What is Density operator: Definition and 41 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. Demystifier

    A Statistical ensemble interpretation done right

    Concerning statistical ensemble interpretation (SEI), a lot of confusion has been created and lot of nonsense has been said, especially in a recent thread on "missed opportunities in Bohmian mechanics". Even Ballentine himself said many things about it that do not always seem perfectly clear and...
  2. M

    I Derivation of two-electron density operator

    Hello, I am going over the derivation for two-electron density. I am having a hard time understanding how the second term in 2.11a seen below is derived. I know this term must eliminate the i=j products but can't seem to understand how. Thanks for the help.
  3. Dario56

    I Density Operators of Pure States

    Quantum states are most often described by the wavefunction ,##\Psi##. Variable ,##\Psi(x_1x_2\dots x_n) \Psi^*(x_1x_2\dots x_n)## defines probability density function of the system. Quantum states can also be described by the density matrices (operators). For a pure state, density matrix is...
  4. Killtech

    I The interpretation of probability

    I am looking for a way to compare the handling of probability in QT with how it's done in classic PT (probability theory) - and their interpretations. QT does have it's own formalism that works, so there isn't much motivation to bring it into a usual representation which makes it hard to find...
  5. Haorong Wu

    I Derivatives for a density operator

    Hi. Suppose I have a state ##\left | \psi (0)\right >=\sum_m C_m \left | m\right >## evolving as $$\left | \psi (0+dz)\right>=\left | \psi (0)\right >+dz \sum_iD_i\left | i\right >=\sum_m C_m \left | m\right >+dz \sum_iD_i\left | i\right >=\sum_m( C_m+dz D_m)\left |m\right >.$$ Then the density...
  6. stephen8686

    A Jaynes-Cummings Density Operator Evolution

    I am studying two level atoms interacting with fields in order to study Dicke Superradiance. From Loudon's book, the Optical Bloch Equations for a two level atom interacting with a field say (with rotating wave approx): $$\frac{d\rho_{22}}{dt}=- \frac{d\rho_{11}}{dt} = -\frac{1}{2}...
  7. Haorong Wu

    I Applying Heisenberg picture to density operator

    Suppose that a particle evolves from point A to point B. The state of the particle can be written as $$\rho=\sum \left | m\right >\rho_{mn}\left< n\right | .$$ Because the basis is evolving as the particle travels, I am considering applying the Heisenberg picture to the density operator. Let...
  8. P

    I Is the current density operator derived from fundamental considerations?

    Hello, I found this article. In equation (1) the authors wrote that the current operator is given by : ## - \frac{\delta H}{\delta A} ##. I just would like to know if this relation is a just definition or if it can be derived from more fundamentals considerations ? Thanks !
  9. T

    Density operator and natural log/trace

    I feel like I'm going around in circles trying to do something with the expression ## tr( \rho *log(\rho)) ##. I thought about a Taylor expansion, but I don't think there's a useful one here because of the logarithm. We learned the Jacobi's formula in class, but I don't think I want a derivative...
  10. Danny Boy

    A Von Neumann Entropy of a joint state

    Definition 1 The von Neumann entropy of a density matrix is given by $$S(\rho) := - Tr[\rho ln \rho] = H[\lambda (\rho)] $$ where ##H[\lambda (\rho)]## is the Shannon entropy of the set of probabilities ##\lambda (\rho)## (which are eigenvalues of the density operator ##\rho##). Definition 2 If...
  11. A

    A Qualitative Explanation of Density Operator

    Hey all! I am prepping myself for a quantum course next semester at the graduate level. I am currently reading through the Cohen-Tannoudji Quantum Mechanics textbook. I have reached a section on the density operator and am confused about the general concept of the operator. My confusion stems...
  12. 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...
  13. Sharkey4123

    A Reduced Density Operator and Entanglement

    I'm having a little bit of trouble getting my head around the idea of the reduced density operator being used to tell us about the entanglement of a state. I understand that if you take the reduced density operator of any of the Bell states, you get a reduced density operator proportional to...
  14. 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|##...
  15. 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|##
  16. Guilherme Vieira

    How to Calculate Probability using Density Operator?

    Hello, I'm trying to understand how to calculate de probability of finding a system in a specific eigenstate using the density operator. In the book of Balian, Haar, Gregg I've found a good definition of it being the expectation value of the projector Pr in the orientation of the eingenstate...
  17. 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
  18. munirah

    Reduced Density operator in matrix form

    Homework Statement I already read book of Quantum Computation and Quantum Information by Nielsen and Chuang according to reduced density operator and I already understand how to do the reduced density using Dirac notation, Ket Bra.Homework Equations [/B] My problem here I want to know the...
  19. 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...
  20. P

    Matrix form of Density Operator

    Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu
  21. L

    Unpacking the Physics Behind the Annihilator-Creator Operator Formula

    a In the formula above, on the left hand side, ρ(0) is a system's density operator in its initial state. a is the annihilator operator of the system, and a+ is the create operator of the system. ρss is the system's density operator in its steady state. But I don't understand why this formula...
  22. 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...
  23. Dale

    Is the Current Density Operator Related to Classical Magnetic Moment?

