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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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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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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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28 views

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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40 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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60 views

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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49 views

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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30 views

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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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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49 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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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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35 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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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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23 views

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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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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50 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 ...
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Solving the von-Neumann equation explicitly [closed]

The von-Neumann equation reads: $$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho]$$ The solution: $$\rho(t)=U\rho U^{\dagger}$$ with $U=e^{-\frac{iHt}{\hbar}}$ is easily obtained when starting from the ...
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Are density matrices symmetric? [closed]

The context is that I want to simplify an expression like $$ \mathrm{Trace}[\rho_1 \rho_2 \rho_3] + \mathrm{Trace}[\rho_2 \rho_1 \rho_3] $$ (Note that the second term is not a cyclic permutation of ...
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Conventions for density matrix and projections

From this link : [http://math.ucr.edu/home/baez/lie/node12.html][1], it is said that, from starting a quantum state, $$\vec{v}=a|\text{up} \rangle+b|\text{down} \rangle,$$ we can define the matrix ...
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Markovian approximation for teleportation? [closed]

Assume a model including a system with time dependent Hamiltonian ( 3 entangled qubits subject to a noisy reservoir) coupled weakly to a thermal bath. in order to study the time evolution of a ...
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Why is the partial trace of this subsystem equal to this? [closed]

I am doing my bachelors dissertation based on an article by David Deutsch. He defines the action of a quantum gate as: $$ U = \sum_{x, y \in \mathcal{Z}_{2}} |x \dot{+}y\rangle|y\rangle\langle x|\...
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Finite temperature quantum mechanics and mixed states

Is it necessarily true that a quantum-mechanical system in thermal equilibrium is in a mixed state? If so, why is this the case? Is there any physical intuition as to why one cannot use a pure state ...
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1answer
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“Contraction Property of thermal density matrix” in the Maldacena's paper of A Bound of Chaos

In the paper, https://arxiv.org/abs/1503.01409 (Maldacena, et al. “A Bound on Chaos.”) in equation (24), the authors write an inequality, $$ Tr( y^{1+\eta} V y^{3-\eta} V ) \leq Tr(y V y^2 V) $$Where $...
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Understanding the degrees of freedom counting argument for complex amplitudes in quantum mechanics

In these lecture notes, Scott Aaronson gives three arguments for why complex Hilbert spaces are more natural than real ones for formulating quantum mechanics. I don't understand his second argument, ...
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Density matrix of a partially polarized beam of electrons

The following problem is from Huang's Statistical Mechanics (2nd edition): 8.1 Find the density matrix for a partially polarized incident beam of electrons in a scattering experiment, in which a ...
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Are there constraints on the density matrix during decoherence?

A privileged basis emerges during measurement. starting from a pure vector state $|V \rangle$ in a n dimensional hilbert space it gives after décoherence a density matrix $p_1 |e_1 \rangle \langle e_1|...
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Why do the density operators span the whole operator space $\mathcal{B}(H)$?

The convex set of density operators on a finite-dimensional Hilbert space $H$ defined by $\mathcal{D}(H):=\{\rho\in\mathcal{B}(H)|\,\rho\geq 0, \text{tr}\rho =1\}$ is said to span the entire space of ...
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Are the diagonals of the density operator always the probabilities of finding the system in various states?

I found lecture notes online that said "Diagonal density matrix elements are the probabilities of finding system in various states." My quantum mechanics textbook doesn't say anything about this, it ...
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How can the symmetry of the quantum state fidelity be shown directly?

Consider the quantum state fidelity $F(\rho,\sigma)$ defined as (I will use the notation used in Nielsen & Chuang here): $$ F(\rho,\sigma) \equiv \operatorname{Tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}} ...
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System-Environment Correlations for a Quantum Dynamical Semigroup

The density matrix for an open bipartite system can be written $$ \rho = \rho_S \otimes \rho_E + \rho_{corr} $$ and it is assumed that the system starts in a separable state $\rho(0) = \rho_S \otimes ...
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Deriving a path-integral expression for a thermal density matrix with position-dependent temperature

I've been fiddling with deriving a path-integral expression for a thermal partition function with a position-dependent temperature but I'm not sure how to get started on this. Concretely, I'm trying ...
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63 views

What are unital maps?

In quantum information, one defines an unital map as the one that preserves the identity operator $\epsilon (I) = I$. A popular example is the so-called amplitude damping channel. My question is, ...
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85 views

Does this density matrix represent a density operator of some single qubit state?

$$\rho = \begin{bmatrix} 1/2 & (1+i)/{\sqrt{2}} \\ (1+i)/{\sqrt{2}} & 1/2 \end{bmatrix} $$ Can this matrix represent a a density operator of some single qubit state? I'm a little confused ...
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Deriving Reduced Density Matrix for Electron Z Coordinate Position from State Vector of Electron Z-Spin Entangled with Z Momentum

Suppose I have a composite state vector that is a product state of z spin and z coordinate momentum of an electron (tensor product of state vectors for z spin and z momentum). Then, at $t_0$, the z ...
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74 views

What are the range of values of the negativity in quantum mechanics?

The negativity of a quantum state, given by $N(\rho)=\frac{||\rho^{\Gamma_A}||-1}{2}$ where $\rho^{\Gamma_A}$ is the partial transpose with respect to subsystem $A$ of the density matrix of a ...
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42 views

Density operators of coherent state plus phase commute?

I am trying to see if $ \left[ \lvert e^{i \theta} \alpha \rangle \langle e^{i \theta} \alpha\rvert, \lvert e^{i \theta'} \alpha \rangle \langle e^{i \theta'} \alpha\rvert \right] = 0 $ with $\...
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State after measurement when outcome is in a given set

I have become quite rusty so please don't mind if you find something stupid. Axiom of QM is that, upon measurement of an observable A for a system described by a state $\rho_t$ at time $t$, ...
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Can the same density matrix represent two (or more) different ensembles?

Given an ensemble i.e, a collection of states and the respective probabilities $\{p_i,|i\rangle\}$, one can uniquely construct the density matrix using $\rho=\sum_ip_i|i\rangle\langle i|$. Is the ...
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Set of states $\{|\phi_n\rangle\}$ in the density operator $\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$

The set of quantum states $\{|\phi_n\rangle\}$ in the definition of the density operator $$\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$$ need not be orthonormal, and need not form a basis. But ...
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Difference between pure quantum states and coherent quantum states

In the post What is coherence in quantum mechanics? and the answer by udrv in this post it seems to imply that a pure quantum state and coherent quantum state are the same thing since any pure state ...
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If I have a two-qubit subsystem and know the reduced density operator for one qubit is maximally mixed, are the two qubits maximally entangled?

I have a two-qubit closed system described by the Hamiltonian $H=A(\sigma_x\otimes\sigma_x+\sigma_y\otimes\sigma_y+\sigma_z\otimes\sigma_z)$ where $A$ is a coupling between the two qubits and $\...
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Is it true that $\mathrm{tr}(\rho^n)$ is monotonically decreasing as a function of $n$?

In the case of mixed states density matrix made out of orthogonal vectors this is clear since we have $$\sum_i p_i^n,$$ but what if we use non-orthogonal vectors or a continuous mixture?
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What is the one body density for $n$ fermions in $d$ dimensions?

What is the definition of the one body density $\rho$ in plain English, i.e. not in dirac notation: Is $\rho(\vec{r}) d \vec{r}$ the probability of observing a particle in a differential volume ...