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Would be helpful to tie this in to density operator as discusssed in quantum logic. CSTAR 23:03, 19 May 2004 (UTC)
Any objections to moving this page back to "Density matrix" where it belongs? --
V79 16:35, 2005 Jun 15 (UTC)
The whole "C*-algebraic formulation of density states" section seems pretty unnecessary to me - right now, it's a lot of jargon that doesn't even make sense to a physicist. Also, a statement like "It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable," definitely needs a reference. I will delete the section unless someone objects. -- 130.126.230.244 17:20, 27 January 2006 (UTC)
"It is now generally accepted" again! This is not good writing style. Was there a vote that I missed
on that acceptance? ** However at last these hidden arguments against the "adjoint operator style" should be made clear.
Is it just me or does this article use “density matrix, ” for the “density operator, ”? —Preceding unsigned comment added by Laser Lars ( talk • contribs) 12:22, 10 September 2008 (UTC)
The operator is the actual mathematical object, where the matrix is one of its many possible representations. Infact, in a different basis, the same operator is represented by a different matrix. This is true for finite and infinite dimensions. So I suggest to change the name of the article to "Density operator", and have "Density matrix" redirected to it. Oakwood ( talk) 01:07, 17 October 2012 (UTC)
I think it's necesary to add to the article the dynamical equation of the density matrix: —The preceding unsigned comment was added by 201.232.162.150 ( talk) 04:28, August 23, 2007 (UTC)
I completely agree: the time evolution of a density matrix is, in general, more general that the von Neumann equation. And this time evolution simply “generates“ density matrices/operators (or mixed states) from pure states. Just adding “Lindblad equation“ to the further references list, is not sufficient, I think. Although I know, of course, that there are lots of discussions which is the “correct” (or “most general form”) of such an equation of motion. DieHenkels ( talk) 10:30, 8 December 2023 (UTC)
so what's the problem with self adjoint operators as observables? in the finite dim case, the C* algebraic and SA operator formulations are the same. if one is to completely abandon the SA operator formulation, how does one come up with the C*-algebra of observables in the first place? As stated in the article, the GNS lets you recover the Hilbert space, is this not the state space you start with? Mct mht 01:41, 22 May 2006 (UTC)
so it seems one needs to use the Heisenberg picture map, between observables, in general. it's probably a good idea to modify the article quantum operation a bit accordingly. Mct mht 02:26, 18 June 2006 (UTC)
Do we really want C* algebra in the wikipedia article. This seems quite technical and probably not of interest to many people. Maybe it needs its own page. — Preceding unsigned comment added by 109.153.172.124 ( talk) 10:12, 2 November 2014 (UTC)
CSTAR: ok, obviously they're not measures in the measure theoretic sense, but one can see the formal resemblence. and they are called quantum probability measures (was probably your contribution, if i have to guess) in their own right anyway. calling them just that makes the heuristic comment on mixed states as probability distribution on states (common in physics literature) more clear. Mct mht 16:57, 22 May 2006 (UTC)
the articles pure state and mixed state can easily be merged with this one. Mct mht 05:20, 22 May 2006 (UTC)
I've made some additions to the article consisting of expansion & clarifications of exisiting material and also linking measurement with entropy. See what you think. -- Michael C. Price talk 20:51, 23 June 2006 (UTC)
For von Neumann Entropy: I think Log should be to the base 2, not natural log?
See
Von_Neumann_entropy#Definition -
Mark Moriarty —Preceding
undated comment added
17:05, 28 April 2012 (UTC).
It states in the context that "Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state." So can anybody comment on how to produce a pure state? Reducing temperature to zero, or get Bosons under critical temperature, any other ways? — Preceding unsigned comment added by 173.20.49.9 ( talk) 16:54, 16 July 2011 (UTC)
Hi, surely there is some thing missing in the text and is bringing confusion to the reader. Confusion is arising because explanation started with the passing of vertical polarized through a circular polarizer as given below: "If we pass (|R\rangle +|L\rangle )/{\sqrt {2}} polarized light through a circular polarizer which allows either only |R\rangle polarized light, or only |L\rangle polarized light, intensity would be reduced by half in both cases. This may make it seem like half of the photons are in state |R\rangle and the other half in state |L\rangle ."
but, suddenly started asserting that photons are abosrbed by vertical linear polarizer as given below:
"But this is not correct: Both |R\rangle and |L\rangle photons are partly absorbed by a vertical linear polarizer, but the (|R\rangle +|L\rangle )/{\sqrt {2}} light will pass through that polarizer with no absorption whatsoever."
