# Questions tagged [density-matrix]

A density matrix is a matrix that can be used to describe a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state.

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### Helstrom Measurement when two quantum states are close

I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true. To summarize the description, suppose we have two ...
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### Does ${\rm tr}(\Pi \rho) = 1$ imply $\Pi\rho\Pi=\rho$?

Suppose I have a density matrix $\rho$ and an orthogonal projector $\Pi$. Is it true that, if $tr(\Pi \rho) = 1$ then it must hold that $$\Pi \rho \Pi = \rho$$? If yes, how can I prove it?
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### Mathematics of Measurement then Partial Trace

Say we have the following quantum state: $$|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|10\rangle)$$ To measure the first qubit and then further trace out the first qubit, my notes have the ...
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### Data processing inequality for relative entropy in the presence of an amplitude damping channel

Consider the single qubit quantum depolarizing channel, given by $$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}.$$ For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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### Density Matrix of 3 Qubit Quantum Circuit (Qiskit Density Matrix)

If I have a one gate circuit like in the following image: I can calculate the density matrix by following steps: However, I want to do it for the Toffoli gate circuit: and I am confused with the ...
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### Are quantum gates superoperators? How to write a quantum circuit as superoperator?

I have some questions related to superoperators: What is the differences between quantum operators and superoperators? For instance quantum gates are also unitary operators but can we say quantum ...
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### Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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### Show that if the Lindblad satisfy $\sum_\mu L_\mu L_\mu^\dagger=\sum_\mu L_\mu^\dagger L_\mu$ then $\rho\propto I$ is a fixed point of an evolution

How can we show that the Lindblad condition: $$\sum_{\mu}L_{\mu} L_{\mu}^{\dagger} = \sum_{\mu} L_{\mu}^{\dagger}L_{\mu},\tag{1}$$ implies that $\rho \propto I$ is the fixed point of the evolution ...
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### QuTiP VS RK45: Which one gives the correct results for time-dependent systems?

I am writing a code for a quantum thermal machine which includes both coherent and dissipative time evolutions in its different stages of operation. However, evolving the system with "mesolve&...
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### Suggest, partly based upon limited numerical results, possible “elegant” exact formulas for Bures two-qubit separability probability

Lovas and Andai (https://arxiv.org/abs/1610.01410) have recently established that the separability probability (ratio of separable volume to total volume) for the nine-dimensional convex set of two-...
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Let $\rho : V_1 \to V_1$ and $\rho_2 : V_2 \to V_2$, where $V_1$ and $V_2$ are Hilbert spaces. Suppose that $U:V_1\otimes V_2 \to V_1\otimes V_2$ is a unitary operator. Define a map $M : L(V_1, ... 1answer 85 views ### SWAP test and density matrix distinguishability Let us either be given the density matrix $$|\psi\rangle\langle \psi| \otimes |\psi\rangle\langle \psi| ,$$ for an$n$qubit pure state$|\psi \rangle$or the maximally ... 1answer 127 views ### How to find a separable decomposition for$|\Psi^+\rangle\!\langle\Psi^+|+|\Phi^+\rangle\!\langle\Phi^+|$? The state $$\frac{1}{2}\left(| \phi^+ \rangle \langle \phi^+ | + | \psi^+ \rangle \langle \psi^+ | \right)$$ where $$| \phi^+ \rangle = \frac{1}{\sqrt2} \left(|00 \rangle + | 11 \rangle \right) ... 2answers 457 views ### What's the 'physical consistency' in the partial trace scenario? I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand: Physical consistency requires that any ... 1answer 31 views ### how to know the appropriate time to know the SWAP gate operation in dipole interaction Consider dipole-dipole interaction between two qubits,H_{int} = g \boldsymbol\sigma_{1}\cdot\boldsymbol\sigma_{2}=g(X_1X_2+Y_1Y_2+Z_1Z_2). How can I show that by turning on this interaction for an ... 2answers 219 views ### Proof for Cardinality of the Clifford Group In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ... 0answers 46 views ### What proportions of certain sets of PPT-two-retrit states are bound entangled or separable? For two particular (twelve-and thirteen-dimensional) sets of two-retrit states (corresponding to 9 x 9 density matrices with real off-diagonal entries), I have been able to calculate the Hilbert-... 2answers 298 views ### How does the probability of measurement turn out to be negative? c) Compute$$\text{Prob}(\uparrow_\hat{n}\uparrow_\hat{m}) \equiv \text{tr}(\pmb{E}_A(\hat{n})\pmb{E}_B(\hat{n})\pmb{p}(\lambda)), \tag{4.164}$$where \pmb{E}_A(\hat{n}) is the projection of Alice's ... 1answer 70 views ### What are the ranges of the four q parameters in the magic simplex of Bell states formula? Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state \rho_d in the "magic simplex" of Bell states \begin{... 0answers 66 views ### Is there a two-qudit Choi entanglement witness W^{(+)}? Example 2 in arXiv:1811.09896 states that the "Choi EW (entanglement witness) W^{(+)} obtained from the Choi map in d=3 \ldots is given by W^{(+)} = \frac{1}{6} \left( \... 2answers 35 views ### Compute {\rm tr}(a_k a_{k'}\rho) with \rho=e^{-\beta H}/Z(\beta) Gibbs state and a_k ladder operators Consider a harmonic oscillator with hamiltonian H=\sum_k\omega_k a_k^\dagger a_k and a state \rho=\frac{e^{-\beta H}}{Z(\beta)} where Z(\beta)=\text{tr}[{e^{-\beta H}}]. The quantity$$A:=\sum_{... 3answers 316 views ### Interpretation of the unitaries involved in the eigenvalue decomposition of a density operator If$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle \psi_{i}|$, this ensemble doesn't require$\langle \psi_{i}|\psi_{j}\rangle$=0. Given that$\rho$is positive semi-definite, by the spectral theorem it ... 0answers 94 views ### Is there a way to write down the eigenstates of this two-qubit density matrix? I am considering the density matrix which represents an arbitrary state for a pair of qubits. When written out in terms of the Pauli operators, this is as follows (certain terms vanish for another ... 1answer 37 views ### Relation between symmetric subspaces and$n$-exchangeable density matrices Let us consider$n$elements, each taken from the set$\{1, 2, \ldots, d\}$and let$S_n$be the set of all permutations on these$n$elements. Define a permutation operator on the set of$n$qudits ... 2answers 387 views ### Quantum teleportation of a mixed state through a pure state? Let's assume we have a register of qubits present in a mixed state $$\rho = \sum_i^n p_i|\psi_i\rangle \langle \psi_i|$$ and we want to teleport$\rho$through a random pure state$|\phi\rangle$. What ... 2answers 44 views ### How to combine/calculate for interference using density matrices? Let's assume following two density matrices are corresponding to the A and B in the Stern-Gerlach apparatus bellow (I know Stern-Gerlach is a more of a physics experiment but I think it can equally be ... 1answer 146 views ### Can all mixed states be written as a convex combination$\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$? States belonging to some space$\mathcal H$can be described by density operators$\rho\in L(\mathcal H)$that are positive and have trace one. Pure states are the ones that can be written as$\rho=|\...
Let $|\psi\rangle = C_1 |0^{n}\rangle$ be a quantum state, such that $C_1$ is a Haar random unitary circuit. Consider a density matrix $\rho$ as follows \rho_1 = \mathbb{E}[|\psi\...
I have a question regarding Kraus operators. Any quantum channel can be written in terms of Kraus operators as $E(\rho)= \sum_{i=0}^n K_i \rho K_i^{\dagger}$ where $\rho$ is the initial density ...