Weinberg: Quantum Mechanics Without State Vectors

[marcus]

Steven Weinberg has made what he calls a "modest proposal":

http://arxiv.org/abs/1405.3483
Quantum Mechanics Without State Vectors
Steven Weinberg
(Submitted on 14 May 2014)
It is proposed to give up the description of physical states in terms of ensembles of state vectors with various probabilities, relying instead solely on the density matrix as the description of reality. With this definition of a physical state, even in entangled states nothing that is done in one isolated system can instantaneously effect the physical state of a distant isolated system. This change in the description of physical states opens up a large variety of new ways that the density matrix may transform under various symmetries, different from the unitary transformations of ordinary quantum mechanics. Such new transformation properties have been explored before, but so far only for the symmetry of time translations into the future, treated as a semi-group. Here new transformation properties are studied for general symmetry transformations forming groups, rather than semi-groups. Arguments are given that such symmetries should act on the density matrix as in ordinary quantum mechanics, but loopholes are found for all of these arguments.
28 pages

He recently posted this followup:

http://arxiv.org/abs/1603.06008
What Happens in a Measurement?
Steven Weinberg
(Submitted on 18 Mar 2016)
It is assumed that in a measurement the system under study interacts with a macroscopic measuring apparatus, in such a way that the density matrix of the measured system evolves according to the Lindblad equation. Under an assumption of non-decreasing von Neumann entropy, conditions on the operators appearing in this equation are given that are necessary and sufficient for the late-time limit of the density matrix to take the form appropriate for a measurement. Where these conditions are satisfied, the Lindblad equation can be solved explicitly. The probabilities appearing in the late-time limit of this general solution are found to agree with the Born rule, and are independent of the details of the operators in the Lindblad equation.
12 pages

 

[marcus]

===excerpt from section 1, pages 2 and 3===
A MODEST PROPOSAL

Two unsatisfactory features of quantum mechanics have bothered physicists for decades. The first is the difficulty of dealing with measurement. The unitary deterministic evolution of the state vector in quantum mechanics cannot convert a definite initial state vector to an ensemble of eigenvectors of the measured quantity with various probabilities. Here we seem to be faced with nothing but bad choices. The Copenhagen interpretation[1] assumes a mysterious division between the microscopic world governed by quantum mechanics and a macroscopic world of apparatus and observers that obeys classical physics. If instead we take the wave function or state vector seriously as a description of reality, and suppose that it evolves unitarily according to the deterministic time-dependent Schrödinger equation, we are inevitably led to a many-worlds interpretation[2], in which all possible results of any measurement are realized. To avoid both the absurd dualism of the Copenhagen interpretation and the endless creation of inconceivably many branches of history of the many-worlds approach, some physicists adopt an instrumentalist position, giving up on any realistic interpretation of the wave function, and regarding it as only a source of predictions of probabilities, as in the decoherent histories approach[3].

The other problem with quantum mechanics arises from entanglement[4]. In an entangled state in ordinary quantum mechanics an intervention in the state vector affecting one part of a system can instantaneously affect the state vector describing a distant isolated part of the system. It is true that in ordinary quantum mechanics no measurement in one subsystem can reveal what measurement was done in a different isolated subsystem, but the susceptibility of the state vector to instantaneous change from a distance casts doubts on its physical significance.

Entanglement is much more of a problem in some modifications of quantum mechanics that are intended to resolve the problem of measurement, such as the general nonlinear stochastic evolution studied in [5]. It is difficult in these theories even to formulate what we mean by isolated subsystems, much less to prevent instantaneous communication between them[6,7]. Polchinski[7] has shown that unless nonlinearities are constrained to depend only on the density matrix, such modified versions of quantum mechanics even allow communication between the different worlds of the many-worlds description of quantum mechanics.

The problem of instantaneous communication between distant isolated systems has been nicely summarized in a theorem of Gisin[6]. It states that in a system consisting of two isolated subsystems I and II, with a prescribed density matrix ρI for subsystem I, it is always possible in a suitable entangled state of the two subsystems to make measurements on subsystem II that put subsystem I in any set of states ΨIr (not necessarily orthogonal) with probabilities Pr, provided only that r PrΛIr = ρI, where ΛIr is the projection operator on the state ΨIr.

