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.
420 questions
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Density Matrix for a Quantum Circuit with Clifford Gates and a $T$ Gate in Qiskit
I am trying to analyze the impact of a single $T$ gate within a quantum circuit that primarily consists of Clifford gates. My goal is to understand the $T$ gate's role in $T$-design and Anti-...
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Depolarizing channel on GHZ-state
Consider a GHZ-state $|\psi\rangle =\frac{1}{\sqrt{2}}(|0\rangle^{n}+|1\rangle^n)$, and consider a depolarizing channel that maps a density matrix
$$\rho\to(1-\lambda)\rho + \frac{\lambda}{2^d}I.$$
...
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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, ...
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Can a CPTP map increase the purity of a state?
I am wondering if there exist CPTP maps $T$ such that the purity of a quantum state $\rho$ can increase, i.e.
$$ \text{tr} ( T ( \rho )^2 ) \geq \text{tr} ( \rho ^2). $$
If so, what are the conditions ...
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Can I obtain the pure state corresponding to a density matrix from its main diagonal?
Suppose we have a bipartite pure state as follows:
$$|\psi\rangle=a_1|00\rangle+a_2|01\rangle+a_3|10\rangle+a_4|11\rangle\,.$$
Then, the density matrix is as follows:
$$|\psi\rangle\langle\psi|=\left(
...
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What is the formula for the matrix representation of a general controlled gate?
Suppose I have $n$-qubit circuit. I have a single-qubit gate (e.g. a Pauli gate) at qubit $a$ and it is controlled by the qubit $b$. What is the matrix representation for this controlled gate? The ...
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How to analyze a system in nonthermal equilibrium?
In quantum information theory, density matrix is one of the main resource for analyzing a system.
I know in general how to obtain density matrix of a system but there is a case that still i dont know ...
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Minimizing $1 - \text{Tr}(\Phi(\rho,U)^2)$
I am looking for a computationally efficient way to minimize the following function. Let
$$\Phi(\rho, U) = \text{Tr}_2(U\rho U^\dagger)$$
be a reduced density matrix where $\rho = \overline{\rho}_1 \...
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Is $\text{Tr}(\text{Tr}_\mathcal{E}(\rho)) = \text{Tr}(\rho)$?
Let $\rho$ be a density matrix over some composite Hilbert space $\mathcal{H}_S \otimes \mathcal{H}_{\mathcal{E}}$. Is partial trace full trace preserving? I.e., is
$$\text{Tr}(\text{Tr}_\mathcal{E}(\...
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Can we de-decohere an open system?
Can a mixed state become pure due to its interaction with a vast environment? Certainly, a strange proposal, and yet let's take a diagonal matrix representing a mixed state, say $$\begin{pmatrix}
p_{1}...
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How is the expression $\frac{\|\rho^{T_B}\|-1}{2}$ obtained from the definition of negativity?
In quantum information theory, negativity is defined as summation of the absolute values of negative eigenvalues of the partial transposed density matrix. The expression of negativity is given as
$$
\...
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If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD
Def: The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^...
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Show that the Choi of a tensor product is the tensor product of the Chois
I have the following problem. Let $N:L(H_A)\rightarrow L(H_A)$ be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is $\Gamma_A^{N}=id\...
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Is it possible to derive a Schmidt decomposition for a mixed state?
It is relatively simple to derive the Schmidt decomposition of a pure state $|{\psi}\rangle \in H_A \otimes H_B$ with the SVD decomposition theorem. There are plenty of examples (lecture notes, books, ...
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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 ...
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What are the eigenvalues of a state in thermal equilibrium?
Suppose the density matrix $\rho$ with eigenvalues $k_{i}$.
Now consider the density matrix $\rho$ in a thermal equilibrium with temperature $T$. Let's show the density matrix with $\rho(T)$ in this ...
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Simplification of a generic quantum state
We are given a generic 2-qubit density matrix
$$\rho=\frac{1}{4}\left[I_4+\Sigma_i a_i \sigma_i \otimes I_2 + \Sigma_i b_i I_2 \otimes \sigma_i + \Sigma_{i,j} c_{ij} \sigma_i \otimes \sigma_j\right]$$
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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\...
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What is the best way to model a polarizer?
If I have a photon reaching a polarizer, I can think of a polarizer as an operator of $P=a^\dagger_Va_V$ where $a^\dagger_V$ creates a photon with vertical polarization (V).
However, on the other ...
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Obtaining the reduced density matrices for both subsystems of a bipartite system [duplicate]
If we have a single copy of a bipartite quantum system with density matrix $\rho$, is it possible to extract the reduced density matrices of the constituent subsystems separately, i.e. to achieve the ...
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Proving CLDM is in QMA, In particular why is it possible to assume that the given input is a product of copies in the soundness section?
I'm wondering about a specific proof for Consistency of Local Density Matrices (CLDM) $ \in $ QMA appearing in "QMA-hardness of Consistency of Local Density Matrices with Applications to Quantum ...
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Initialize circuit by density matrix (i.e. mixed state) by Cirq, Qiskit, QuTip
I want to do a simulation involving:
mid-circuit measurement (i.e. based on the measurement result of some qubits to append further gates on other qubits; e.g., Pauli error correction in entanglement ...
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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 ...
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How to find density matrix of 3 qubit W state?
Given a state in bra-ket notation as $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix of this state written using Pauli's spin operator?
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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 ...
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QISKIT: ValueError: too many subscripts in einsum DensityMatrix()
I am trying to compute the entanglement entropy of a partition of a quantum system on qiskit. To do this, I call the function DensityMatrix(). If I go above 10 sites (e.g. 12), I get an error like:
...
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Derivation of Choi-Jamiolkowski isomorphism
I'm following the course Mathematical Methods of Quantum Information Theory by Reinhard Werner. In lecture 6, he gave a derivation of Choi-Jamiolkowski isomorphism, and I'm struggling to understand ...
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How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?
I am confused as to how we get $M_{0} = \sqrt{1-p} I$ and the following $M_{1}, M_{2}, M_{3}$.
The above notes say that we should partially trace over the environment in the $|{0}\rangle, |{1}\rangle, ...
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Why is a density matrix an orthogonal projector?
Suppose I have a density matrix like $\rho = \frac{1}{2}[I + \hat{n}\vec{\sigma}]$.
The claim is that $\rho$ is an orthogonal projector for the state $|+\rangle$ in an arbitrary direction $\hat{n}$.
...
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I have two Choi matrix I suspect be equivalent. Can I manipulate them?
I am performing a process tomography over a protocol I suspect to be equivalent to the $CS$ gate.
To compare basic operators I usually compute the Choi matrix of the target gate -- which in this case ...
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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 ...
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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}, \...
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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\...
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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| +...
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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 ...
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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/...
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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 ...
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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 ...
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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}[\...
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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 ...
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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 ...
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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 ...
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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} | \...
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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\...
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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...
...
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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:
$$\...
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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^...
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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.
...
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