What is Density matrix: Definition and 124 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. Rayan

    I Compute density matrix

    If we for example have such a bipartite state: $$ | \phi > = \frac{1}{2} [ |0>|0> + |1>|0> + |0>|1> + |1>|1> ] $$ What is the easiest way to compute a density matrix of bipartite states? Should I just compute it as it is? i.e: $$ \rho = | \phi > < \phi | $$ Or should I convert to matrix form...
  2. T

    I Mixed State Density Matrix

    Hi all, I am having trouble visualizing the matrix representation of the mixed density matrix from the following post (specifically from the accepted answer): https://quantumcomputing.stackexchange.com/questions/21561/swap-test-and-density-matrix-distinguishability That is, for...
  3. T

    I Density Matrix of Multiple Qubits

    Hey all, I am having trouble relating probabilities with the density matrix of multiple qubits. Consider we have a system of 3 qubits: the first qubit is in the state ##\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{0}+\ket{1})## and the remaining 2 qubits are prepared in the state described by the...
  4. Demystifier

    A States & Observables: Are They Really Different?

    Usually states and observables are treated as fundamentally different entities in quantum theory. But are they really different? A state can always be represented by a density "matrix", which is really a hermitian (or self-adjoint) operator. Since observables are also hermitian (or self-adjoint)...
  5. richard_andy

    A Relation between the density matrix and the annihilation operator

    This question is related to equation (1),(3), and (4) in the [paper][1] [1]: https://arxiv.org/abs/2002.12252
  6. C

    I Ways of measuring open quantum systems

    At the heart of the theory of open quantum systems is the idea that the measurement statistics of many-body systems can be expressed in terms of a reduced density matrix, obtained by tracing over degrees of freedom that are irrelevant to the system of interest. In general, given a pure state...
  7. P

    A Purification of a Density Matrix

    I'm trying to find the purification of this density matrix $$\rho=\cos^2\theta \ket{0}\bra{0} + \frac{\sin^2\theta}{2} \left(\ket{1}\bra{1} + \ket{2}\bra{2} \right) $$ So I think the state (the purification) we're looking for is such Psi that $$ \ket{\Psi}\bra{\Psi}=\rho $$ But I'm not...
  8. yucheng

    I Partial trace and the reduced density matrix

    From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix: $$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...
  9. H

    I Interferences (with diagonal density matrix)

    suppose that elecrons are in a state described by a diagonal density matrix for their spin (we are not interested in their spatial matrix). They are used in the double slit experiment. will we get fringes. I ask the question because when Bob ans Alice share pairs of electrons (the total spin of...
  10. Haorong Wu

    I Improper density matrix with negative eigenvalues

    Hi, there. I am working with a model, in which the dimension of the Hilbert space is infinite. But Since only several states are directly coupled to the initial state and the coupling strength are weak, then I only consider a subspace spanned by these states. The calculation shows that the...
  11. jk22

    I Question about density matrix

    With ##\rho=\sum_i p_i|\Psi_i\rangle\langle\Psi_i|##If the ##p_i=|\langle\Psi|\lambda_i\rangle|^2## are taken as joint probabilities given by quantum mechanics for the singlet state in EPRB then this cannot represent a statistical mix (classical) of those states because of Bell's theorem ?
  12. dRic2

    I Minimize grand potential functional for density matrix

    I'd like to show that, by minimizing this functional $$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$ I get the well known expression $$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$ I'm...
  13. F

    A Relation between Dirac's equation density matrix and current with spin

    After computind dirac 1D equation time dependant for a free particle particle I get 2 matrixs. From both,them I extract: 1) the probablity matrix P =ps1 * ps1 + psi2 *psi2 2) the current matrix J = np.conj(psi1)*psi2+np.conj(psi2)*psi1 I think that current is related to electricity, and...
  14. C

    A Off-diagonal elements in a density matrix

    Hello everyone, I have a math / physics question that has been with me for a while. I would be grateful if someone could help me. Given a density matrix, what is the minimum value a sum of some of its off-diagonal elements can assume (or the most negative value)? Remark: if one collect an...
  15. A

    Density Matrix of Unpolarized Light

    I have unfortunately no attempt for 31.i. I don't know how to start the problem. 31.ii is attached. Can someone give me a tip on how to start 31.i? I also have troubles solving 29.iii. I have attached that as well.
  16. Diracobama2181

    I Possible simple Density Matrix Question

    If I am given that the density matrix of an incoming beam of spin 1 particles of the form $$p_o=(1/3)[|1><1|+|0><0|+|-1><-1|]$$, aand I needed to find the fraction of particles that would be found with a spin x component of zero, how would I go about solving this problem? My hunch is that I...
  17. Jamister

    I What is really that density matrix in QM?

