 )$ represent electron spin qubits, and the last three $( n_1, n_2, n_3 $) represent ...
Aparna Gupta's user avatar
4 votes
1 answer
181 views

Prove that spectral decomposition is the minimal ensemble decomposition

I understand that the spectral decomposition of a density matrix $\rho$ expresses it in terms of its eigenvalues and eigenvectors: $$\rho=\sum_i\lambda_i\left|\psi_i\middle\rangle\!\middle\langle\psi\...
SK SAIF IBNA EZHAR ARKO's user avatar
-1 votes
1 answer
49 views

how to mix (or time average) two density matrix?

Given two density matrix $\rho_1,\rho_2$ with the same size, how to get a mix state of the two matrix, $$ \rho = \frac12 (\rho_1+\rho_2)? $$ e.g. there are two quantum channel both of them have 4 ...
Boyuan Wang's user avatar
1 vote
1 answer
46 views

Computing the expected value of a spin - 1 particle component given density matrix

I have a density matrix $\rho$ where $$\rho = \frac{1}{4} \cdot \begin{pmatrix} 2 & 1 & 1\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{pmatrix}$$ and the x component of a spin - 1 particle ...
James's user avatar
  • 33
2 votes
1 answer
144 views

Does the trace distance between these averages of pure states satisfy $|\rho - \sigma|_1 \geq \frac{N_a - N_b}{N_a}$?

Consider two $n$ qubit density matrices: $$ \rho = \frac{1}{N_a} \left(\sum_{i=1}^{N_a} |\psi_i\rangle \langle \psi_i| \right). $$ $$ \sigma = \frac{1}{N_b} \left(\sum_{i=1}^{N_b} |\phi_i\rangle \...
BlackHat18's user avatar
  • 1,515
0 votes
0 answers
36 views

How to generate random quantum states in matlab?

I was wondering if there is some academic standard/any way of generating random n times n q-states/density matrices in Matlab without using any other package then QETLAB.
Pink Elephants's user avatar
3 votes
1 answer
110 views

Density Matrices for states $|+\rangle$ and state represented by $\rho = \frac{|0\rangle \langle0| + |1\rangle \langle1|}{2}$

As per my understanding, the first one is a "pure state" and represents a system with one qubit having equal probability of being measured as $|0\rangle$ or $|1\rangle$ (standard basis ...
Jasjyot Gulati's user avatar
4 votes
1 answer
204 views

Is the closest diagonal state to a given state always the dephased original state?

This question is about the following optimization problem: Given some density matrix $\rho\in\mathbb C^{n\times n}$ find the diagonal state which is closest to it in trace norm. More precisely, find $...
Frederik vom Ende's user avatar
1 vote
0 answers
41 views

Help to understand a QFI derivation

Can anyone help me understand the QFI derivation being done in Appendix C of this paper? The density matrix $\rho = \frac12(|\psi_1\rangle\langle\psi_1|+|\psi_2\rangle\langle\psi_2|)$. I understand ...
Som's user avatar
  • 11
1 vote
0 answers
42 views

Alice and Bob play a Multi-Qubit game

Well I am quite new to this so excuse me if the question is absurd Alice and Bob each can "measure" variables A and B respectively: Alice can use $a_1$ and $a_2$ methods of measurement while ...
qinnairen's user avatar
4 votes
0 answers
57 views

What is the minimum number of separable states (not necessarily pure) needed to decompose arbitrary separable states?

For a bipartite separable quantum state $\rho$ acting on Hilbert space $H\otimes H'$ with $\dim H=D$ and $\dim H'=D'$, what is the minimum number of separable state needed for a decomposition? That ...
Yujie Zhang's user avatar
0 votes
1 answer
80 views

How to convert from choi to chi matrix in qiskit

I have done a quantum process tomography experiment on a two qubit system. ...
idislikecoding's user avatar
2 votes
1 answer
119 views

Is $\rho = \sum_{j} p_j|n_j\rangle\langle n_j|$ a valid construction for any mixed state?

