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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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### What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?

$|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$ I absolutely cannot figure out the Schmidt decomposition of this state. I have looked at a ton of ...
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### 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 ...
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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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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 $\rho = \overline{\rho}_1 \... • 131 1 vote 2 answers 67 views ### Is$\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 $$\text{Tr}(\text{Tr}_\mathcal{E}(\... • 131 1 vote 1 answer 37 views ### Can we de-decohere an open system? Can a mixed state become pure due to its interaction with a vast environment? Certainly, a strange proposal, and yet let's take a diagonal matrix representing a mixed state, say$$\begin{pmatrix} p_{1}... 2 votes 1 answer 41 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 $$\... 1 vote 1 answer 54 views ### If \rho_{AB} is a separable then the partial transpose w.r.t to A is PSD Def: The partial transpose of a linear operator \rho_{AB} over a Hilbert space H_A \otimes H_B w.r.t A is defined for a linear operator \rho_{AB}=\rho_A \otimes\rho_B as \rho^{T_A}_{AB}=\rho_A^... 0 votes 0 answers 93 views ### Show that the Choi of a tensor product is the tensor product of the Chois I have the following problem. Let N:L(H_A)\rightarrow L(H_A) be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is \Gamma_A^{N}=id\... 6 votes 1 answer 406 views ### Is it possible to derive a Schmidt decomposition for a mixed state? It is relatively simple to derive the Schmidt decomposition of a pure state |{\psi}\rangle \in H_A \otimes H_B with the SVD decomposition theorem. There are plenty of examples (lecture notes, books, ... • 173 4 votes 1 answer 97 views ### confusion on the LCU method regarding the normalization Let A = \sum_{k} a_k U_k where a_k are real, positive coefficients U_k are unitary matrices. I have realized that \sigma = A \rho A can be implemented on a quantum computer by using the LCU ... • 169 -1 votes 1 answer 59 views ### What are the eigenvalues of a state in thermal equilibrium? Suppose the density matrix \rho with eigenvalues k_{i}. Now consider the density matrix \rho in a thermal equilibrium with temperature T. Let's show the density matrix with \rho(T) in this ... • 771 1 vote 1 answer 52 views ### Simplification of a generic quantum state We are given a generic 2-qubit density matrix$$\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]$$... 3 votes 1 answer 46 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\...
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 ...