Kindly, address this issue. — Preceding unsigned comment added by Sharma yamijala ( talk • contribs) 11:18, 14 February 2014 (UTC)
Quite briefly, I don't believe it. Is there a citation? The assumption is that ensembles are in 1-1 correspondence with operators of the form $A U$ where $A$ is non-negative trace class of trace $1$ and $U$ unitary.
Why? Ensembles have the structure of a convex set. One might naturally associate to an ensemble to a probability measure supported on the convex set of density operators, but from there to the characterization given in the most recent edit, is in my view unjustified. -- CSTAR 04:56, 29 June 2006 (UTC)
In conclusion, then the non-uniqueness follows from the multiplicity of representations of the form
by probability measures μ supported on the compact convex set of states S. This can be described as non-uniqueness of convex representations. What does this have to do with non-uniquess of factorizations (except in very special cases?) -- CSTAR 00:45, 30 June 2006 (UTC)
BTW, in general the set of states ain't compact, no? Mct mht 03:14, 30 June 2006 (UTC)
Re:In all my comments, ensemble elements are pure states (vectors). i thought that was clear You thought that was clear? Um, it was not clear to me. Now with that caveat, the result you are claiming is equivalent to the following: Representations of the form
where the are selfadjoint projections (that are however not assumed to be pairwise orthogonal) are uniquely determined up to unitaries. That is if
then there is a unitary U such that
Frankly, I don't believe this is true. (If the projections are assumed pairwise orthogonal, of coure it's just the spectral theorem). This claim would imply that all probability measures on the compact convex set of states, supported on the extreme points, are "unitarily equivalent" provided they have the same center of mass. I would be surprised f that were true in anything but the abelian case. However, if you provide me with a reference, I'll believe it.-- CSTAR 03:49, 30 June 2006 (UTC)
CSTAR, i don't know, man, it's probably in most linear algebra books. It's pretty standard. For instance, Choi used this to show how the Kraus operators of a CP map are related by a unitary matrix. Again, the claim is simply the following: Let A = M M* be a positive semidefinite matrix. Then N satisfies A = N N* iff N = M U for some unitary U, where all matrices are square (If we drop the square assumption, then U need not be square, but it is isometric). Mct mht 04:53, 30 June 2006 (UTC)
This applies to couple other related objects as well. Besides Kraus operators, purifications of a given mixed state are related in a similar way.
Mct mht
05:31, 30 June 2006 (UTC)
CSTAR, i am done with this particular discussion. one can't get much more explicit than the explanations i've given. i am sorry you're not convinced. this is completely elementary and standard. Mct mht 02:05, 1 July 2006 (UTC)
First of all definition of an ensemble: this is an indexed family of pairs
with
and
a unit ket vector. The corresponding density matrix is the convex combination of rank one projections
As Nielsen and Chuang point out, it is actually more convenient to absorb the probability coefficient into the state by replacing thekets with "renormalized" kets:
Thus with this renormalized formulation, the corresponding density matrix is
Theorem. Two ensembles ψ, ψ' define the same density state iff there is a unitary matrix
such that
This is Theorem 2.6 of Nielsen and Chuang. -- CSTAR 04:43, 1 July 2006 (UTC)
Question: Is it implicitly assumed that the states in this definition are orthogonal (i.e. states in some orthonormal basis), as in the example with and polarization? If so, I think this should be stated explicitly. If not, it might be better to change the example to one with non-orthogonal states. 72.229.22.106 ( talk) 21:50, 23 October 2019 (UTC)Anonymous User
OK it may be standard, but here is the the explicit relationship:
By definition, a positive rank one operator on H is one of the form
Recall that an ensemble representing a density matrix is a sequence of positive rank one operators which add up to A.
Fix an orthonormal basis for H. The matrix of such an operator is
Conversely any operator whose matrix has that form is rank 1. Now suppose
Thus
Consider the sequence of rank one operators with matrices
Then clearly
This is an ensemble representing A. Conversely revcersing the argument, one sees that any ensemble representing A has this form. -- CSTAR 21:21, 1 July 2006 (UTC)
The page includes the usual method of obtaining the density matrix from spinors but does not show the less well known method of obtaining spinors from density matrices. I'll go ahead and add it in. If you haven't seen the method, see Julian Schwinger's "Quantum Kinematics and Dynamics" or www.DensityMatrix.com dead link . CarlAB 01:16, 6 October 2006 (UTC)
So far this article has ignored the idea of various orders of density matrices. Also, some of the particular expressions are in fact for reduced density operators where the true expressions for the reduced density *matrices* is not given.
