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Questions tagged [density-operator]

The density operator describes a quantum system in an (in general mixed) state.

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What does the energy-resolved spin density averages show?

What does the energy-resolved spin density averages show? Spin density refers to the density of states for spin up and down, my question is what does energy-resolved refer and how can be the energy-...
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Matrix Product State - entanglement entropy as function of location / position on lattice [on hold]

I have a hw problem that consists of implementing MPS decomposition algorithm (based on https://arxiv.org/abs/quant-ph/0301063 paper) and a further "- Try out your code on $N = 10$ GHZ states for ...
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Why is the partial differentiation of density operator with respect to time zero, for an ensemble in thermal equilibrium?

Sakurai initially says that density operator evolves with time because state kets evolve with time. But for an ensemble in thermal equilibrium, its partial differentiation is zero. As far as I know, ...
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Explanation of why this derivation of Schmidt decomposition works

I'm following Preskill's notes and he derives the Schmidt decomposition in the following way: Let a bipartite state be $\psi_{AB} = \sum_{i,j}\lambda_{ij}\vert i\rangle\vert j\rangle = \sum_{i} \vert ...
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Conditioning a quantum state: why does it have this particular form?

Can someone please explain the phenomenon of conditioning a quantum state, and why it has this particular form? An observable $A$ of a quantum mechanical system, described by the density operator $\...
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How to obtain $Y$ rotation with only $X$ and $Z$ rotation gates on the Bloch sphere?

Let's say you have a system with which you can perform arbitrary rotations around the $X$ and $Z$ axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the $Y$ ...
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Density operator in second quantization form [closed]

In first quantization the particle density operator is $$n(x)=\sum_{\alpha}\delta^{3}(\vec{x}-\vec{x}_{\alpha})$$ In second quantization I have: $$ n(\vec{x})=\sum_{\alpha,i,j}\langle i|_{\alpha}\...
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Programmatically generate valid density matrices of arbitrary dimension

I would like to generate a random $N\times N$ density matrix for a program. My current technique works for qubits but I suspect there are much more elegant ways. For a single qubit state, I write $\...
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What is the source of discontinuity of heat capacity?

Here's something that's been bothering me. Given the density matrix of a system: $$ \rho = \sum_{ij} p_{ij} |\psi_i \rangle \langle \psi_j |$$ Now, we can define it's entropy given by: $$ S= \text{...
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Can a single-qubit state be nontrivially extended to a non-pure state?

Consider a generic single-qubit state $$\rho=\lambda_1\lvert \lambda_1\rangle\!\langle \lambda_1\rvert+\lambda_2\lvert \lambda_2\rangle\!\langle \lambda_2\rvert\in\mathcal H_S.$$ I am interested in ...
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Schwinger-Keldysh path integral and RG flow ambiguity

I am wondering about how renormalization group (RG) flow works on the Schwinger-Keldysh (SK) contour. In particular, it seems like there is an ambiguity in how to define RG flow depending on what ...
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Entangled state density matrix: Defintion and example

I was looking at this question but still don't fully understand the distinction between classically correlated mixed states and entangled mixed states. I understand that a pure state is considered ...
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Proof of strong convexity of trace distance

I'm trying to follow the Nielsen and Chuang proof (equation 9.49 of Chapter 9, page 408). I reproduce it here for completeness. With trace distance defined as $D(\rho, \sigma) = \frac{1}{2}tr(|\rho - ...
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1answer
63 views

Can we compute the entropy of a subsystem of a quantum state?

Suppose there is some entangled system with 5 subsystems labeled 1 to 5. Can we write the density operator for the subspace of subsystems 1 to 4 and compute entropy and the other relations with it ...
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Does the anti-diagonal of a density matrix have any special interpretation?

Suppose a density matrix $$ \rho= \begin{bmatrix} x_{11} & x_{12} & x_{13} & \cdots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \cdots & x_{2n} \\ \...
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Complete positivity: why is the condition sufficient for quantum maps?

I know that when we define quantum maps, we need the map to be completly positive, to ensure that if our system $A$ is entangled with some extra system $B$, the evolution on $H_A \otimes H_B$ will ...
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Is partial trace the inverse operation of Kronecker product?