I have a mixed state $\rho$ and its hamiltonian $H$. Firstly, I find the eigenvalues $\{p_j\}$ of $\rho$, and orthonormal basis of $H$. I write $\rho$ in terms of $H$'s eigenstates and $\rho$'s ...
Việt Nguyễn's user avatar
1 vote
0 answers
87 views

State tomography in Qiskit on a subset of qubits of real QPU

Could anyone please explain how should I carry out a state tomography on a real device in Qiskit (version 0.43.2)? I have access to devices with 127 qubits, but I want to perform a simulation using ...
Andrea's user avatar
  • 11
2 votes
1 answer
62 views

Existence of a two-outcome measurement $M$ such that the induced distributions differs between different density matrices

Let $\rho \neq \sigma$ be density matrices. I want to show that there exists a two-outcome measurement $M$ such that the induced distributions $M(\rho)$ and $M(\sigma)$ differ. From what I learned, ...
Gabi G's user avatar
  • 239
2 votes
1 answer
67 views

How is the expression $\frac{\|\rho^{T_B}\|-1}{2}$ obtained from the definition of negativity?

In quantum information theory, negativity is defined as summation of the absolute values of negative eigenvalues of the partial transposed density matrix. The expression of negativity is given as $$ \...
Anindita Sarkar's user avatar
4 votes
1 answer
146 views

confusion on the LCU method regarding the normalization

Let $A = \sum_{k} a_k U_k$ where $a_k$ are real, positive coefficients $U_k$ are unitary matrices. I have realized that $\sigma = A \rho A$ can be implemented on a quantum computer by using the LCU ...
Hailey Han's user avatar
1 vote
1 answer
58 views

Simplification of a generic quantum state

We are given a generic 2-qubit density matrix $$\rho=\frac{1}{4}\left[I_4+\Sigma_i a_i \sigma_i \otimes I_2 + \Sigma_i b_i I_2 \otimes \sigma_i + \Sigma_{i,j} c_{ij} \sigma_i \otimes \sigma_j\right]$$ ...
Anindita Sarkar's user avatar
3 votes
1 answer
48 views

Can a generic 2-qubit state be unitarily converted into one of the form $I_2\otimes I_2+\lambda\sigma_z\otimes\sigma_z$?

Suppose I have a general 2-qubit state written in a basis consisting of tensor products of Pauli matrices: $\rho=\frac{1}{4}\left[I_2\otimes I_2+\Sigma_{i} a_i \sigma_i\otimes I_2+\Sigma_{i} b_i I_2\...
Anindita Sarkar's user avatar
2 votes
1 answer
217 views

Finding the eigenvalues of a qutrit state

I am interested in the state: $\frac{1}{\sqrt{2}} (\left|11\right> + \left|22\right>)$ If I find the density matrix of this, I find the $9 \times 9$ matrix $\rho$. If I want to find the reduced ...
am567's user avatar
  • 739
4 votes
1 answer
219 views

Is possible to write a separable state as a finite or countable infinite sum of product states?

Let us consider the tensor product of two finite Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_1$. According to Watrous book, the set of separable states is the convex hull of the set of product ...
raskolnikov's user avatar
3 votes
1 answer
148 views

calculate the reduced density matrix of a 2 qubit state and compare the eigenvalues

So I have the exercise to apply a Cz gate to the following 2 Qubit state $|a\rangle \otimes |b\rangle = (a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)\\\\$ Afterwards, I ...
Ruebli's user avatar
  • 31
1 vote
1 answer
294 views

Statevector from Density matrix of non-pure state

I have a state vector of a 16 qubit system. I want to get the wave function (in the form of a state vector) for half and quarter of this system. When I try to make a ...
VladislavOkatev's user avatar
3 votes
2 answers
300 views

Prove the fidelity equals $F( \rho , \sigma) = |\langle \psi_{\rho} | \psi_{\sigma}\rangle|^2$ for pure states

I am trying to learn by myself quantum computing and information and I have a very simple question concerning the demonstration of the following equality: $F( \rho , \sigma) = |\langle \psi_{\rho} | \...
X0-user-0X's user avatar
2 votes
0 answers
193 views

Density matrix and State vector give different result in mesolve in QuTiP

qutip mesolve gives me different population evolve depending on that initial state is state vector or density matrix. And, in some situation, it gives me negative population. It doesn't make sense... ...
eechiki's user avatar
  • 21
2 votes
1 answer
612 views

How many dimensions does an n-qubit system have?

How many dimensions does an $n$-qubit system have? What is definition of dimension for a quantum state? Is it related to the number of rows and columns of a density matrix? My guess is that it has $2^...
reza's user avatar
  • 791
1 vote
2 answers
63 views

How to perform a state density modification for a single targeted state only?