    Is there a current density operator or something equivalent? If so, how does it relate to other operators like momentum and angular momentum? Basically, the classical picture of a magnetic moment is a little loop of current, I would like to understand the quantum analog.
  24. A

    Particle density operator

    There is something I do not understand. One way to define the current density operator is through the particle density operator Ï(r). From the fundamental interpretation of the wavefunction we have: Ï(r)= lψ(r)l2 But my book takes this a step further by rewriting the equality above...
  25. sunrah

    Density operator of unknown system

    Homework Statement What is the density operator (statistical operator) of a system about which nothing is known? Homework Equations \hat{\rho} = \sum p_{i} |i\rangle\langle i| The Attempt at a Solution If nothing is known about a system we must assume something in order to make...
  26. P

    Density operator quantum mechanics

    I have the following situation: About the polarization of the photon, I introduce the basis: Horizontal polarization $|\leftrightarrow>=\binom{1}{0}$ Vertical polarization $|\updownarrow>=\binom{0}{1}$ The density matrix in this problem is: $$\rho =\frac{1}{2}\begin{pmatrix} 1+\xi...
  27. C

    Complex coefficents in density operator expansion?

    Hey, I recently had an exam where the quantum state were on the form |\psi\rangle = \frac{1}{\sqrt{2}} ( |+\rangle + i |-\rangle ) Here I formed the density operator for the pure state \rho(t) = |\psi\rangle \langle \psi| = \frac{1}{2} ( |+\rangle + i |-\rangle )( \langle +| - i \langle...
  28. S

    The definition of the density operator in Pathria

    Hello Everybody, I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator: \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, where N is the number of systems in the ensemble and the...
  29. X

    Trace and its square of mixed state density operator using integral

    Homework Statement I want to show that tr\left(\hat{\rho}_{mixed}\right)=1 tr\left(\hat{\rho}_{mixed}^{2}\right)<1 when \hat{\rho}_{mixed}=\frac{1}{2\pi}\int_{0}^{2\pi}d \alpha \hat{\rho}(\psi) Homework Equations tr\left(\psi\right)= \sum_{n}\langle n|\psi|n\rangle...
  30. Z

    Density operator for one part of a two-party state

    Homework Statement Let |\psi\rangle_{AB}=\sum_{i}\sum_{j}c_{ij}|\varphi_{i}\rangle_{A}\otimes|\psi_{j}\rangle_{B} be a normalized two party state, being \{|{\varphi_{i}}\rangle_{A}\} and \{|{\psi_{j}}\rangle_{B}\} basis of H_{A} and H_{B} respectively, with dimensions N and M. Find \rho_{A}...
  31. E

    Commutator of density operator with kinetic energy operator

    Hey guys, maybe you can help me with the following problem. I have to calculate the commutator relations in position representation: a) [V,ρ] b) [p,ρ] c) [p^2,ρ] Note that <q'|ρ|q>=ρ(q',q) is a matrix element of the density operator I already solved the first one. You just have to...
  32. J

    Entropy as a function of a density operator?

    I've got this problem about a harmonic oscillator with energy eigenstates |n>, which is prepared in the state |\psi> = \frac{1}{cosh r}\sum_n tanh^nr|n\rangle where r is any real number and the sum is from n=0 to infinity. I'm asked to calculate the entropy S(\rho) = -Tr(\rho ln\rho) if...
  33. L

    Understanding the Two Types of Density Operators in Quantum Mechanics

    I'm confused about the two density operators: \rho=\sum_{i}\delta(r-r_{i}) and \rho=\sum_{i}|\psi_{i}>\rho_{ii}<\psi_{i}| Is there anyone explaining this question to me? Thanks very much.
  34. G

    Construct Density Operator from Ensemble Average of Sx, Sy and Sz

    Homework Statement I have been given a problem. The density matrix can be constructed if the ensemble average of Sx, Sy and Sz are given. But I have no idea on how to construct the density matrix from these Si's. Any help is most welcome. Homework Equations Ensemble...
  35. D

    Microcanonical ensemble, density operator

    hi, usually the density operator for the microcanonical ensemble is given by \rho = \sum_n p_n|n><n| where |n> are energy eigenstates and p_n is the probability that our system is in this state. p_n = const. if the energy corresponding to |n> is in the energy inteval (E,E+∆E)...
  36. K

    Commutator of the density operator

    Hello all! I hope some of you are more proficient in juggling with bra-kets... I am wondering if/when the density operator commutes with other operators, especially with unitaries and observables. 1. My guess is, that it commutes with unitaries, but I am not sure if my thinking is correct...
  37. K

    Why Must the Density Operator Be Hermitian? Exploring Its Physical Significance

    Hi there, In all text of QM I have, they tells that the density operator is hermitian. But without considering the math, from the physics base, why density operator must be hermitian? What's the physical significane of the eigenvalue of density matrix? Thanks
  38. K

    Density Operator: Probability of Transition |0,e> to |1,g>

    Assume there is a two level system, two eigenstates are written as |\psi_1\rangle = \cos\theta |1, g\rangle + \sin\theta |0, e\rangle and |\psi_2\rangle = -\sin\theta |1, g\rangle + \cos\theta |0, e\rangle For the density operator of the system is written as \rho =...
  39. K

    Question about number operator and density operator

    In quantum harmonic oscillator, we define the so called number operator as \hat{N} = \hat{a}^\dagger\hat{a} Apply \hat{N} to the state with n number of particles, it gives \hat{N}|n\rangle = n |n\rangle so \langle n| \hat{N}|n\rangle = \langle n| n |n\rangle = n But in other...
  40. C

    Questions about the Density Operator Formulation of Quantum Mechanics

    so I've been reading about the density operator formulation of quantum mechanics and I have some questions what is the density operator analog of the schrodinger equation that determines the time evolution? and how do you perform a projection measurement on a quantum system in the density...
  41. P

    Density Operator for 2 Systems: Pure State

    We are given that 2 systems can only be found in the states |00\rangle, |01\rangle, |10\rangle, |11\rangle. We are also given that the density operator is \rho=\frac{1}{2}\left(|00\rangle \langle 00|+|11\rangle \langle 00|+|00\rangle \langle 11|+|11\rangle \langle 11|\right). a)Write the...
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