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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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Can we use a Werner state for quantum teleportation? [duplicate]

Some background: The quantum teleportation protocol requires first that Alice and Bob share an entangled state, say a Bell state $|\psi^{+}\rangle_{AB}$. There is another state $|\psi\rangle_{A'}$ to ...
Physkid's user avatar
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How is Pauli twirling so powerful?

So the Pauli twirling approximation gives us a quantum channel $\Phi$ that transforms a density matrix $\rho$ to: $\Phi(\rho)\mapsto\sum_{i=0}^3 \sigma^i \rho \sigma^i,$ where $\sigma^0 = \mathbb{I}, \...
JoJo P's user avatar
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General Bell state expression: What condition for mixture of Bell states to be entangled?

Convention: $|qubit_{A}, qubit_{B}\rangle$ The general Bell state equation: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k\oplus b\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\...
Physkid's user avatar
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Finding entanglement in matrix that is a sum of 4 Bell states

A general Bell state: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\rangle + (-1)^{a}|1,1 \oplus b \rangle]$ $|\beta(0,0)\rangle = \frac{1}{2}[|00\rangle \langle 00| + |00\rangle \langle 11| +...
librarian_'s user avatar
1 vote
2 answers
51 views

Does separability of a matrix implies the matrix is a density matrix?

Suppose I have a matrix that is unknown whether it is a density matrix and assume that finding the eigenvalues of it is difficult because the matrix is expressed generally. However, suppose that this ...
Physkid's user avatar
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2 answers
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Define a traceless part of $\rho$ [closed]

I saw in a paper: $|\bar{\rho}\rangle\rangle=|\rho\rangle\rangle-|\hat{I}\rangle\rangle / 2^{n / 2}$ for the $4^n$-dimensional vector representing the traceless part of $\rho$. https://arxiv.org/abs/...
karry's user avatar
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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
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5 votes
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156 views

Moments of Pauli coefficients of Haar-random states

I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is ...
Feng Pan's user avatar
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How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = \text{Tr}_{2}[\rho_{12}] = \sum_{i}\langle i_{2} | \rho | i_{2} \rangle$?

Let the system be a 2 - qubit system and let $\rho_{12}$ be a density matrix of some state for this 2 - qubit system. How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\...
Physkid's user avatar
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How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?

I have the following problem and have attempted to find a solution to it, but to no avail. Alice and Bob have one qubit each, say $|\psi\rangle$ with Alice and $|\phi\rangle$ with Bob. They also share ...
requiemman's user avatar
2 votes
1 answer
124 views

Given that for every valid density matrix $\rho$, $\text{Tr}(M\rho) = 1$; what can we conclude about matrix $M$?

My intuition says that $M$ has to be the identity matrix, but I am not able to show it rigorously. I tried playing around using spectral decomposition. If $$ \rho = \sum_i \lambda_i |\lambda_i \rangle ...
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How to compute marginal probabilities of Alice's qubit (in density operator language)?

Let $| \psi \rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{2}|01\rangle + \frac{\sqrt{3}}{4} |10\rangle + \frac{1}{4}|11\rangle$ be a state vector describing a closed quantum mechanical system. ...
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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
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What does the product of two density matrices represent physically?

A quantum state, pure or mixed, can be described by a density matrix that encodes the Bloch vector $\hat{m}$ analog of a quantum state like $\rho = \frac{1}{2}[\mathbb{I} + \hat{m}.\vec{\sigma}]$. Let ...
Physkid's user avatar
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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
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1 answer
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How to compute the partial trace of the state $|\psi\rangle = \sum_{k}c_k |k\rangle\otimes|k\rangle\otimes |k\rangle$?

Suppose we have a quantum system defined on a Hilbert space of $H=H_A\otimes H_B\otimes H_C$, and there is a state defined in $H$ of the form: \begin{eqnarray} |\psi\rangle = \sum_{k}c_k |k\rangle\...
Zarathustra's user avatar
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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
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Create qnode with density matrix on pennylane

I'm using pennylane. What I want to do is Create a qnode with the 2*2 density matrix of a single qubit one. It has the parameter as phi Given density matrix: $$\...
Donguk kim's user avatar
2 votes
1 answer
387 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
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Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?

My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung. Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states. ...
Physkid's user avatar
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2 answers
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Induced measure on the set of density matrices defined through the Ginibre ensemble

I am defining a density matrix via $\rho = \frac{X^\dagger X}{\textrm{tr}(X^\dagger X)}$, where $X$ belongs to the Ginibre ensemble. This results in an induced distribution on the set of density ...
Ghost-of-PPPF's user avatar
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What is the probability of a state $|0\rangle$ being in another state $\alpha|0\rangle+\beta|1\rangle$?

I am trying to calculate the probability of a state (density matrix) being in a specific other state. Lets say I have a 2-dimensional state with the states given by the orthonormal basis states $|0\...
TTa's user avatar
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Can density matrices of multipartite systems always be written as product states?

suppose the density matrix $\rho_{ABC}$ with the subsystems {A,B,C} can we write $\rho_{ABC}$ as below? $\rho_{ABC}=\rho_A \otimes \rho_B \otimes \rho_C $ if the answer is yes please share a ...
reza's user avatar
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1 vote
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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
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Is Klein's inequality due to Klein?

You may be familiar with "Klein's inequality"; one form of it is $$ -\operatorname{tr}(\rho \log \sigma) + \operatorname{tr}(\rho \log \rho) \ge 0, $$ stating that relative entropy is ...
echinodermata's user avatar
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how to find a quantum gate matrix from RHO before and RHO after evolve

to evolve a 4x4 density matrix I use this method: rhoafter = np.dot(np.dot(gate,rhobefore),np.conjugate(gate.T)) And I want to find the gate from rhobefore and ...
Luis ALberto's user avatar
3 votes
1 answer
129 views

On the probability of agreeing with different density matrices?