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:47, 10 November 2007 (UTC)
The article claims a density matrix is non-negative, which the link defines as all the elements are greater than zero. This is clearly not true of a density matrix. Also shouldn't there be a condition on the density matrix that keeps the coherences (off-diagonals) smaller than the populations (on-diagonal elements). Otherwise the matrix isn't physical?-- J S Lundeen ( talk) 23:20, 18 January 2008 (UTC)
The article says that the density matrix was first developed by von Neumann, Landau and Bloch in 1927, but Bloch's articles only go as far back as 1945 on ISI web of knowledge, and I can't find anything by von Neumann. Can someone please provide the references for where the density matrix first developed. —Preceding unsigned comment added by Dr. Universe ( talk • contribs) 00:04, 26 September 2010 (UTC)
Mixed states redirects here, but I can't find any definition here of what a mixed state is, or an explanation of what a mixed state physically means. Could someone who understands it better than me either add one, or redirect mixed state to somewhere that explains it?-- Physics is all gnomes ( talk) 21:54, 26 December 2010 (UTC)
Btw.: a state is pure if and only if ! so how does this artificial distinction of a mixed state and a superposition make sense? If I calculate for the suggested "pure" state I arrive at ! to put it in another way: a pure state is a normalized, positive, weakly continuous functional on the kinematical algebra which is represented by the scalar product of a unique vector in Hilbert Space so how can we represent by a unique vector? —Preceding unsigned comment added by 130.75.25.219 ( talk) 14:00, 2 May 2011 (UTC)
The Manual of Style says that an article should have an accessible introduction. This lead, particularly the first sentence, failed this requirement to an almost comical degree. I have rearranged the text and streamlined it to improve readability without changing any of the content. However, someone should take this even further and replace some of the technical terminology in the last paragraph by a more accessible discussion. RockMagnetist ( talk) 16:31, 5 April 2012 (UTC)
I would suggest to include at least a reference to the Löwdin paper that explains density matrices and natural orbitals. It has helped me a lot in my PhD work. Here it is: http://prola.aps.org/abstract/PR/v97/i6/p1474_1 Jocasa ( talk) 11:42, 5 June 2012 (UTC)
Unfortunately it's not useful to most readers because it's behind a paywall. Longitude2 ( talk) 10:53, 17 February 2021 (UTC)
I don't see why this is necessary as density matrices apply just as well to pure states: If I have the superposition, , I can write
which in the basis gives me the matrix: which is fully valid and something we do in quantum optics all the time.
Thanks (sorry, couldnt be bothered to rewrite the page during exam period, might try during the summer) — Preceding unsigned comment added by 142.150.226.247 ( talk) 17:12, 26 March 2013 (UTC)
CONTRADICTION
The very first sentence of this topic contradicts the section on 'Pure and mixed states'. One says the density matirx describes mixed states; the other says it describes pure states or mixed. — Preceding
unsigned comment added by
Stephiefaulkner (
talk •
contribs)
08:50, 10 January 2017 (UTC)
It may be better to alter the first sentence to say that the density matrix describes a state prior to measurement, which in practice is always a mixed state, since measurement can never be guaranteed to exactly align with the singularly pure aspect (orientation) of the state prepared.
The formula which I have deleted from the article is the following:
Where
This formula is clearly incorrect. Consider the case of a pure state: . The Von Neumann entropy is zero for this case (you can regard this density matrix to have an eigenvalue of 1, with the rest zero). On the other hand looking at the right hand side we can obtain a non zero result by expanding (for example): Such that the Shannon entopy on the r.h.s is , and the Von Neumann ones on the r.h.s are still zero. -- Nomadbl ( talk) 15:19, 24 July 2018 (UTC)
There are all kinds of criteria under which a density matrix represents a pure state. The one currently used (), however, I believe is wrong, which is the reason I removed it. My argument is as follows: Let be two mutually orthogonal, normed vectors. The operator is an orthogonal projection on the plane spanned by . It cannot be written as a pure state (in the form ) but it is idempotent. Thus, is a counter-example. — Preceding unsigned comment added by 217.95.160.47 ( talk • contribs)
I'd like to add something like the following to the lead:
The density matrix for an experimental system becomes a diagonal matrix when a measurement happens, and it is even possible to see the matrix gradually becoming diagonal as the system naturally decoheres due to connection with the environment without a measurement happening.
