What is Density matrix: Definition and 125 Discussions

In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.
Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information.

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  1. A

    How Do Eigenvalues and Eigenvectors Determine a Quantum Density Operator?

    Hi, How do eigenvalues and eigenvectors relate to the density operator. Given the eigenvalues of a matrix, can they help to find the density operator ? I have seen the formula within http://www.quantiki.org/wiki/Density_matrix but given a matrix how do I work it out to get the density operator...
  2. H

    Coherence in density matrix formalism

    Hello, In the density matrix formalism I have read in numerous places that coherence is identified with the off-diagonal components of the density matrix. The motivation for this that is usually given is that if a state interacts with the environment in such a way that the basis state...
  3. K

    Density Matrix for Spin 1/2 particle in a magnetic field

    Hi everyone! I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part. Before starting I...
  4. R

    Derive the motion equation of density matrix

    Homework Statement density matrix : ρ(t)=0.5+0.5*a(t)⋅σ (a is a 3 dimensional vector and σ is paul victor) H=-μ*σ⋅B (B is a three dimensional magnetic field ) and also assume that in t=0 , ρ(0)=0.5+0.5*a(0)⋅σ whats is the motion equation of a(t)? Homework Equations whats is the motion equation...
  5. S

    Microcanonical ensemble density matrix

    Ref: R.K Pathria Statistical mechanics (third edition sec 5.2A) First it is argued that the density matrix for microcanonical will be diagonal with all diagonal elements equal in the energy representation. Then it is said that this general form should remain the same in all representations. i.e...
  6. S

    Microcanonical ensemble density matrix

    Ref: R.K Pathria Statistical mechanics (third edition sec 5.2A) First it is argued that the density matrix for microcanonical will be diagonal with all diagonal elements equal in the energy representation. Then it is said that this general form should remain the same in all representations. i.e...
  7. M

    Density matrix in the canonical ensemble

    Homework Statement We have a quantum rotor in two dimensions with a Hamiltonian given by \hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2} . Write an expression for the density matrix \rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle Homework Equations...
  8. V

    Superposition in the density matrix formalism

    Suppose I have a two level system with the states labeled ##|0\rangle## and ##|1\rangle##. In this basis, these correspond to density matrices: ## \rho_0 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad \rho_1 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} ## I can create a coherent...
  9. J

    Single particle Density Matrix meaning

    Hey guys! In an n-electron system, The second order reduced DM is defined as \Gamma (x_{1},x_{2}) = \frac{N(N-1)}{2}\int{\psi(x_{1},x_{2}...,x_{n})\psi^{*}(x_{1},x_{2}...,x_{n})}dx_{3}...dx_{n} It can be intepreted as the probability of finding two electrons at...
  10. A

    Density matrix formalism and Poincaré invariance

    The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space ##\mathcal{H}##. Composition is defined through the tensor product and reduction through partial trace. Operations on the system are...
  11. fluidistic

    Density matrix, change of basis, I don't understand the basics

    Homework Statement Hello people, I am trying to understand a problem statement as well as the density operator, but I still don't get it, desperation is making me posting here. The problem comes as The problem then continues with other questions but I'm having troubles with the very first one...
  12. C

    How BEC being described by the single-particle density matrix?

    Hello everybody, this is my first time being here. I am a beginner learning some introductions on Bose-Einstein Condensation (BEC) on my own. Often times in the literature (say, [1], [2] (p.409) ) it comes the one-body(single-particle) density matrix, as...
  13. V

    Expected Value Partial Trance Density Matrix

    Hey I am currently studying Quantum Mechanics and I have difficulty grasping a concept. I don't understand the following step in the derivation: \langle X_{A} \rangle=tr\left[\left(X_{A}\otimes I_{B}\right)\rho_{AB}\right] =tr_{A}\left[X_{A} tr_{B}\left[\rho_{AB}\right]\right] Thanks
  14. L

    Solution to Density Matrix Pure State Problem

    Homework Statement Find condition for which ##\hat{\rho}## will be pure state density operator? ##\hat{\rho} = \begin{bmatrix} 1+a_1 & a_2 \\[0.3em] a_2^* & 1-a_1 \end{bmatrix}## Homework Equations In case of pure state...
  15. S

    Exploring the Possibility of Deducing Quantum States from a Density Matrix

    Is it not possible to deduce quantum states from a density matrix?
  16. N

    QM: Density matrix evolution

    Homework Statement Hi The density matrix evolves as \dot \rho = -\frac{i}{\hbar}[H,\rho] but is this equation written in the Schrödinger or Heisenberg picture? I'm not entirely sure how to figure this out. In my book it just mentions the equation, not how it is derived (which may have given...
  17. V

    Quantum teleportation and the density matrix

    I'm re-reading some course notes on quantum teleportation, and something isn't making sense. In the description my instructor gave, we used the Bell state ##|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)## for the entangled pair. So, suppose the state we want to teleport is...
  18. B