Since any statement that a system is in an ensemble of states with definite probabilities can thus be changed instantaneously by a measurement at an arbitrary distance, keeping only the density matrix fixed, it seems reasonable to infer that such statements are meaningless, and that only the density matrix has meaning. That is, it seems worth considering yet another interpretation of quantum mechanics: The density matrix rather than the state vector or wave function is to be taken as a description of reality.

Taking the density matrix as the description of reality is very different from giving the same status to an ensemble of state vectors with various probabilities, because the density matrix contains much less information. If we know that a system is in anyone of a number of states Ψr, with probabilities Pr, then we know that the density matrix is ρ = r PrΛr, where Λr is the projection operator on state Ψr, but this does not work in reverse. As is well known, for a given density matrix ρ there are any number of ensembles of not necessarily orthogonal or even independent state vectors and their probabilities that give the same density matrix. (An exception is discussed in Section II.) The density matrix is of course a Hermitian operator on Hilbert space, a vector space. In speaking of “quantum mechanics without state vectors” I mean only that a statement that a system is in anyone of various state vectors with various probabilities is to be regarded as having no meaning, except for what it tells us about the density matrix.
...
...
If the density matrix is not to be defined in terms of ensembles of state vectors, then what is it? We may define it by postulating a physical interpretation: The average value A of any physical quantity represented by an Hermitian operator A is Tr(Aρ), which since it applies also to powers of A allows us to find from the density matrix the probability distribution for values of A. (These may be regarded as objective probabilities, independent of whether or not anything is actually being measured.) This postulate leads to all the properties of the density matrix that are usually derived from its interpretation in terms of an ensemble of states with various probabilities...
...
It may seem like a mere matter of language to say that it is the density matrix rather than an ensemble of state vectors with various probabilities that should be taken as the description of a physical system. Already many studies of the interpretation of quantum mechanics and of quantum information theory are based on the density matrix rather than the state vector, without needing a new interpretation of quantum mechanics. What difference does it make?

There is one big difference, that is our chief concern in this paper. Giving up the definition of the density matrix in terms of state vectors opens up a much larger variety of ways that the density matrix might respond to various symmetry transformations. ...
==endquote==

Taking an ensemble of state vectors as the description of reality, I believe he is saying, takes on the excess baggage of a lot of unnecessary (not even uniquely determined or physically real) information. This extra is responsible for quantum weirdnesses and paradoxes we are all familiar with. The density matrix is a LEANER description of reality which still contains the essential information--probabilities about the outcomes of measurements--but which leaves out the liability of extra info.

 

[marcus]

I'd like to hear some other views on this. I would say it's an initiative to mathematically REBUILD Quantum Mechanics . That is why the symmetries issue is so important to Weinberg. it is not just another proposal to re-INTERPRET the conventional state vector math in an attempt to wiggle out of the paradoxes that plague it.
He wants to rebuild QM on the basis of something (the density matrix) he thinks might turn out to be physically real. The enterprise may fail, some people here may already be able to point out serious flaws. But I think it is remarkably interesting and stands a chance of turning out right.

As I see it, he wants to say there is a density matrix state that, in contrast to the vector state, is real. The density matrix is the real state, the physically real description of the system regardless of who or what is observing, or where they are looking from, or what measurements are being contemplated.

The density matrix state contains the following information: for any measurement A, what the probabilities are of each possible result. See back in the previous post where it says:
"The average value A of any physical quantity represented by an Hermitian operator A is Tr(Aρ),..."

https://en.wikipedia.org/wiki/Steven_Weinberg
Weinberg was born in May 1933, he is 82.

 

[strangerep]

As I see it, he wants to say there is a density matrix state that, in contrast to the vector state, is real.

I'll have to read the papers for more carefully, but my initial reaction is to wonder why these ideas are perceived as new... [Edit: I see now -- he's interested in the possibility of new symmetries, as you said.]

Certainly, the density matrix (state operator) is more physically important, since it encapsulates all that can (in principle) be measured about the system, via (variations on) the $Tr(A\rho)$ functional. Also, the fact that the representation of a density matrix in terms of state vectors is never unique (cf. Ballentine sect 2.3) suggests that the latter contain extra unphysical degrees of freedom.

Also, the density matrix for composite systems "contains" the correlations among the subsystems (in a sense), hence is a more useful vehicle expressing relational QM ideas, and the view that correlations are more fundamental than correlata. (The usual QM/QFT entities can be recovered from the correlations via reconstruction theorems.)