Let's say I have a density matrix and I (person $1$) suspect it to be of the form: $$ \rho_1 = p_1 |\psi \rangle \langle \psi | + p_2 |\phi \rangle \langle \phi |$$ $|\psi \rangle$ and $| \phi \rangle$...
More Anonymous's user avatar
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33 views

Prove that $Tr(\chi(\rho_A) \log(\chi(\rho_A)) = Tr(\rho_A \log(\chi(\rho_A))$

I am trying to see how the following statement about trace $Tr$ is true. $$ Tr(\chi(\rho_A) \log(\chi(\rho_A)) = Tr(\rho_A \log(\chi(\rho_A)), $$ for some quantum state $\rho_A$, Where, $$ \chi(.) = \...
QuestionEverything's user avatar
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0 answers
48 views

Tf.einsum vs matmul for computing density matrix from a set of Cholesky decomposed matrices

I am trying to construct a density matrix of shape 256x256 from a set of T matrices. These T matrices are all Cholesky-decomposed matrices. But I am not sure if the ...
Dimitri's user avatar
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0 answers
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Is it possible to apply a quantum gate to a density marix from a partial trace?

To apply a gate(matrix) to a 2 qubit partial trace(4x4 matrix) I have this function: ...
Luis ALberto's user avatar
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1 answer
108 views

Operator qubit ordering not matching circuit qubit ordering

I tried constructing a cx gate manually using tensor products and one using QuantumCircuit in qiskit followed by converting it ...
Zee's user avatar
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0 votes
1 answer
73 views

How to perform below operation in Qiskit?

I want to implement the below equation in Qiskit. $(A \otimes B).\rho.(B^\dagger \otimes A^\dagger)$ where $\rho$ is a density matrix and $A$ and $B$ are CNOT gates. $$ A=\begin{bmatrix} 1 & 0 &...
joy Jaganath's user avatar
2 votes
2 answers
126 views

Compatibility of partial_transpose in Qiskit

I need to calculate the negativity of a density matrix; in doing so on Qiskit I stuck on the problem of computing the partial transpose for a problem of compatibility. Basically I extract my density ...
Giulia Tranquillini's user avatar
1 vote
0 answers
35 views

How to Find a circuit that evolves from one density matrix to another(qiskit or cirq)

given two density matrices, dmBefore and dmAfter, I want to generate(find) a circuit in Qiskit or Cirq that starting initaliazed with dmBefore ends with dmAfter after it's execution. Is it possible?. ...
Luis ALberto's user avatar
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0 answers
26 views

Observable for Absolute Overall Magnetization of an Ising Model

I am currently following this tutorial for generating a phase transition plot that has been generated in the same tutorial. In this tutorial's magnetization ...
Zee's user avatar
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2 votes
2 answers
1k 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
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3 votes
1 answer
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What trace properties are used in the identity ${\rm tr}_A{\rm tr}_B(\rho\Pi)={\rm tr}_A(\rho_A{\rm tr}_B(\rho_B\Pi))$?

To turn the probability of the projection over the Hilbert space $\mathcal H_A \otimes \mathcal H_B$ into the POVM probabilty over $\mathcal H_A$ we we use this equality: $$tr_Atr_B(ρΠ_i)=tr_A(ρ_Atr_B(...
catmousedog's user avatar
1 vote
1 answer
58 views

Non trace-preserving map in axiomatic approach to quantum operations

In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations). A quantum operation is ...
EugeneB's user avatar
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2 votes
1 answer
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How does Qiskit/Qasm simulate the density matrix of up to 30 qubits?

The full density matrix of 30 qubits contain $2^{30}$ states. How does qiskit/qasm implement this without storing and computing the full $2^{30}$ density matrix of possible state coefficients?
James's user avatar
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2 votes
1 answer
162 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
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2 votes
2 answers
168 views

what is square root of a density matrix power two?

I know that in algebra for a variable we have $ \sqrt {x^2} = |x|$ What if $x$ is a density matrix? Please share resource for your answer.
reza's user avatar
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1 vote
1 answer
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How to reconstruct the density matrix $\rho$ from the overlap matrix $T_{a,a'}={\rm Tr}(M^{(a)}M^{(a')})$?

Suppose we have $N$-qubit POVM $${\bf M} = \{M^{(a_1)} \otimes M^{(a_2)} \otimes \cdots \otimes M^{(a_N)}\}_{a_1, \ldots, a_N}.$$ Given an $N$-qubit state $\rho$, the measurement outcome ${\bf a} = (...
MonteNero's user avatar
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2 votes
1 answer
90 views

Converting a Matrix to a Gate in OpenQasm 2

I am a beginner when it comes to quantum computing so forgive me if this is a dumb question. Does anyone know how to create a gate from any matrix on OpenQasm2? Specifically, can anyone provide any ...
Sam's user avatar
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1 vote
1 answer
198 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
67 views

Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?

In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says: A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i, The ...
Guangliang's user avatar
1 vote
1 answer
97 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
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2 votes
2 answers
99 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
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2 votes
1 answer
75 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
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4 votes
1 answer
397 views

Collapse of the density operator

I am a bit confused by the collapse of the density operator. Consider a system described by the density operator $$ \hat{\rho}=\sum_{m}P_{m}|\psi_{m}\rangle\langle\psi_{m}| $$ and a measurement of the ...
Adrien Amour's user avatar
1 vote
1 answer
256 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
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