I will give this a week for feedback before I edit the article, since I am not an expert in the field, and this might include a mistake or two. David Spector ( talk) 14:38, 29 November 2019 (UTC)
True. How about, "we know the matrix is gradually becoming diagonal"? We do know that there is a transition period between the existence of a quantum state and its replacement by a classical state (the diagonal matrix). Collapse doesn't happen in a femtosecond, right? David Spector ( talk) 19:07, 2 December 2019 (UTC)
Since a week has elapsed, I will edit the article as I indicated. But if an expert visits here, please feel free to improve what I've added. David Spector ( talk) 12:02, 7 December 2019 (UTC)
An article on a matrix, but there is no matrix to be found!!! Poor again. — Preceding unsigned comment added by Koitus~nlwiki ( talk • contribs)
The topic is "Density Matrix". So the very least this article needs to do is to answer basic questions such as:
-- Density of what?
-- What are the indices of the matrix?
-- What are the values associated with the coordinates in the matrix?
Taking a guess, I suspect "density" refers to "probability density", and is thus related to Probability density function. Is that right?
So are the values at the coordinates of the matrix samples of some large multi-dimensional probability density function, within which one can interpolate? Or are the values at each matrix coordinate actually independent probability density functions in their own right?
And what are these "extreme points" and entire "set of density matrices" that we already meet in the 3rd sentence? What is even a point in this context? Perhaps a particular coordinate of the matrix? But then what is extreme? Gwideman ( talk) 03:33, 24 February 2021 (UTC)
what i find most concerning about literature post 1970 (lecture notes and such) is the remarkably brief discussion of these important concepts with each lecturer varying in notation.
i consider myself fortunate that i was starting with Richard Bader's work, since his approach follows Lowdin and McWeeny and uses bra-ket sparingly. but when you step out of that comfortable spot, things look quite scary
my proposal is that, if we do propose a rewrite, we adopt a more mathematical notation. i don't mind either McWeeny (1955) [1] and Lowdin (1955) [2]. i found davidson [3] to be a great source that adopted McWeeny and Lowdin for bra-ket, but it's not a drop-in replacement for what we have right now. 198.53.159.44 ( talk) 23:56, 6 March 2021 (UTC)
References
{{
cite journal}}
: Cite journal requires |journal=
(
help)
For explaining 'projection', which is essential to the motivation, this article links only to projection operator (linear algebra). So does Measurement in quantum mechanics, and Quantum mechanics doesn't yet link anywhere. I believe readers need more explanation since it's so intimately related to measurement and so often used in QM. And to define, for example projective measurement.
Q: Is it advisable to make a new article Projection (quantum mechanics) or Projective measurement (quantum mechanics)? Seems like a linkable section of one of these QM articles would work well. Thoughts? -- MadeOfAtoms ( talk) 23:01, 19 May 2021 (UTC)
It's lacking an explanation here. The two references date back to 1963 and 1972, is this "now"? or the "now" more recent? would it be possible to give an example? — Preceding unsigned comment added by 89.3.212.183 ( talk) 01:23, 9 December 2022 (UTC)
The last sentence in the introduction feels quite non-pedagogic:
....physical system which is entangled with another, as its state can not be described by a pure state.
If the other system is a known state, ie together they make a pure entangled state, then per def they are a pure state. The intro makes it sound like this is not the case. Perhaps a better phrasing is something like..
...physical system which is entangled with another system, without also describing the other systems state.
Thanks 2001:9B1:26FD:8D00:81D:350A:F27A:DE70 ( talk) 12:44, 10 December 2022 (UTC)
This is also explained in the article it seems. The section "Example: light polarization" contains an explanation of this (see esp. the last sentence in that section). So, it seems the intro is in contradiction with the rest of the article? — Preceding unsigned comment added by 2001:9B1:26FD:8D00:81D:350A:F27A:DE70 ( talk) 13:08, 10 December 2022 (UTC)
Ok, i updated the text in the intro myself now. Why burdon others when i can do the labor myself. If you get upset about this or think i am wrong plz undo and also plz clarify. Thanks — Preceding unsigned comment added by 2001:9B1:26FD:8D00:81D:350A:F27A:DE70 ( talk) 13:35, 10 December 2022 (UTC)
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Would be helpful to tie this in to density operator as discusssed in quantum logic. CSTAR 23:03, 19 May 2004 (UTC)
Any objections to moving this page back to "Density matrix" where it belongs? --
V79 16:35, 2005 Jun 15 (UTC)
The whole "C*-algebraic formulation of density states" section seems pretty unnecessary to me - right now, it's a lot of jargon that doesn't even make sense to a physicist. Also, a statement like "It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable," definitely needs a reference. I will delete the section unless someone objects. -- 130.126.230.244 17:20, 27 January 2006 (UTC)
"It is now generally accepted" again! This is not good writing style. Was there a vote that I missed
on that acceptance? ** However at last these hidden arguments against the "adjoint operator style" should be made clear.