Indeed, one of the paragraphs you quoted, i.e.,

Since any statement that a system is in an ensemble of states with definite probabilities can thus be changed instantaneously by a measurement at an arbitrary distance, keeping only the density matrix fixed, it seems reasonable to infer that such statements are meaningless, and that only the density matrix has meaning. That is, it seems worth considering yet another interpretation of quantum mechanics: The density matrix rather than the state vector or wave function is to be taken as a description of reality.

is extremely similar to the way Mermin expresses his "Ithaca" interpretation of QM. Yet Weinberg seems unaware of this (and doesn't cite Mermin, afaict).

(BTW, this thread should probably be moved to the quantum forum.)

 

[marcus]

I want to stress how deep this transformation is---it is a proposed rebuilding of QM into something that is BEYOND THE STANDARD quantum mechanics we are used to. A lot of puzzling stuff goes away. Correct me if I'm wrong, here's what I think:
The EPR paradox goes away.
The BH information loss puzzle goes away.
Firewalls.
The measurement problem.
The Many Worlds branching.
The weakening of full realism you get with Mermin or with RELATIONAL QM, where there is observer dependence---that goes away.
The problems related to unitary evolution and pure state vs mixed state. There aren't pure state vectors any more.

I don't see this rebuilding as philosophical. Like eg. an instrument to make predictions which you aren't supposed to believe in, just use to make bets with. It is not sophisticated reinterpretation of accepted math. To succeed Weinberg's proposal will require fundamentally new math results. That is why Weinberg is worried about the SYMMETRIES of the density matrix (as he defines it). If there turn out to be obviously unphysical new symmetries this will doom the enterprise. Because he wants to attribute full physical reality to the density matrix. And he wants to deny physical meaning to individual vector states and ensembles thereof.

Or? Show me what I'm missing.

It is not surprising that there is overlap between other people's research and the details of Weinberg's argument that ordinary state vector QM is wrong (no matter how you interpret it). The problems have been described over and over again. In fact he cites quite a lot. But the important thing is not this or that problem but the fact that he goes ahead boldly and says OK we've had it with this mare's nest, let's get a whole new theory to take its place.

 

[DrDu]

I wonder whether anybody gives a dime about the wavefunction as such today. On the other hand, each state (whether mixed or not), can be described by a vector in a Hilbert space via the GNS construction. So this is not the problem.

Weinberg seems interested in finding some generalisation of the unitary time evolution to arrive at some special case of the Lindblad equation.

 

[Demystifier]

The BH information loss puzzle goes away.

I think my own paper
http://arxiv.org/abs/1502.04324 [JCAP 04 (2015) 002]
would also be relevant in that context.

 

[marcus]

Thanks Demy! I'll post the abstract--it does look relevant:
http://arxiv.org/abs/1502.04324
Violation of unitarity by Hawking radiation does not violate energy-momentum conservation
H. Nikolic
(Submitted on 15 Feb 2015)
An argument by Banks, Susskind and Peskin (BSP), according to which violation of unitarity would violate either locality or energy-momentum conservation, is widely believed to be a strong argument against non-unitarity of Hawking radiation. We find that the whole BSP argument rests on the crucial assumption that the Hamiltonian is not highly degenerate, and point out that this assumption is not satisfied for systems with many degrees of freedom. Using Lindblad equation, we show that high degeneracy of the Hamiltonian allows local non-unitary evolution without violating energy-momentum conservation. Moreover, since energy-momentum is the source of gravity, we argue that energy-momentum is necessarily conserved for a large class of non-unitary systems with gravity. Finally, we explicitly calculate the Lindblad operators for non-unitary Hawking radiation and show that they conserve energy-momentum.
18 pages
http://inspirehep.net/record/1345006?ln=en (4 citations already)

 

[marcusl]

This sounds very exciting to me, since it offers a way our of the weirdness of entanglement and action at a distance. I must confess to being naive regarding quantum mechanics, however, having not used it since leaving grad school some decades ago. Accordingly, many of the subtleties mentioned above (like what the Lindblad equation is) are beyond me. Please forgive me, then, if the following question is somehow ridiculous:

I recall that E. T. Jaynes was a big fan of density matrices. Looking back at his second 1957 paper, he asserts that the density matrix does not, in fact, contain all relevant information for time-evolving systems but must be supplemented with information related to states
bayes.wustl.edu/etj/articles/theory.2.pdf
Does this play against Weinberg's attempt?