    Summary: What are the basic assumptions of QM about the density matrix? The subject of density matrix in quantum mechanics is very unclear to me. In the books I read (for example Sakurai),they don't tell what are the basic assumptions and how you derive from them the results of the density...
  18. L

    Density matrix of an ammonia molecule

    In ##t = 0##, we have ##\rho (0) = | + \rangle \langle + |##. The time evolution of the density matrix is given by ##\rho(t) = e^{-i\hat{H}t} \rho (0) e^{i\hat{H}t}## (I am considering ##\hbar = 1##). I can write the state ##| + \rangle ## as a linear combination of the eigenstates of the...
  19. Morbidly_Green

    Expressing the density matrix in matrix form

    Homework Statement Given the above lambda system, is it wrong to say that the density matrix is of the form ## \rho = c_1|1> + c_2|2> + c_3|3> ## ? Hence when written in matrix form (basis of ##|i>##), ## \rho ## is a diagonal matrix who's elements are the ##c_i##s?
  20. J

    I Density Matrix Quantum Mechanics Help - Hi, I'm Confirming!

    Hi, I am wanting to confirm my understanding of the density matrix in quantum mechanics. Is it the wave function co-efficients squared - in other words the wave amplitudes squared which in turn are the probabilities and then these turn out to be placed into a matrix form with the squared wave...
  21. N

    I What is the meaning of coherent states of mean photon number

    I am studying Quantum Cryptography and I am quite new in Quantum area. I have read an article and I found this confusing statement: My questions: 1. The three stage protocol implementing multiphoton. What is the meaning of coherent states of mean photon number? 2. How to describe the quantum...
  22. AwesomeTrains

    Density matrix for a mixed neutron beam

    Homework Statement A beam of neutrons (moving along the z-direction) consists of an incoherent superposition of two beams that were initially all polarized along the x- and y-direction, respectively. Using the Pauli spin matrices: \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\...
  23. D

    I Indices of a Density Matrix

    I am reading Leonard Susskind's Theoretical Minimum book on Quantum Mechanics. Excercise 7.4 is as follows: Calculate the density matrix for ##|\Psi\rangle = \alpha|u\rangle + \beta|d\rangle##. Answer: $$ \psi(u) = \alpha, \quad \psi^*(u) = \alpha^* \\ \psi(d) = \beta, \quad \psi^*(d) =...
  24. Tspirit

    Quantum Textbook for density matrix and trace?

    I want a QM textbook which introduces detail knowledge of density matrix and trace (i.e. the average), who can recommend one for me? Thank you.
  25. G

    Expectation values as a phase space average of Wigner functions

    Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...
  26. L

    Partial traces of density operators in the tensor product

    Homework Statement Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...
  27. G

    Relation between the matrix elements of the density matrix

    Hi. I must prove that, in general, the following relation is valid for the elements of a density matrix \rho_{ii}\rho_{jj} \geq |\rho_{ij}|^{2}. I did it for a 2x2 matrix. The density matrix is given by \rho = \left[ \begin{array}{cc} \rho_{11} & \rho_{12} \\ \rho^{\ast}_{12} & \rho_{22}...
  28. G

    I Decomposing a density matrix of a mixed ensemble

    I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...
  29. P

    Density Operator to Matrix Form

    Homework Statement Write the density operator $$\rho=\frac{1}{3}|u><u|+\frac{2}{3}|v><v|+\frac{\sqrt{2}}{3}(|u><v|+|v><u|, \quad where <u|v>=0$$ In matrix form Homework Equations $$\rho=\sum_i p_i |\psi><\psi|$$ The Attempt at a Solution [/B] The two first factors ##\frac{1}{3}|u><u|##...
  30. V

    I Density matrix on a diagonal by blocks Hamiltonian.

    If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?
  31. M

    I Qubit mixed state density matrix coordinates on a Bloch ball

    What are the coordinates on the 3D Bloch ball of a qubit's mixed state of the form: ##\rho=p_{00}|0\rangle \langle 0|+p_{01}|0\rangle \langle 1|+p_{10}|1\rangle \langle 0|+p_{11}|1\rangle \langle 1|##
  32. stevendaryl

    I On uniqueness of density matrix description as mixed state

    If you have a density matrix \rho, there is a basis |\psi_j\rangle such that \rho is diagonal in that basis. What are the conditions on \rho such that the basis that diagonalizes it is unique? It's easy enough to work out the answer in the simplest case, of a two-dimensional basis: Then \rho...
  33. ShayanJ

    A Entanglement and density matrix in QFTs

    I'm reading this paper. But I haven't read anything on how to calculate the density operator in a QFT or how to calculate its trace. Now I can't follow this part of the paper. Can anyone clarify? Thanks
  34. Dishsoap

    Density matrix of spin 1 system

    Homework Statement Consider an ensemble of spin 1 systems (a mixed state made of the spin 1 system). The density matrix is now a 3x3 matrix. How many independent parameters are needed to characterize the density matrix? What must we know in addition to Sx, Sy and Sz to characterize the mixed...
  35. munirah

    From density matrix, how can I know what state it belongs to

    Homework Statement Given a density matrix of three qubit pure state, how can I know after do some transformation, this state belong to what class?. Class I mean here, either separable state, biseparable, GHZ state or W state? I mean here what is the indicator to me classify it? It is the...
  36. L

    Density matrix spin half, Pauli vector

    A nice discussion of the density operator for a qubit can be found here: http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf
  37. munirah

    I How reduced density matrix obtained from the matrix.