Is it just me or does this article use “density matrix, ” for the “density operator, ”? —Preceding unsigned comment added by Laser Lars ( talk • contribs) 12:22, 10 September 2008 (UTC)
The operator is the actual mathematical object, where the matrix is one of its many possible representations. Infact, in a different basis, the same operator is represented by a different matrix. This is true for finite and infinite dimensions. So I suggest to change the name of the article to "Density operator", and have "Density matrix" redirected to it. Oakwood ( talk) 01:07, 17 October 2012 (UTC)
I think it's necesary to add to the article the dynamical equation of the density matrix: —The preceding unsigned comment was added by 201.232.162.150 ( talk) 04:28, August 23, 2007 (UTC)
I completely agree: the time evolution of a density matrix is, in general, more general that the von Neumann equation. And this time evolution simply “generates“ density matrices/operators (or mixed states) from pure states. Just adding “Lindblad equation“ to the further references list, is not sufficient, I think. Although I know, of course, that there are lots of discussions which is the “correct” (or “most general form”) of such an equation of motion. DieHenkels ( talk) 10:30, 8 December 2023 (UTC)
so what's the problem with self adjoint operators as observables? in the finite dim case, the C* algebraic and SA operator formulations are the same. if one is to completely abandon the SA operator formulation, how does one come up with the C*-algebra of observables in the first place? As stated in the article, the GNS lets you recover the Hilbert space, is this not the state space you start with? Mct mht 01:41, 22 May 2006 (UTC)
so it seems one needs to use the Heisenberg picture map, between observables, in general. it's probably a good idea to modify the article quantum operation a bit accordingly. Mct mht 02:26, 18 June 2006 (UTC)
Do we really want C* algebra in the wikipedia article. This seems quite technical and probably not of interest to many people. Maybe it needs its own page. — Preceding unsigned comment added by 109.153.172.124 ( talk) 10:12, 2 November 2014 (UTC)
CSTAR: ok, obviously they're not measures in the measure theoretic sense, but one can see the formal resemblence. and they are called quantum probability measures (was probably your contribution, if i have to guess) in their own right anyway. calling them just that makes the heuristic comment on mixed states as probability distribution on states (common in physics literature) more clear. Mct mht 16:57, 22 May 2006 (UTC)
the articles pure state and mixed state can easily be merged with this one. Mct mht 05:20, 22 May 2006 (UTC)
I've made some additions to the article consisting of expansion & clarifications of exisiting material and also linking measurement with entropy. See what you think. -- Michael C. Price talk 20:51, 23 June 2006 (UTC)
For von Neumann Entropy: I think Log should be to the base 2, not natural log?
See
Von_Neumann_entropy#Definition -
Mark Moriarty —Preceding
undated comment added
17:05, 28 April 2012 (UTC).
It states in the context that "Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state." So can anybody comment on how to produce a pure state? Reducing temperature to zero, or get Bosons under critical temperature, any other ways? — Preceding unsigned comment added by 173.20.49.9 ( talk) 16:54, 16 July 2011 (UTC)
Hi, surely there is some thing missing in the text and is bringing confusion to the reader. Confusion is arising because explanation started with the passing of vertical polarized through a circular polarizer as given below: "If we pass (|R\rangle +|L\rangle )/{\sqrt {2}} polarized light through a circular polarizer which allows either only |R\rangle polarized light, or only |L\rangle polarized light, intensity would be reduced by half in both cases. This may make it seem like half of the photons are in state |R\rangle and the other half in state |L\rangle ."
but, suddenly started asserting that photons are abosrbed by vertical linear polarizer as given below:
"But this is not correct: Both |R\rangle and |L\rangle photons are partly absorbed by a vertical linear polarizer, but the (|R\rangle +|L\rangle )/{\sqrt {2}} light will pass through that polarizer with no absorption whatsoever."