EDIT: corrected the link

 

[eloheim]

Sorry if this is a really obvious question. How can the density matrix for each part of two entangled states coordinate in order to produce the bell state measurement results if they are separate? Or are they both part of one larger matrix state? (And if so how is this different than the standard description?)

 

[A. Neumaier]

"The average value A of any physical quantity represented by an Hermitian operator A is Tr(Aρ),..."

I'd like to hear some other views on this. I would say it's an initiative to mathematically REBUILD Quantum Mechanics .

I rebuilt quantum mechanics along these lines in 2008; see my online book. Most relevant in the present context are the Chapters 1, 8, and 10 of the current version from 2011. A revised version of the book will be published next year.

See also the recent discussion here and my Thermal Interpretation FAQ.

 

[haushofer]

Your "online book"-link doesn't work.

 

[A. Neumaier]

Your "online book"-link doesn't work.

Corrected.

 

[vanhees71]

I also don't understand, what's new, because the state in quantum theory is described by statistical operators, i.e., positive semidefinite trace-class operators. Pure states are special cases and were never described by Hilbert-space vectors but rays in Hilbert space, and this can be equivalently formulated as the special case, where the statistical operator is a projection operator. Of course, it's always good to have clear and physical descriptions by Weinberg. I guess he got involved in this when he wrote his textbook and that's why he's publishing now papers on fundamental issues of quantum theory.

 

[A. Neumaier]

like what the Lindblad equation is

See https://en.wikipedia.org/wiki/Lindblad_equation
It is a dissipative version of the quantum Liouville equation $\dot\rho=i[\rho,H]/\hbar$. The latter describes unitary (conservative) evolution of the density operator; the former relates to the latter like the equations for a damped classical harmonic oscillator to its conservative version. It arises from the unitary dynamics of a larger system including the coupling to its environment by an approximation process similar to the derivation of friction in classical mechanics.

 

[bhobba]

I'll have to read the papers for more carefully, but my initial reaction is to wonder why these ideas are perceived as new.

That was my thought as well. I have always thought the density matrix was more fundamental.

Thanks
Bill

 

[naima]

Density matrices may be pure or mixed. Many people think that they are useless because the Schmidt decomposition allows us to compute every thing with pure states (vectors). As they only think in term of vectors, they say that the cosmological state of the universe is pure because there is nothing outside to purify it. This is no more valid when we cease to consider pure states as more fondamental.
In QM we may have to add probability amplitudes and to square them (pure states). We may have to add the probabilities (when the paths are known).
There are many other cases where 0 < fringe visibility < 1.
We have to use other inner composition laws for the density matrices.
Man'ko gives it them xxx.tau.ac.il/pdf/quant-ph/0207033v1.pdf

 

[A. Neumaier]

He recently posted this followup:
http://arxiv.org/abs/1603.06008
What Happens in a Measurement?

This paper contains nothing new. He assumes a reduced Lindblad dynamics for an open system and investigates conditions on the coefficients of the Lindblad equation that make it an acceptable description of a measurement process. (What he calls collapse is usually called decoherence, and has been frequently discussed in a Lindblad setting.)

 

[A. Neumaier]

http://arxiv.org/abs/1405.3483
Quantum Mechanics Without State Vectors

This paper indeed contains a new direction of thought, though what remains after the dust settled is known (but very little known) stuff.

The new thing is not so much to give reality (and priority) to the density operator, but to use this as a starting point for restricting the set of admissible observables (p.8 bottom) and density operators (p.19), rather than allowing all self-adjoint operators as observables and all trace 1 positive semidefinite Hermitian operators as density operators. In particular, Weinberg wants to exclude rank one operators in both cases, implying that there are no pure states and no tests for being in a pure state. It is this feature that would change the foundations. Thus he looks for a class of observables and density operators that would be suitable - by allowing symmetry operations more general that unitary conjugation as in the traditional framework but still respecting the relevant structure.

Weinberg's quest can be rephrased as the search for a suitable C^*-algebra of observables and a collection of nice states on it. One expects to get as symmetries nothing more than C^*-algebra automorphisms, which are a little more general than unitary (or antiunitary) transformations but only in the case of an infinite number of degrees of freedom. Indeed, in search of these new symmetries he shows that the most natural attempts (for finite-dimensional matrices) don't work. This means that whatever loopholes (his word!) there are none of it would be natural - which in my opinion leaves only an open door to awkward patchwork, comparing unfavorably with the elegance of the standard quantum setting. Thus I consider his attempt to get more general symmetry operations to be doomed to fail.