    Time evolution of density matrix

    Homework Statement Hi there. just working on a problem from sakurai's modern quantum mechanics. it is: A) Prove that the time evolution of the density operator ρ (in the Schrodinger picture) is given by ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0}) B) Suppose that we have a pure ensemble at...
  19. M

    Density matrix in Momentum Space

    I have an one-body density matrix in a Sine wave basis set (Thus psi = psi*). Unfortunately, these are not the natural orbitals (I have correlated particles), so I have off-diagonal elements. I believe I know how to extract the charge density from this density matrix \rho(x;x') = \sum_{ij}...
  20. P

    Density matrix for bell states

    Hi I have three states (I believe bell states) and want to find the density matrix, am I right in thinking: 1) \frac{|00> + |11>}{\sqrt{2}} \rightarrow \rho = \left( \begin{array}{cc} \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} \\ \end{array} \right) (because it is pure) 2)...
  21. A

    Density matrix off-diagonal elements

    The possible values of the diagonal elements of a density matrix are restricted by the condition \mathrm{Tr}~\rho = 1 . Are there any restrictions on the possible values of off-diagonal elements, apart from the obvious \mathrm{Re}~\rho_{nm} = \mathrm{Re}~\rho_{mn}, \mathrm{Im}~\rho_{nm} = -...
  22. B

    Change of basis of density matrix

    I have a density matrix in one basis and need to change it to another. I know the eigenvectors and eigenvalues of the basis I want to change to. How do I do this? Any help really appreciated- thanks!
  23. C

    Partial trace of the density matrix

    Hi, I am trying to work out the atomic inversion of the Jaynes cummings model using the density matrix. At the moment i have a 2x2 matrix having used the Von neumann equation (technically in Wigner function in x and y). Each of my matrix elements are 1st order pde's describing the...
  24. C

    Partial trace of density matrix

    I am unsure how to (mathematically) do the partial trace of a density matrix so that I can find the expectation value of an observable. I am working on a model similar to the Jaynes cummings model. My density matrix is of the form; \rho = [\rho_{11}, \rho_{12}, \rho_{21}, \rho_{22}]...
  25. C

    Problem with Hermicity of Density matrix

    Hi, I am trying to solve a modified Jayne's cummings model using the Von Neumann equation and Wigner function but am having a problem with Density matrix hermicity; I am trying to solve in Schrodinger picture. I have my system Hamiltonian as; H_{0} = \frac{1}{2}\hbar \Omega \sigma_{z}...
  26. J

    Charge Density Matrix: Calculating & Evaluating for N Electrons

    Firstly, I have been able to find almost nothing on this kind of question in textbooks or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could...
  27. J

    Problem with density matrix

    How would you define density matrix for an ensemble of identical harmonic oscillators in thermal equilibrium? For example, consider N atoms in a crystalline lattice. I would like to find density matrix to calculate the average dipole moment of the ensemble and also its standard deviation...
  28. C

    Partial trace of a density matrix?

    Hi, I'm working on a modified version of the Jayne's Cummings model and am a little confussed. I have: -Taken modified version of JCM Hamiltonian in Schrodinger picture. -Used Von Neumann equation to get evolution of density matrix -Converted to Wigner function. I want to run...
  29. C

    Decoherence and the Density Matrix

    Hi all, I've been reading the seminal Zurek papers on decoherence but there is one (crucial) point on which I am confused. I understand the mathematical demonstrations that over very short timescales the superpositions of states represented as off-diagonal terms in the density matrix can be...
  30. F

    Density matrix elements, momentum basis, second quantization

    Hello everyone, I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise. Specifically, I'm having a hard time understanding the matrix elements of the density matrix...
  31. L

    Non-radiative density matrix EOM

    Dear all, Could anyone please explain to me for a 3 level syste, coupled with 2 lasers, what is the equation describing the disspative term of the equation of motion of the density matrix? I am looking for a non-radiative decay, namely the overall population should conserved. I have found...
  32. Z

    Negative off-diagonal elements in density matrix?

    I have a quick question. I've been trying to search for an answer, but I'm probably looking in the wrong places. Is it valid to have negative off-diagonal elements in a density matrix? Thanks!
  33. S

    A question on reduced density matrix

    I have worked out a problem on reduced density matrix, but my final expression looks kind of weird to me. I would appreciate someone can point out where I have gone wrong. Homework Statement The problem goes like this: Consider a two spin 1/2 particles. Their Hilbert space is spanned bu...
  34. J

    How Do You Calculate the Density Matrix for a Diatomic Molecule?