Computer science student here, who is interested in quantum information theory. Suppose I have these pure states: \begin{bmatrix}1&0\\0&0\end{bmatrix} and \begin{bmatrix}0&0\\0&1\end{...
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Spontaneous symmetry breaking in fluids

BACKGROUND: One can think of solids as spontaneously breaking translational symmetries in the sense that each atom in a lattice has to pick a particular position. Yet, as with everything in our ...
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Quantum map and preservation of trace

I am currently learning about quantum maps, ie maps that transform a density matrix into another one. Assume we are in the Hilbert space : $H_A \otimes H_B$. I call the quantum map on the density ...
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Reduced density matrix of two spins

I am reading this (https://arxiv.org/abs/1209.0062) article about constructing order parameters from reduced density matrix. The author is discussing long-range order by taking antiferromagnetic spin ...
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General properties on reduced density matrix with assumption on the global density matrix

Let's consider $\mathcal{H_1} \otimes \mathcal{H_2}$ the space of the problem. I call $\rho$ a density matrix of the full space and : $\rho_1=Tr_2(\rho)$ the reduced density matrix in $\mathcal{H_1}$...
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How to calculate correlation functions for fermionic operators?

In the paper by Peschel (2003) https://arxiv.org/pdf/cond-mat/0212631.pdf How does one derive the following relation: $$ \langle c_{n}^\dagger c_{m}^\dagger c_{k}c_{l}\rangle = \langle c_{n}^\...
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Can we say that density matrix gives probability?

In statistical physics Boltzmann probability is given by $$P= \frac{\exp(-\beta E_i)}{\sum_i\exp(-\beta E_i)}$$ whereas we can also write it $$\rho= \frac{\exp(-\beta H_0)}{\sum_i\exp(-\beta H_0)}.$$...
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Why we ignore off-diagonal elements in partition function?

In quantum statistical mechanics, the density operator is $$ \rho = \exp(-\beta H_0)/Z $$ where $$Z = \text{Tr} (\exp(-\beta H_0)) \, .$$ Why do we take the trace over only diagonal elements and ...
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Given density matrices of the subsystems is it in general possible to recover density matrix of the whole system?

Let us consider bipartite system in an entangled mixed state. Since its density matrix can always be diagonalized we can write it in following ways: $$\rho_{AB} = \sum_{j} p_{j} | \psi_j\rangle \...
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von Neumann measurement model of a qubit with continuous detector

I have a two state qubit system with initial state $|\psi_s\rangle_i = a|0\rangle+b|1\rangle$ and a detector with initial state $$|\psi_d\rangle_i = \int_{-\infty}^{\infty}\left(N \exp[-\frac{q^2}{2\...
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1answer
63 views

Non-Hermitian measurement operators

I am familiar with the von Neumann projection postulates but I don't know how one can write a non-Hermitian measurement operator using von Neumann measurement model. Does anyone can help?
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1answer
25 views

Density matrix for system and surroundings

In my QM lecture it was claimed that if you have a system with degrees freedom $\vec{s}$ and its surroundings which have degrees of freedom $\vec{u}$ then every density matrix for the combined system ...
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Classical correlations in bipartite entangled mixed state

I have recently asked somewhat related question and got very illuminating answer. After some thinking however I have realized that (at least) one more point is unclear to me: How can we check ...
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3answers
67 views

Deriving or building a Hamiltonian from a Density Matrix

Is it possible to create a Hamiltonian if given a Density Matrix. If you already the the Density Matrix, then is the Partition Function (Z) even needed? This Q is not about physics. Its about an ...
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1answer
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Quantum superposition in density matrix formalism

I was thinking about quantum superposition and stumbled into something that made me quite uncomfortable. Consider a qubit with Hamiltonian eigenstates $|0\rangle$ and $|1\rangle$. To each of these ...
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1answer
81 views

Calculating the Probability of Measuring One Hydrogen Atom in the Given States

I'm currently enrolled in a statistical mechanics course and am a bit stuck on how to calculate the probabilities of a hydrogen atom in a given state. I'll post the exact question I'm working on and ...
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Classical and quantum correlations in bipartite system

I would like to know how to answer following questions: Is there classical/quantum correlations in given bipartite pure/mixed state? I have gathered several definitions. Some of them (it seems) ...
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1answer
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Lindbladian and Dynamical semigroups