This leaves the standard (von Neumann) C^*-algebraic setting of quantum mechanics in terms of a C^*-algebra of observables and its states. From this perspective it is worth noting that results from algebraic quantum field theory (cf. Yngvason 2014) imply that all local algebras induced by a relativistic QFT on a double cone (generalizing a 2D diamond to 4D) are (Y., p.12) factors of type $III_1$ in von Neumann's classification refined by Connes. Picking such a double cone containing our present planetary system implies that we may assume the algebra of observables currently accessible to mankind to be such a factor of type $III_1$. Remarkably, such a C^*-algebra has no pure states (Y., p.14).

This means that for local algebras it is not meaningful to interpret statistical mixtures as “classical” probability distributions superimposed on pure states having a different “quantum mechanical” probability interpretation, as sometimes done in textbooks on non-relativistic quantum mechanics.

This may be the necessary change in the foundations that Weinberg was looking for. On the other hand (Y., p.18),

On the other hand, the framework of LQP does not per se resolve all “riddles” of quantum physics. Those who are puzzled by the violation of Bell’s inequalities in EPR type experiments will not necessarily by enlightened by learning that local algebras are type III. Moreover, the terminology has still an anthropocentric ring (“observables”, “operations”) as usual in Quantum Mechanics. This is disturbing since physics is concerned with more than designed experiments in laboratories. We use quantum (field) theories to understand processes in the interior of stars, in remote galaxies billions of years ago, or even the “quantum fluctuations” that are allegedly responsible for fine irregularities in the 3K background radiation. In none of these cases “observers” were/are around to “prepare states” or “reduce wave packets”! A fuller understanding of the emergence of macroscopic “effects” from the microscopic realm, without invoking “operations” or “observations”, and possibly a corresponding revision of the vocabulary of quantum physics is still called for.

My own proposal to resolve these issues (based on the reality of the density matrix of the universe) is my thermal interpretation discussed here.

 

[haushofer]

Corrected.

Offtopic: I see you wrote that book with Dennis Westra, which is an old (study) acquintance of mine. :P

 

[Agrippa]

See https://en.wikipedia.org/wiki/Lindblad_equation
It is a dissipative version of the quantum Liouville equation $\dot\rho=i[\rho,H]/\hbar$.

Why is the rhs of your equation positive? In your link it is negative.

 

[ShayanJ]

Why is the rhs of your equation positive? In your link it is negative.

$[\rho,H]=-[H,\rho]$

 

[vanhees71]

The von Neumann equation for the density operator in the Schrödinger picture is
$$\frac{1}{\mathrm{i}\hbar} [\hat{\rho},\hat{H}]+(\partial_t \hat{\rho})_{\text{expl.}}=0.$$

 

[Zafa Pi]

Taking an ensemble of state vectors as the description of reality, I believe he is saying, takes on the excess baggage of a lot of unnecessary (not even uniquely determined or physically real) information. This extra is responsible for quantum weirdnesses and paradoxes we are all familiar with. The density matrix is a LEANER description of reality which still contains the essential information--probabilities about the outcomes of measurements--but which leaves out the liability of extra info.

So by reformulating quantum theory the weirdness of the double slit experiment and the violation of Bell's inequality (= the negation of local realism) disappear?
That's pretty weird.

 

[marcus]

So by reformulating quantum theory the weirdness of the double slit experiment and the violation of Bell's inequality (= the negation of local realism) disappear?
That's pretty weird.

Maybe someone else would like to comment. That's how I read what is said in the abstract
"With this definition of a physical state, even in entangled states nothing that is done in one isolated system can instantaneously effect the physical state of a distant isolated system. This change in the description of physical states opens up a large variety of new ..."

 

[Zafa Pi]

Maybe someone else would like to comment. That's how I read what is said in the abstract
"With this definition of a physical state, even in entangled states nothing that is done in one isolated system can instantaneously effect the physical state of a distant isolated system. This change in the description of physical states opens up a large variety of new ..."

I think you're reading too much into that quoted passage. Run of the mill QM doesn't say that "one isolated system can instantaneously effect the physical state of a distant isolated system." That is just a bit of quantum flapdoodle [Gell-Mann]. QM merely says the results of Alice and Bob's measurements are correlated and those correlations will not disappear with a reformulated theory.
How those correlations take place is indeed a mystery, but no more so than how masses make gravitational attraction.