Kindly, address this issue. — Preceding unsigned comment added by Sharma yamijala ( talk • contribs) 11:18, 14 February 2014 (UTC)
Quite briefly, I don't believe it. Is there a citation? The assumption is that ensembles are in 1-1 correspondence with operators of the form $A U$ where $A$ is non-negative trace class of trace $1$ and $U$ unitary.
Why? Ensembles have the structure of a convex set. One might naturally associate to an ensemble to a probability measure supported on the convex set of density operators, but from there to the characterization given in the most recent edit, is in my view unjustified. -- CSTAR 04:56, 29 June 2006 (UTC)
In conclusion, then the non-uniqueness follows from the multiplicity of representations of the form
by probability measures μ supported on the compact convex set of states S. This can be described as non-uniqueness of convex representations. What does this have to do with non-uniquess of factorizations (except in very special cases?) -- CSTAR 00:45, 30 June 2006 (UTC)
BTW, in general the set of states ain't compact, no? Mct mht 03:14, 30 June 2006 (UTC)
Re:In all my comments, ensemble elements are pure states (vectors). i thought that was clear You thought that was clear? Um, it was not clear to me. Now with that caveat, the result you are claiming is equivalent to the following: Representations of the form
where the are selfadjoint projections (that are however not assumed to be pairwise orthogonal) are uniquely determined up to unitaries. That is if
then there is a unitary U such that
Frankly, I don't believe this is true. (If the projections are assumed pairwise orthogonal, of coure it's just the spectral theorem). This claim would imply that all probability measures on the compact convex set of states, supported on the extreme points, are "unitarily equivalent" provided they have the same center of mass. I would be surprised f that were true in anything but the abelian case. However, if you provide me with a reference, I'll believe it.-- CSTAR 03:49, 30 June 2006 (UTC)
CSTAR, i don't know, man, it's probably in most linear algebra books. It's pretty standard. For instance, Choi used this to show how the Kraus operators of a CP map are related by a unitary matrix. Again, the claim is simply the following: Let A = M M* be a positive semidefinite matrix. Then N satisfies A = N N* iff N = M U for some unitary U, where all matrices are square (If we drop the square assumption, then U need not be square, but it is isometric). Mct mht 04:53, 30 June 2006 (UTC)
This applies to couple other related objects as well. Besides Kraus operators, purifications of a given mixed state are related in a similar way.
Mct mht
05:31, 30 June 2006 (UTC)
CSTAR, i am done with this particular discussion. one can't get much more explicit than the explanations i've given. i am sorry you're not convinced. this is completely elementary and standard. Mct mht 02:05, 1 July 2006 (UTC)
First of all definition of an ensemble: this is an indexed family of pairs
with
and
a unit ket vector. The corresponding density matrix is the convex combination of rank one projections
As Nielsen and Chuang point out, it is actually more convenient to absorb the probability coefficient into the state by replacing thekets with "renormalized" kets:
Thus with this renormalized formulation, the corresponding density matrix is
Theorem. Two ensembles ψ, ψ' define the same density state iff there is a unitary matrix
such that
This is Theorem 2.6 of Nielsen and Chuang. -- CSTAR 04:43, 1 July 2006 (UTC)
Question: Is it implicitly assumed that the states in this definition are orthogonal (i.e. states in some orthonormal basis), as in the example with and polarization? If so, I think this should be stated explicitly. If not, it might be better to change the example to one with non-orthogonal states. 72.229.22.106 ( talk) 21:50, 23 October 2019 (UTC)Anonymous User
OK it may be standard, but here is the the explicit relationship:
By definition, a positive rank one operator on H is one of the form
Recall that an ensemble representing a density matrix is a sequence of positive rank one operators which add up to A.
Fix an orthonormal basis for H. The matrix of such an operator is
Conversely any operator whose matrix has that form is rank 1. Now suppose
Thus
Consider the sequence of rank one operators with matrices
Then clearly
This is an ensemble representing A. Conversely revcersing the argument, one sees that any ensemble representing A has this form. -- CSTAR 21:21, 1 July 2006 (UTC)
The page includes the usual method of obtaining the density matrix from spinors but does not show the less well known method of obtaining spinors from density matrices. I'll go ahead and add it in. If you haven't seen the method, see Julian Schwinger's "Quantum Kinematics and Dynamics" or www.DensityMatrix.com dead link . CarlAB 01:16, 6 October 2006 (UTC)
So far this article has ignored the idea of various orders of density matrices. Also, some of the particular expressions are in fact for reduced density operators where the true expressions for the reduced density *matrices* is not given.