I am attempting to learn a bit more about open quantum systems. Often we derive master equations or Heisenberg-Langevin equations where we have something like \begin{align} \dot{\rho}(t) = \mathcal{...
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Mathematical formulation of density matrix on hypersurface in 3+1 formalism

Of course we have a notion of qft in curved spacetime, though I'm not sure how one can represent a particle state on curved spacetime without a timelike Killing vector field (i.e. a particle should ...
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Is entanglement *not* intrinsic to state, but dependent on division into subsystems? (Susskind QM)

I'm working through Susskind's "Quantum Mechanics" book (TTM series), which I quite like. Background In Lecture 7 (Chapter 7), he studies a 2-spin system. A single spin has eigenvectors: $$|u\...
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Why is it not possible to describe a mixed quantum state by a Hilbert space vector?

I read (for instance in Landau/Lifshitz III) that if I know the wave function of a quantum state, I have the maximal information of the state available, in different words, the description of the ...
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Is purification physicaly meaningful?

Consider a quantum system with Hilbert space $\mathscr{H}$ and suppose the quantum state is specified by a density operator $\rho$. Since it is hermitian, it has a spectral decomposition: $$\rho = \...
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Lower bound on quantum relative entropy

In my research this summer, I have become interested in lower bounds on the standard "Umegaki quantum relative entropy". For two non-negative matrices $X$ and $Y$, the Umegaki quantum relative ...
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1answer
102 views

Reconcile a pair of two-qubit boundary-state separability probability analyses

It is now clearly well-established--though formalized proofs are still largely lacking—that the probability, with respect to Hilbert-Schmidt measure, that a generic two-qubit state is separable/...
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60 views

Displacement transformation of Liouvillian superoperator

The displacement operator $D(\alpha)$ has the property $D^{\dagger}(\alpha) \hat{a} D(\alpha) = \hat{a} + \alpha$. We obtain the Hamiltonian $\hat{H}'$ in the displaced frame from the transformation $...
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1answer
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Quantum mechanics in terms of density operators as the fundamental object? [duplicate]

So I've done a two courses in undergrad quantum mechanics, the first began with wave mechanics and then went on to bras and kets, the second course went more into detail regarding bras, kets, hilbert ...
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48 views

Question about the true significance of the partial trace

Consider a composite system whose Hilbert space is $\mathcal{H}_{AB}=\mathcal{H}_A\otimes \mathcal{H}_B$, where $\{|0_A\rangle, |1_A\rangle\}$ and $\{|0_B\rangle, |1_B\rangle\}$ are orthonormal bases ...
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If a wave function represents knowledge, what does a density matrix mean, then?

I'm curious about this. I've heard of interpretations of quantum theory in which the wave function $\psi$ is taken as representing knowledge, or information (e.g. "Quantum Bayesianism"), about the ...
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52 views

Density Matrix approach in Density Functional Theory - interpretation

In a paper describing a Kohn-Sham Density Functional Theory implementation, the authors describe the use of the density matrix for e.g. the calculation of the electronic density and for efficiency ...
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Time evolution of a projected mixed state

Suppose a quantum system (non-interacting) at finite temperature ($\beta^{-1}$). I want to know how to compute the transition probability between two degrees of freedom ($u$ and $v$) at two different ...
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52 views

Phase space representations and coherent state representation of $\rho$, W, P and Q functions in quantum optics and related questions

What is the difference between phase space representations (with real x & p as variables) and coherent state representation (with complex $\alpha$ as a variable) of $\rho$, W, P and Q functions in ...
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56 views

Evolution of reduced density matrix

Suppose we have two density matrices of an n-partite system, $\rho$, $\rho'$, with $\rho$ $\neq$ $\rho'$, but $\rho_A$ = $\rho_A'$, where A is a certain subset of the n parties. Is it true that $(U$$\...
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Proof of factorization at late times for chaotic systems

While reading the paper "A bound on Chaos - Maldacena et. al", https://arxiv.org/abs/1503.01409 in equation (23) of the paper they factorize a correlator of the form, $$ Tr [\rho^{1/2} W(t) V \rho^{1/...
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Estimate of trace of powers of density matrix

Given a very generic, lower bounded Hamiltonian, is there a estimate on how $Tr(\rho^{1/k})$ grows as $k>0$ increases? Does this quantity diverge as a function of $N$, the degrees of freedom of the ...