 

[Agrippa]

It would be great to better understand Weinberg's proposal. But I'm not used to density matrix formulation. I only have basic understanding of the state vector formalism he is criticizing. What is the physical significance of the extra terms in his Lindblad equation? First we have $L_n ρ(t) L_m^\dagger$. To me this looks similar to $O_{ij} = <A_i | \hat{O} | A_j>$ which just defines the matrix elements for operator $\hat{O}$ in a basis $|A_1>,...|A_N>$. So is $L_n ρ(t) L_m^\dagger$ doing nothing more than casting the density matrix of the system in the L basis? And then I wonder what the physical significance of subtracting $½(ρ(t) L_m^\dagger L_n + L_m^\dagger L_n ρ(t))$ is. Is the part that's added to the quantum Liouville equation doing nothing more than constructing a matrix out of the unspecified $h_{n,m}$ elements in the $L$ basis? Thanks!

 

[A. Neumaier]

What is the physical significance of the extra terms in his Lindblad equation?

pp.21-24 of my slides http://arnold-neumaier.at/ms/lightslides.pdf might be of help in interpreting the Lindblad equation in a concrete situation. In a microscopic derivation, all three terms arise together; it therefore does not make sense to interpret the three terms separately. The combination is the generator of a completely positive map - that's what counts.

 

[Agrippa]

pp.21-24 of my slides http://arnold-neumaier.at/ms/lightslides.pdf might be of help in interpreting the Lindblad equation in a concrete situation. In a microscopic derivation, all three terms arise together; it therefore does not make sense to interpret the three terms separately. The combination is completely positive - that's what counts.

Thanks for this. A concrete illustration of the Lindblad equation is certainly helpful and your slides are clearly written. But I'm still puzzled.

One prima facie difference between your equation and the general Lindblad equation is that the sum (the sigma) is not present in yours. For an N-dimensional system there is a sum of $N^2-1$ terms. Why is that missing in yours? Why isn't there such a sigma in front of each of your three additional terms (corresponding to a, b, and c)?

Your equation also appears to be missing the $h_{n,m}$ term, i.e. the term for the hermitian matrix elements - the constants that determine the dynamics. What you instead have outside each bracket is just one constant. For example outside the first set of brackets you have 0.002 x 138 - the cavity loss rate. Shouldn't the $h_{n,m}$ term be specified so that it's clear that the matrix elements are continuous functions of the relevant constant (such as cavity loss rate)?

In trying to make sense of your claim that the terms make no sense separately, but together function to guarantee "complete positivity", I stumbled across this. It helped to define $L_n ρ(t) L_m^\dagger$ as a test for whether ρ(t) is positive definite: ρ(t) is positive definite relative to operator $L$ if $L_n ρ(t) L_m^\dagger$ yields a positive number. Is the idea that ($L_n ρ(t) L_m^\dagger - ½(ρ(t) L_m^\dagger L_n + L_m^\dagger L_n ρ(t))$ just functions to ensure all eigenvalues are positive?

 

[A. Neumaier]

One prima facie difference between your equation and the general Lindblad equation is that the sum (the sigma) is not present in yours. For an N-dimensional system there is a sum of $N^2-1$ terms. Why is that missing in yours? Why isn't there such a sigma in front of each of your three additional terms (corresponding to a, b, and c)?

Your equation also appears to be missing the $h_{n,m}$ term, i.e. the term for the hermitian matrix elements - the constants that determine the dynamics. What you instead have outside each bracket is just one constant.

https://en.wikipedia.org/wiki/Lindblad_equation gives the most general form. The number $N$ of terms could be arbitrary but one can show that this many are enough. My example just has three nonzero diagonal terms and no nonzero off-diagonal term.

In the initial wikipedia form, only the complete expression generates a completely positive map; thus even the individual expressions whose coefficients are the $h_{n,m}$ don't mean anything. But there is an equivalent diagonal form, derived in the subsection https://en.wikipedia.org/wiki/Lindblad_equation#Diagonalization
which is of the same type as mine, just with more terms, and which can be interpreted physically.

I stumbled across this. It helped to define $L_n ρ(t) L_m^\dagger$ as a test for whether ρ(t) is positive definite

ρ(t) is positive definite iff $x^*ρ(t)x>0$ for all nonzero $x$. But this is unrelated to complete positivity.