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:47, 10 November 2007 (UTC)
The article claims a density matrix is non-negative, which the link defines as all the elements are greater than zero. This is clearly not true of a density matrix. Also shouldn't there be a condition on the density matrix that keeps the coherences (off-diagonals) smaller than the populations (on-diagonal elements). Otherwise the matrix isn't physical?-- J S Lundeen ( talk) 23:20, 18 January 2008 (UTC)
The article says that the density matrix was first developed by von Neumann, Landau and Bloch in 1927, but Bloch's articles only go as far back as 1945 on ISI web of knowledge, and I can't find anything by von Neumann. Can someone please provide the references for where the density matrix first developed. —Preceding unsigned comment added by Dr. Universe ( talk • contribs) 00:04, 26 September 2010 (UTC)
Mixed states redirects here, but I can't find any definition here of what a mixed state is, or an explanation of what a mixed state physically means. Could someone who understands it better than me either add one, or redirect mixed state to somewhere that explains it?-- Physics is all gnomes ( talk) 21:54, 26 December 2010 (UTC)
Btw.: a state is pure if and only if ! so how does this artificial distinction of a mixed state and a superposition make sense? If I calculate for the suggested "pure" state I arrive at ! to put it in another way: a pure state is a normalized, positive, weakly continuous functional on the kinematical algebra which is represented by the scalar product of a unique vector in Hilbert Space so how can we represent by a unique vector? —Preceding unsigned comment added by 130.75.25.219 ( talk) 14:00, 2 May 2011 (UTC)
The Manual of Style says that an article should have an accessible introduction. This lead, particularly the first sentence, failed this requirement to an almost comical degree. I have rearranged the text and streamlined it to improve readability without changing any of the content. However, someone should take this even further and replace some of the technical terminology in the last paragraph by a more accessible discussion. RockMagnetist ( talk) 16:31, 5 April 2012 (UTC)
I would suggest to include at least a reference to the Löwdin paper that explains density matrices and natural orbitals. It has helped me a lot in my PhD work. Here it is: http://prola.aps.org/abstract/PR/v97/i6/p1474_1 Jocasa ( talk) 11:42, 5 June 2012 (UTC)
Unfortunately it's not useful to most readers because it's behind a paywall. Longitude2 ( talk) 10:53, 17 February 2021 (UTC)
I don't see why this is necessary as density matrices apply just as well to pure states: If I have the superposition, , I can write
which in the basis gives me the matrix: which is fully valid and something we do in quantum optics all the time.
Thanks (sorry, couldnt be bothered to rewrite the page during exam period, might try during the summer) — Preceding unsigned comment added by 142.150.226.247 ( talk) 17:12, 26 March 2013 (UTC)
CONTRADICTION
The very first sentence of this topic contradicts the section on 'Pure and mixed states'. One says the density matirx describes mixed states; the other says it describes pure states or mixed. — Preceding
unsigned comment added by
Stephiefaulkner (
talk •
contribs)
08:50, 10 January 2017 (UTC)
It may be better to alter the first sentence to say that the density matrix describes a state prior to measurement, which in practice is always a mixed state, since measurement can never be guaranteed to exactly align with the singularly pure aspect (orientation) of the state prepared.
The formula which I have deleted from the article is the following:
Where
This formula is clearly incorrect. Consider the case of a pure state: . The Von Neumann entropy is zero for this case (you can regard this density matrix to have an eigenvalue of 1, with the rest zero). On the other hand looking at the right hand side we can obtain a non zero result by expanding (for example): Such that the Shannon entopy on the r.h.s is , and the Von Neumann ones on the r.h.s are still zero. -- Nomadbl ( talk) 15:19, 24 July 2018 (UTC)
There are all kinds of criteria under which a density matrix represents a pure state. The one currently used (), however, I believe is wrong, which is the reason I removed it. My argument is as follows: Let be two mutually orthogonal, normed vectors. The operator is an orthogonal projection on the plane spanned by . It cannot be written as a pure state (in the form ) but it is idempotent. Thus, is a counter-example. — Preceding unsigned comment added by 217.95.160.47 ( talk • contribs)
I'd like to add something like the following to the lead:
The density matrix for an experimental system becomes a diagonal matrix when a measurement happens, and it is even possible to see the matrix gradually becoming diagonal as the system naturally decoheres due to connection with the environment without a measurement happening.
