What is the QM description of a macroscopic event?

[Stephen Tashi]

TL;DR Summary

In QM, is there an agreed upon mathematical interpretation of macroscopic events?

In classical probability theory, a probability space has a set of "point" or "outcomes" which may be multidimensional vectors. "Events" are sets of these points. By analogy, in QM, it seems a macroscopic event would defined (in principle and mathematically) by as a set of possible measurement outcomes. For example, "The cat is alive" would include many different outcomes since whether the cat is alive or not doesn't depend only on whether one atom of the cat is here or there.

Is it correct to think that macroscopic events can (in principle) be defined this way? - correct in the sense that our sensation that a macroscopic event has definitely happened corresponds to one vector of measurement outcomes being realized from a set of such outcomes.

 

[EPR]

In terms of the mathematics involved, macroscopic events are treated as probabilities. With large degrees of freedom, the probabilities tend to average out toward classicality.

 

[anuttarasammyak]

Hi. In a definition of measurement in QT, e.g.,

a. A measurement is an interaction with the quantum system that transforms a state with multiple possible outcomes into a “collapsed” state that now has only one outcome: the measured outcome.
- Key Concepts for Future QIS Learners qis-learners.research.illinois.edu

points of micro or macro is not contained. What is your intention to make distinction ?

 

[Stephen Tashi]

In terms of the mathematics involved, macroscopic events are treated as probabilities.

I think of an event as something that has a probability rather than something that is a probability.

 

[Morbert]

Summary:: In QM, is there an agreed upon mathematical interpretation of macroscopic events?

In classical probability theory, a probability space has a set of "point" or "outcomes" which may be multidimensional vectors. "Events" are sets of these points. By analogy, in QM, it seems a macroscopic event would defined (in principle and mathematically) by as a set of possible measurement outcomes.

The standard formalism does not distinguish between macroscopic properties and microscopic properties. Both "the spin-x of the electron is up" and "the cat is alive" are represented by projectors. The projector for the macroscopic property will of course project onto a much larger subspace, but there is no demarcation of microscopic and macroscopic.

For example, "The cat is alive" would include many different outcomes since whether the cat is alive or not doesn't depend only on whether one atom of the cat is here or there.

The subspace corresponding to the macroscopic property could be fine-grained into subspaces of smaller microscopic properties yes, but you have to be careful that your fine-graining stays consistent. In CM this is guaranteed. In QM complementarity limits the fine-graining you can do.

Is it correct to think that macroscopic events can (in principle) be defined this way? - correct in the sense that our sensation that a macroscopic event has definitely happened corresponds to one vector of measurement outcomes being realized from a set of such outcomes.

Standard QM formalism does not assert in principle that an event has definitely happened due to reversibility*. At best a physicist can identify practical irreversibility and intersubjective outcomes (outcomes all observers will agree upon)

*Fundamental irreversibly can be introduced by relativity, and I believe Froehlich's EHT formalism also introduces it.

 

[stevendaryl]

The standard formalism does not distinguish between macroscopic properties and microscopic properties. Both "the spin-x of the electron is up" and "the cat is alive" are represented by projectors.

The standard formalism is ambiguous, though. It states something along the lines of

When a measurement is performed, the result is an eigenvalue of the operator corresponding to the observable being measured, with probabilities given by blah, blah, blah.

There are at least two fuzzy concepts there: (1) When is a measurement performed? (2) What does "the result of the measurement" mean? For the quantum formalism to make testable predictions, we have to have at least a rough answer to those two questions. I guess there is a third question: (3) How do we know what observable is being measured by the measurement?

If you look at what people actually do when they perform a measurement, and try to formalize it, it seems something like the following:

You have an observable/operator $\hat{O}$ with eigenvalues $\lambda_1, \lambda_2, ...$. To measure this observable, you build an apparatus with states $S_1, S_2, ...$ (plus maybe a neutral state corresponding to "measurement not successful") such that:

• If the observable is initially in a state with eigenvalue $\lambda_n$, then a successful measurement will result in the apparatus in state $S_n$

For it to count as a measurement, states $S_n$ and $S_m$ must be macroscopically distinguishable.

Of course, this is a simplified description. In real experiments, there might be an enormous amount of data processing required before you can figure out what the measurement result was. But the simplest example is the Stern Gerlach experiment: If the electron has spin-up, then a black spot is made on the left photographic plate, and if the electron has spin-down, then a black spot is made on the right photographic plate.