    Firstly, I have been able to find almost nothing on this kind of question in textbooks or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could...
  35. maverick280857

    Trace of higher powers of Density Matrix

    Hi, The Quantum Liouville Equation is \dot{\rho} = \frac{i}{\hbar}[\rho, H] where the dot denotes the partial derivative with respect to time t. We take \hbar = 1 hereafter for convenience. Tr(\dot{\rho}) = 0 Consider Tr(\rho^2) Differentiating with respect to time...
  36. jfy4

    Density matrix and averages

    I'm really excited to get this as a homework problem. I have wanted to feel good about this formalism is quantum mechanics for a while now but my own stupidity has been getting in the way... With this homework problem hopefully I can move on to a new level. Homework Statement The most...
  37. J

    Conditions for a density matrix; constructing a density matrix

    Homework Statement What are the conditions so that the matrix \hat{p} describes the density operator of a pure state? Homework Equations http://img846.imageshack.us/img846/2835/densitymatrix.png p=\sum p_{j}|\psi_{j}><\psi_{j} The Attempt at a Solution I know that tr(\rho)=1 for...
  38. V

    Quantum mechanics: density matrix purification

    Homework Statement Given a matrix M(a) = (a -(1/4)i ; (1/4)i a) (semicolon separates rows) a) Determine a so that M(a) is a density matrix. b) Show that the system is in a mixed state. c) Purify M(a) The Attempt at a Solution a) from conditions for a density matrices...
  39. M

    Can a Qubit's Mixed State Density Operator Be Expressed Using Pauli Matrices?

    1. Show that an arbitrary density operator for a mixed state qubit may be written as 2. \rho = \frac{I+r^i\sigma_i}{2}, where ||r||<1 (Nielsen and Chuang pg 105) 3. So my attempt was as follows Given that a \rho is hermitian it may be written as a linear combination of the pauli...
  40. E

    Density matrix and von Neumann entropy - why does basis matter?

    Density matrix and von Neumann entropy -- why does basis matter? I'm very confused by why I'm unable to correctly compute the von Neumann entropy S = - \mathrm{Tr}(\rho \log_2{\rho}) for the pure state | \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2 Now, clearly the simplest...
  41. N

    Density Matrix: Theorem & Normality Conditions

    I have a question regarding the slide: http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture3.pdf On page 18-21 it gives the proof of the theorem that | \psi_i^{~} \rangle and |\phi_{i}^{~}\rangle generate the same density matrix iff |\psi_{i}^{~}\rangle = \sum_{j} u_{ij}...
  42. P

    Is there a guide to using the density matrix formalism in quantum mechanics?

    Hello, I am looking for a guide to quantum mechanics and the density matrix formalism which uses the Einstein summation convention. Does such a guide exist?
  43. D

    Reduced density matrix and entanglement

    Homework Statement I have in my textbook (QM by Auletta) the example of the polarization-entangled state of a pair of photons given by: |\psi\rangle_{12}=\frac{1}{\sqrt{2}}\left(|h\rangle_1\otimes|v\rangle_2+|v\rangle_1\otimes|h\rangle_2\right) The density matrix associated to such a...
  44. T

    Density Matrix (pure state) Property

    Hello. I need some help to prove the first property of the density matrix for a pure state. According to this property, the density matrix is definite positive (or semi-definite positive). I've been trying to prove it mathematically, but I can't. I need to prove that |a|^2 x |c|^2 +...
  45. N

    What Conditions Must a Matrix Meet to Be a Density Matrix?

    What are the conditions for some matrix to be a density matrix ? I know of these conditions: 1.) \rho=\rho^{2} 2.) Tr(\rho)=1 (for pure state) Is this all ?
  46. S

    Notation question for density matrix

    This is probably my misunderstanding of the notation... The definition of a density matrix is in the attached file. (Sorry, the latex editor is not rendering properly when I preview my post). This definition is a sum over only one index 'j', which will invariably lead to a diagonal matrix...
  47. P

    Time dependent perturbation theory for density matrix

    Does anyone kown how to apply time dependent perturbation theory to densities matricies (I'm interested in first order)? Thanks.
  48. B

    Eigenvalues of a reduced density matrix

    My lecturer keeps telling me that if a density matrix describes a pure state then it must contain only one non-zero eigenvalue which is equal to one. However I can't see how this is true, particularly as I have seen a matrix \rho_A = \begin{pmatrix} 1/2 & - 1/2 \\ -1/2 & 1/2 \\ \end{pmatrix} for...
  49. B

    Proof of trace of density matrix in pure/mixed states

    Can someone help me prove that tr(\rho^2) \leq 1 ? Using that \rho = \sum_i p_i | \psi_i \rangle \langle \psi_i | \rho^2 = \sum_i p_i^2 | \psi_i \rangle \langle \psi_i | tr(\rho^2) = \sum_{i, j} p_i^2 \langle j | \psi_i \rangle \langle \psi_i | j \rangle Where do I go from here? Thanks guys.
  50. D

    Mixed State QM: Unpacking the Non-Diagonal Density Matrix

    what do the non-diagonal components of a density matrix tell us about the mixed state we are in?
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