    Can any expert help me in explaining how this example below get the reduced density matrix from the density matrix in bipartite system. $$\rho =\frac{1}{4}\begin{pmatrix} 1 & 1 & cos(\frac{\alpha}{2})-sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{2})+sin(\frac{\alpha}{2}) \\ 1 & 1 &...
  38. D

    A Expectation values and trace over the environment

    I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I think is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would...
  39. entropy1

    I Expectation value in terms of density matrix

    It says in Susskind's TM: ##\langle L \rangle = Tr \; \rho L = \sum_{a,a'}L_{a',a} \rho_{a,a'}## with ##a## the index of a basisvector, ##L## an observable and ##\rho## a density matrix. Is this correct? What about the trace in the third part of this equation?
  40. L

    Statistical physics. Density matrix

    Homework Statement A system is subject to Hamiltonian ##\hat{H}=-\gamma B_z \hat{S}_z##. Write down the density matrix.[/B]Homework Equations For canonical ensemble ##\hat{\rho}=\frac{1}{Tr(e^{-\beta \hat{H}})}e^{-\beta \hat{H}}## In general ##\rho=\sum_m |\psi_m\rangle \langle \psi_m|## The...
  41. entropy1

    I Density matrices, pure states and mixed states

    I got (very) confused about the concept of states, pure states and mixed states. Is it correct that a linear combination of pure states is another pure state? Can pure (and mixed) states only be expressed in density matrices? Is a pure state expressed in a single density matrix, whereas mixed...
  42. J

    Density Matrix and State Alpha

    There is something that I don't quite understand or want clarification. See John Wheeler article "100 years of the quantum" http://arxiv.org/pdf/quant-ph/0101077v1.pdf refer to page 6 with parts of the quotes read "so if we could measure whether the card was in the alpha or beta-states, we...
  43. S

    Decoherence in the long time limit of density matrix element

    For a state |\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle , the density matrix elements in the energy basis are \rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar} How is it that in the long time limit, this reduces to \rho_{ab}(t) \approx |c_a|^2 \delta_{ab} ? Is there some...
  44. F

    Mean values & density matrix

    Homework Statement A system's state of spin 1/2 is represented at t=0 by C*exp[-a2(p-p0)2]*{{1,0},{0,1}} where the density matrix is represented in the base of eighenvalues of Sz and the spatial vector is represented in the continuum base of statesPx, Py, Pz. Find <X>, <Px> and <ΔX>, <ΔPx>...
  45. T

    Second Quantization Density Matrix

    Homework Statement Homework Equations and attempt at solution I think I got the ground state, which can be expressed as |\Psi \rangle = \prod_{k}^{N}\hat{a}_{k}^{\dagger} |0 \rangle . Then for the density matrix I used: \langle...
  46. M

    Sakurai Question regarding density matrix

    Homework Statement Sakurai Modern Quantum Mechanics Revised Edition. Page 81. density matrix p = 3/4 [1 0; 0 0] + 1/4 [1/2 1/2; 1/2 1/2]. We leave it as an exercise to the reader the task of showing this ensemble can be decomposed in ways other than 3.4.24Homework Equations 3.4.24 w( sz +...
  47. J

    Reduced Density Matrix Entropy in 1D Spin Chain

    Good afternoon all, I'm investigating typical values of entropy for a subsystem of a 1D (non-interacting) spin chain. Most of the problem is essentially solved I've shown that a typical pure state of the entire chain is close (trace norm) to the state ##\Omega_S## when reduced. \Omega_S =...
  48. A

    How to calculate density matrix for the GHZ state

    The GHZ state is: |\psi> = \frac{|000> + |111>}{\sqrt2} To calculate density matrix we go from: GHZ = \frac{1}{2}(|000> + |111>)(<000| + <111|) GHZ = \frac{1}{2}( |000><000| + |111><111| + |111><000| + |000><111|) To: GHZ = 1/2[ \left( \begin{array}{cc} 1 & 0 & 0 & 0 & 0 &...
  49. A

    Operators change form for density matrix equations?

    Imagine applying an operator to a wave-function: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||} Where ## \psi _t(x_1, x_2, ..., x_n) ## is initial system state vector, denominator is normalization factor, and Ln(x) is a...
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