# 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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### measurement probability from density operator?

I've been through this before but I can't fully get my head round this upon review. So the density operator $\hat{\rho}=\sum_j p_j|\psi_j\rangle\!\langle \psi_{j}|$ for pure states $|\psi_{j}>$ at ...
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### 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-...
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### Show that while calculating partial traces the probability is independent of the basis of one of the measurements

Consider calculating the probability of the outcome m alone of some composite system $AB$. $p_A(m) = \sum_{n=0}^{d_B-1} p_{AB}(m,n)$ $= \sum_{n=0}^{d_B-1}(⟨α_m|⊗⟨β_n|)\rho_{AB}(|α_m⟩⊗|β_n⟩)$ I'm ...
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### 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 $B$...
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Say $M$ is a matrix acting on $C^r \otimes C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$. With $|\psi\rangle$ sampled from Haar, how can we show that $$\int (... • 153 3 votes 3 answers 63 views ### How to show that a density matrix \rho is extreme iff \rho=|\psi\rangle\!\langle\psi|? A density matrix ρ is called extreme if the only way to write ρ as ρ = p σ + (1 − p) τ , with σ ∈ S_d, τ ∈ S_d, and p ∈ (0, 1) is to have ρ = σ = τ . I want to show that a density matrix ... • 313 3 votes 1 answer 35 views ### What is an "x-type density matrix"? i want to know what is a x-type density matrix structure? i want to know the general case of it. is this something like this? can one of matrix elements be 0? unfortunately there is no info about it ... • 55 3 votes 3 answers 114 views ### Explain the comparison between a state and density matrix The density matrix \rho = \frac{1}{2}(|0\rangle \langle0|+|1\rangle \langle1|) describes a system which is in state |0\rangle and in |1\rangle with equal probability. Also the state |\psi\... 7 votes 2 answers 151 views ### Given an orthogonal projection \Pi, is \|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1 true? Suppose I have an arbitrary orthogonal projector \Pi and two density operators \rho, \sigma. Is it true that:$$ ||\Pi (\sigma - \rho) \Pi||_1 \le || \sigma - \rho ||_1 $$where ||\cdot||_1 ... 0 votes 1 answer 78 views ### What does I represent in the expression A\otimes I? This paper states: Suppose a two qubit system is in the state |\psi\rangle=a|00\rangle+b|11\rangle, and consider the expectation value of any observable A \otimes I that is nontrivial only on the ... • 21 1 vote 0 answers 56 views ### Is \rho | \psi \rangle invariant in the Wigners friend thought experiment? Background Let's say I have a gas of N particles where I cannot distinguish between the particles at a temperature T. Its density matrix is given by \rho. Note, if my friend happens to measure ... 1 vote 2 answers 74 views ### Why can't the purity of a single qubit be less than 1/2? The density matrix of a single qubit system can be defined as,$$ \rho= \frac{1}{2}(\hat I+ \vec r.\hat{\vec \sigma}) $$From here we can derive,$$ Tr(\rho^2)= \frac{1}{2}(1+r^2) $$Since  0\leq r^... 1 vote 1 answer 70 views ### Qiskit DensityMatrix.from_instruction when snapshots are present I have a quantum circuit in which I apply snapshots like this during setup: ... • 111 5 votes 3 answers 2k views ### What is a "maximally mixed state"? What is meant by maximally mixed states? Does this mean that there are partially mixed states? For example, consider \rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right| and \rho_W =... • 255 3 votes 1 answer 27 views ### Relating upper bound on the total variation distance with a bound on a Pauli observable Consider an n qubit state |\psi\rangle. Let us say we have the following upper bound on the total variation distance between the distribution generated by measuring |\psi\rangle in the standard ... • 943 2 votes 1 answer 160 views ### How to find the Kraus operators from the process matrix? I am trying to find the Kraus operator from process matrix. For instance, suppose that for single qubit identity gate, I have the following process matrix: ... • 446 0 votes 1 answer 102 views ### How to write the Kraus representation for many-qubit states? The most general formula of Kraus operator on density matrix is:$$\rho\to \sum_k A_k^\dagger\rho A_k.$$If I want to write this equation for one qubit, the most general way will be: \rho_f = (a^*I+b^... • 446 3 votes 1 answer 65 views ### Helstrom Measurement when two quantum states are close I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true. To summarize the description, suppose we have two ... 4 votes 4 answers 108 views ### Why does \rho^A=\mathrm{tr}_B(\rho^{AB}) guarantee that \mathrm{tr}(M\rho^A)=\mathrm{tr}((M\otimes I_B)\rho^{AB})? Niesen and Chuang, 2nd edition, page 107, Box 2.6, in its motivation for partial trace, says that if M is an observable on system A and \tilde{M} is the corresponding observable on system AB, then ... • 645 5 votes 2 answers 80 views ### Does {\rm tr}(\Pi \rho) = 1 imply \Pi\rho\Pi=\rho? Suppose I have a density matrix \rho and an orthogonal projector \Pi. Is it true that, if tr(\Pi \rho) = 1 then it must hold that$$\Pi \rho \Pi = \rho$$? If yes, how can I prove it? 1 vote 1 answer 49 views ### Mathematics of Measurement then Partial Trace Say we have the following quantum state:$$ |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|10\rangle)$$To measure the first qubit and then further trace out the first qubit, my notes have the ... 