I have a question about single target state modification... Suppose we have a 3 qubit state density distribution as follows (prenormalized): $$\begin{bmatrix} |000\rangle & 3 \\ |001\rangle & ...
James's user avatar
  • 541
2 votes
2 answers
2k views

What is the density matrix of a pure state?

By definition of the density matrix for example the density matrix of $|0\rangle$ state (pure state) is: $$\rho=|0\rangle \langle 0| = \begin{pmatrix} 1 & 0 \\ ...
Curious's user avatar
  • 267
3 votes
1 answer
305 views

Representation of maximally entangled states of $2n$ qubits with Pauli matrices?

I'm reading this paper while the author states in the eq(A1) that, for a $2n$ qubits maximally entangled state $|\Psi ^+\rangle \langle \Psi ^+|$, we can write it with Pauli operators $P_u\in\left\{ I,...
Sherlock's user avatar
  • 715
1 vote
1 answer
288 views

Simple proof that entangled pure states are not separable

I am trying to understand more about the notion of separable states. For clarity, I will only use the word entangled for pure states, even if a non-separable state is sometimes called entangled too. ...
user8622655's user avatar
1 vote
1 answer
106 views

Can we use purity for separable states?

Purity is a measure of how much a state is pure. Suppose $\rho$ is a density matrix. Then purity $p$ is defined as $$ p = \mathrm{tr}(\rho^2). $$ I wonder if we can use purity for separable states? Or ...
reza's user avatar
  • 791
2 votes
2 answers
102 views

Does the inequality $\mathrm{tr}(L^\dagger L \rho^2)-\mathrm{tr}(L^\dagger \rho L\rho )\geq 0$ hold generally?

Does the inequality $$\mathrm{tr}(L^\dagger L \rho^2)-\mathrm{tr}(L^\dagger \rho L\rho )\geq 0$$ hold for any density matrix $\rho$ and any non-Hermitian Lindblad operator $L$?
Kochan's user avatar
  • 41
2 votes
1 answer
89 views

Upper bounding the trace distance between a noisy and noiseless quantum state

Consider a quantum state $$ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \\ \end{pmatrix}. $$ Now, consider the effect of the amplitude damping noise $\mathcal{N}$ of ...
BlackHat18's user avatar
  • 1,515
1 vote
1 answer
425 views

How to write a two qubit state as "diagonal" in the basis of Pauli matrices?

Any two qubit density matrix can be written as $$ \rho = \frac{1}{4} \sum_{n,m = 0}^{3} M_{nm} (\sigma_n \otimes \sigma_m), $$ where $\sigma_\mu$'s are the identity and Pauli matrices. Is it possible ...
Bard's user avatar
  • 327
0 votes
1 answer
50 views

What it is represent if my density_matrix is mixed matrix and it's a diagonal matrix,Can i get the state vector from it?

i have a pqc: ...
qin guang's user avatar
3 votes
1 answer
286 views

Finding a unitary transformation for a qubit mixed state that projects onto a pure state

Suppose we have a single qubit mixed state described by a density matrix $\rho$, and we want to find a unitary transformation that brings $\rho$ to the pure state $|0\rangle\langle 0|$. Is there a ...
Zarathustra's user avatar
1 vote
2 answers
282 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_{...
G Frazao's user avatar
  • 155
3 votes
1 answer
432 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 ...
joy Jaganath's user avatar
3 votes
1 answer
534 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 $\...
Tristan Nemoz's user avatar
  • 7,969
1 vote
1 answer
209 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 ...
reza's user avatar
  • 791
1 vote
1 answer
80 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 ...
pip's user avatar
  • 113
2 votes
1 answer
486 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 ...
Borja Aizpurua's user avatar
1 vote
1 answer
156 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 ...
Radu M.'s user avatar
  • 228
1 vote
0 answers
211 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-...
anand_quanta's user avatar
0 votes
1 answer
810 views

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 ...
Josh's user avatar
  • 427
0 votes
0 answers
36 views

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 $B$...
SVMteamsTool's user avatar
3 votes
2 answers
381 views

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 $|\...
username9's user avatar
4 votes
3 answers
491 views

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\...
Soumyabrata hazra's user avatar
9 votes
3 answers
8k views

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 =...
Bekaso's user avatar
  • 295
3 votes
1 answer
68 views

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 ...
BlackHat18's user avatar
  • 1,515