I will give this a week for feedback before I edit the article, since I am not an expert in the field, and this might include a mistake or two. David Spector ( talk) 14:38, 29 November 2019 (UTC)
True. How about, "we know the matrix is gradually becoming diagonal"? We do know that there is a transition period between the existence of a quantum state and its replacement by a classical state (the diagonal matrix). Collapse doesn't happen in a femtosecond, right? David Spector ( talk) 19:07, 2 December 2019 (UTC)
Since a week has elapsed, I will edit the article as I indicated. But if an expert visits here, please feel free to improve what I've added. David Spector ( talk) 12:02, 7 December 2019 (UTC)
An article on a matrix, but there is no matrix to be found!!! Poor again. — Preceding unsigned comment added by Koitus~nlwiki ( talk • contribs)
The topic is "Density Matrix". So the very least this article needs to do is to answer basic questions such as:
-- Density of what?
-- What are the indices of the matrix?
-- What are the values associated with the coordinates in the matrix?
Taking a guess, I suspect "density" refers to "probability density", and is thus related to Probability density function. Is that right?
So are the values at the coordinates of the matrix samples of some large multi-dimensional probability density function, within which one can interpolate? Or are the values at each matrix coordinate actually independent probability density functions in their own right?
And what are these "extreme points" and entire "set of density matrices" that we already meet in the 3rd sentence? What is even a point in this context? Perhaps a particular coordinate of the matrix? But then what is extreme? Gwideman ( talk) 03:33, 24 February 2021 (UTC)
what i find most concerning about literature post 1970 (lecture notes and such) is the remarkably brief discussion of these important concepts with each lecturer varying in notation.
i consider myself fortunate that i was starting with Richard Bader's work, since his approach follows Lowdin and McWeeny and uses bra-ket sparingly. but when you step out of that comfortable spot, things look quite scary
my proposal is that, if we do propose a rewrite, we adopt a more mathematical notation. i don't mind either McWeeny (1955) [1] and Lowdin (1955) [2]. i found davidson [3] to be a great source that adopted McWeeny and Lowdin for bra-ket, but it's not a drop-in replacement for what we have right now. 198.53.159.44 ( talk) 23:56, 6 March 2021 (UTC)
References
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For explaining 'projection', which is essential to the motivation, this article links only to projection operator (linear algebra). So does Measurement in quantum mechanics, and Quantum mechanics doesn't yet link anywhere. I believe readers need more explanation since it's so intimately related to measurement and so often used in QM. And to define, for example projective measurement.
Q: Is it advisable to make a new article Projection (quantum mechanics) or Projective measurement (quantum mechanics)? Seems like a linkable section of one of these QM articles would work well. Thoughts? -- MadeOfAtoms ( talk) 23:01, 19 May 2021 (UTC)
It's lacking an explanation here. The two references date back to 1963 and 1972, is this "now"? or the "now" more recent? would it be possible to give an example? — Preceding unsigned comment added by 89.3.212.183 ( talk) 01:23, 9 December 2022 (UTC)
The last sentence in the introduction feels quite non-pedagogic:
....physical system which is entangled with another, as its state can not be described by a pure state.
If the other system is a known state, ie together they make a pure entangled state, then per def they are a pure state. The intro makes it sound like this is not the case. Perhaps a better phrasing is something like..
...physical system which is entangled with another system, without also describing the other systems state.
Thanks 2001:9B1:26FD:8D00:81D:350A:F27A:DE70 ( talk) 12:44, 10 December 2022 (UTC)
This is also explained in the article it seems. The section "Example: light polarization" contains an explanation of this (see esp. the last sentence in that section). So, it seems the intro is in contradiction with the rest of the article? — Preceding unsigned comment added by 2001:9B1:26FD:8D00:81D:350A:F27A:DE70 ( talk) 13:08, 10 December 2022 (UTC)
Ok, i updated the text in the intro myself now. Why burdon others when i can do the labor myself. If you get upset about this or think i am wrong plz undo and also plz clarify. Thanks — Preceding unsigned comment added by 2001:9B1:26FD:8D00:81D:350A:F27A:DE70 ( talk) 13:35, 10 December 2022 (UTC)