2 votes 2 answers 290 views ### Density Matrix of 3 Qubit Quantum Circuit (Qiskit Density Matrix) If I have a one gate circuit like in the following image: I can calculate the density matrix by following steps: However, I want to do it for the Toffoli gate circuit: and I am confused with the ... • 446 4 votes 1 answer 80 views ### Data processing inequality for relative entropy in the presence of an amplitude damping channel Consider the single qubit quantum depolarizing channel, given by$$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 ... • 943 0 votes 1 answer 74 views ### 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 ... • 446 3 votes 2 answers 60 views ### Are the states in the convex decomposition of a density matrix necessarily orthogonal? In Nielsen and Chuang's QC&QI, I do not see a statement one way or another. In Steeb and Hardy's Problems and Solutions, orthogonality is asserted. If the p_i in \sum_i p_i |\psi_i\rangle\... • 645 1 vote 1 answer 80 views ### Given a three-qubit state, how do you obtain the density matrix for the third qubit I have a quantum simulator that yields a three-qubit final state. However, I need to measure the first two qubits and apply a one-qubit gate (x,y or z) to the third qubit. How do you reduce a three-... • 645 2 votes 2 answers 176 views ### How to compute the unitary from the \chi matrix obtained from QPT I am trying to do quantum process tomography for one qubit and obtain the unitary for the gates that are applied on the qubit. I have studied the theory on process tomography from mike and ike and the ... 7 votes 4 answers 293 views ### Simulating a quantum circuit with decoherence and noise Based on the answers given here and here, it is pretty clear that an arbitrary quantum circuit can be simulated with matrix algebra. The difficulty is that this assume perfect fidelity. I am unsure ... • 645 7 votes 1 answer 188 views ### 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 ... • 2,139 1 vote 0 answers 88 views ### 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&... • 111 1 vote 0 answers 48 views ### Evaluation of Wigner function representation of a Bloch Sphere Consider Wigner function representation of a qubit in the basis labeled by \sigma_z and \sigma_x eigenvalues. A general single qubit mixed state has the Bloch representation,\rho = 1/2 (I + r.\... 3 votes 1 answer 45 views ### 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}$$implies that \rho \propto I is the fixed point of the evolution ... 5 votes 1 answer 163 views ### Show that if the Lindblad satisfy \sum_\mu L_\mu L_\mu^\dagger=\sum_\mu L_\mu^\dagger L_\mu then the von Neumann entropy increases monotonically How can we show that when the Lindblad operators satisfy the condition:$$\sum_{\mu}L_{\mu} L_{\mu}^{\dagger} = \sum_{\mu} L_{\mu}^{\dagger}L_{\mu},\tag{1}$$the master equation evolution ... 3 votes 2 answers 80 views ### Closeness of \rho such that \text{Tr}(|\psi\rangle\langle\psi|\rho)\le1/2^n+{\cal O}(2^{-2n} ) for all |\psi\rangle to the maximally mixed state Consider an n qubit density matrix \rho such that$$\text{Tr}(|\psi\rangle\langle \psi| ~\rho) \leq \frac{1}{2^{n}} + \mathcal{O}\left(\frac{1}{2^{2n}} \right), $$for every n qubit pure state |... • 943 3 votes 1 answer 186 views ### Schmidt vectors for random quantum states Consider a random quantum circuit U over n qubits, drawn from the Haar measure. Consider the quantum state$$|\psi\rangle = U |0^{n}\rangle.$$Now, partition n into two and consider the Schmidt ... • 943 0 votes 1 answer 116 views ### Show that the trace of squared density matrix gives {\rm tr}(\rho^2)=\frac12(1+\|\mathbf n\|^2) [duplicate] Equation 7.7 is given below:$$\hat\rho = \frac12(I +n_x(\hat X)+n_y(\hat Y)+n_z(\hat Z)) $$Where I is the identity matrix and \hat X,\hat Y,\hat Z are Pauli matrices. Now my attempt of this was ... • 101 0 votes 1 answer 68 views ### Is it possible to find a 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio? [closed] Is it possible to find 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio and if it is how is it done? • 51 3 votes 0 answers 136 views ### Reduced density matrix of a Haar random state and its Schmidt decomposition Consider a Haar random quantum state |\psi\rangle. Note that$$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}. $\mathbb{I}_n$ is the identity operator on $n$ qubits. ...
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How can we find a Wigner function for the four maximally entangled Bell states $(|00\rangle \pm |11\rangle)/\sqrt{2}$, $(|01\rangle \pm |10\rangle)/\sqrt{2}$? I have used the basis where labels for ...
Consider a tripartite quantum system with the three subsystems labeled $A, B,$ and $C$. Now take two states $\rho_{AB}$ on the joint system $AB$ and $\rho_{BC}$ on the joint system $BC$. Under what ...