# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

24. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...
50. ### DMRG. Density matrix renormalization group theory

Why density matrix renormalization group theory works only for 1D systems?