Actually my original statement was slightly incorrect - I had meant ''generates a completely positive map'' not ''is a completely positive map''. (I corrected my statement.)

 

[A. Neumaier]

Picking such a double cone containing our present planetary system implies that we may assume the algebra of observables currently accessible to mankind to be such a factor of type III_1. Remarkably, such a C^*-algebra has no pure states

More information about this is in the discussion in a related thread, starting here.

 

[greypilgrim]

Sorry if this is a really obvious question. How can the density matrix for each part of two entangled states coordinate in order to produce the bell state measurement results if they are separate?

This question hasn't been answered yet, and it puzzles me as well. In the second paper, Weinberg says
"Whether in open systems in ordinary quantum mechanics or in closed systems in some modified version of quantum mechanics, in order to avoid instantaneous communication at a distance in entangled states, it is important to require that the density matrix at one time depends on the density matrix at any earlier time, but not otherwise on the state vector at the earlier time."

Since the density operators of both subsystems in a Bell experiment are completely mixed, how can local measurements (be it his suggested Lindblad evolution or any other kind) be correlated?

 

[kith]

The reduced density matrix contains all information which is relevant for experiments which are performed on the subsystem alone.

Weinberg stresses that if the outcomes of such a measurement on subsystem A depended on the reduced density matrix of subsystem B, FTL communication would be possible by altering the reduced density matrix of B.

What Weinberg's statement is silent about is whether the reduced density matrices contain all information about the combined system of A and B. And indeed, they don't contain information about how the outcomes of individual measurements on the subsystems are correlated.

The information about the correlations is only contained in the density matrix of the combined system and not in the reduced density matrices of the subsystems.

 

[maline]

So by reformulating quantum theory the weirdness of the double slit experiment and the violation of Bell's inequality (= the negation of local realism) disappear?
That's pretty weird.

Indeed, treating $\rho$ as "real" does nothing to remove the "weirdness". The crucial point is that the statement

With this definition of a physical state, even in entangled states nothing that is done in one isolated system can instantaneously effect the physical state of a distant isolated system

applies only to time-evolution of a given density matrix, but not to the process of the experimenter updating the density matrix in response to measurement results, a.k.a. "collapse", "reduction", etc. - our favorite bugbear.

In this context, the EPR paradox is expressed as follows:
Suppose Alice and Bob, at spacelike separation, share an entangled state described by $\rho$. Alice performs a measurement on her subsystem, views the results, and uses them to describe the post-measurement state with a new density matrix $\rho'$. Bob, on the other hand, although we will assume he knows what measurement Alice planned to perform, cannot know the measurement results. Therefore he must continue to use $\rho$, evolving it with time in accordance with the interactions that take place as part of Alice's experimental protocol. The content of No Signalling Property is that the reduced density matrix $\rho_B$, describing Bob's subsystem, is unaffected by Alice's activities at spacelike separation. But the same cannot be said of $\rho'_B$. In general, since Alice uses new observations to write $\rho'$, $\rho'_B$ will likely contain more information (lower entropy) than $\rho_B$. For the standard EPR case of a singlet pair of spin-1/2 particles, $\rho_B$ is the completely mixed state while $\rho'_B$ is the pure state with spin opposite to Alice's result.

As long as we think of density matrices as describing knowledge- basically the instrumentalist viewpoint- there is nothing strange about this: Alice has more knowledge and so she can write a "better" density matrix. But if density matrices are "real", then which one, $\rho$ or $\rho'$, should we consider to be the "true" state? Alice's new knowledge is certainly correct and "part of reality", so it seems her update must be called a "real change of the state"- good old collapse. And of course, this change is nonlocal. Alice's measurement has instantaneously generated new information constraining Bob's subsystem; information that was not part of the previous "state of reality".

The only way out is MWI, which Weinberg dislikes: $\rho$ indeed remains the true state of the whole system, now a "multiverse", while $\rho'$ is Alice's description of the particular "branch" she now finds herself in. "Probabilites" (whatever that means in MWI) for experiments Bob may perform are described by $\rho_B$, while $\rho'_B$ describes "probabilities" for the branches where those results are eventually compared with Alice's particular result.

All this is well known. I do not understand what Weinberg hopes to gain by assuming the density matrix is real. In particular, in approaches like GRW (objective stochastic collapse) which is the direction he seems to prefer, the Lindblad equation describes the evolution of $\rho$ only from the perspective of one who has not seen the measurement results, so it certainly should not be taken to represent "reality".