 

[stevendaryl]

Hi. In a definition of measurement in QT, e.g.,

a. A measurement is an interaction with the quantum system that transforms a state with multiple possible outcomes into a “collapsed” state that now has only one outcome: the measured outcome.
- Key Concepts for Future QIS Learners qis-learners.research.illinois.edu

points of micro or macro is not contained. What is your intention to make distinction ?

The macro/micro distinction and the collapsed/uncollapsed distinction are very closely related, in that we never see evidence of collapse in microscopic interactions and we never see evidence of superpositions of macroscopically distinguishable states.

 

[stevendaryl]

The macro/micro distinction and the collapsed/uncollapsed distinction are very closely related, in that we never see evidence of collapse in microscopic interactions and we never see evidence of superpositions of macroscopically distinguishable states.

Mathematically, the distinction between a mixture, "The system is either in state $A$ or state $B$, but we don't know which" and "The system is in a superposition of state $A$ and state $B$" is a matter of interference effects: superpositions can have interference effects, but a mixture cannot. For practical matters, if $A$ and $B$ are macroscopically distinguishable, then interference effects are unobservable. So we can consistently treat the system as being in a "collapsed" state---either $A$ or $B$ (although we may not know which without further investigation).

 

[anuttarasammyak]

Hi.

I just wonder word meaning of macro-micro, classical-quantum.

Say quantum means interference, we observe photon interference in macro scale as in laser experiment: quantum and macro. Electron interference is observed by electron microscope: quantum and micro, or macro in Davidson Germer experiment because it was done before the invention of electron microscope.

Say quantum means opposite to classical, photon interference is classical because it was well analyzed by Maxwell equation of classical physics (and macro?) , electron interference is quantum micro because de Broglie stated it in 20th century. photon particle feature is quantum ( and micro?) because it cannot be derived from classical Maxwell equation. electron particle feature is classical ( and macro??)

We should be in good accordance in these word meanings to avoid unnecessary confusion.

 

[Stephen Tashi]

If you look at what people actually do when they perform a measurement, and try to formalize it, it seems something like the following:

You have an observable/operator $\hat{O}$ with eigenvalues $\lambda_1, \lambda_2, ...$. To measure this observable, you build an apparatus with states $S_1, S_2, ...$ (plus maybe a neutral state corresponding to "measurement not successful") such that:

• If the observable is initially in a state with eigenvalue $\lambda_n$, then a successful measurement will result in the apparatus in state $S_n$

Relative to that description, a realistic observation of macroscopic physical system like "The cat is alive" by a person seems to involve

1) An observation that produces an incomplete description of an eigenvector (an outcome) with respect to its components (e.g. Not all atoms of the cat are observed.)

2) An observation that produces an incomplete description of an eigenvector due to the resolution of the measurements. (e.g. some observed values may be statistics such as mean values of a collection of components of the vector. The individual values of each component may not be observed).

Assume that the "The cat is alive" is defined (theoretically) by a set of possible eigenvectors. If we assume there exists a measurement technique that is incomplete (in the sense of 1) or 2) ) but which can nevertheless determine the macroscopic property "The cat is alive", what do we assume happens when such an incomplete measurement is made?

Possibilities:
1) The incomplete measurement causes a collapse of all components of the eigenvector, even though we do not observe all these values. (e.g. The cat changes from a superposition to a mixture.)

2) Not all components of the vector collapse, but those that don't must have their values restricted to the subset that defines the macroscopic outcome. (e.g. We determine the cat is alive, but some some aspects of the cat remain in a superposition of substates. )

...we never see evidence of superpositions of macroscopically distinguishable states.

That could be explained by possibility 1). To explain it in terms of possibility 2) seems to require showing that incomplete measurements cannot detect interference.

 

[atyy]

In classical probability theory, a probability space has a set of "point" or "outcomes" which may be multidimensional vectors. "Events" are sets of these points. By analogy, in QM, it seems a macroscopic event would defined (in principle and mathematically) by as a set of possible measurement outcomes. For example, "The cat is alive" would include many different outcomes since whether the cat is alive or not doesn't depend only on whether one atom of the cat is here or there.

Is it correct to think that macroscopic events can (in principle) be defined this way? - correct in the sense that our sensation that a macroscopic event has definitely happened corresponds to one vector of measurement outcomes being realized from a set of such outcomes.

Yes and no. A macroscopic event is something that may "really happen" and to which a probability can be assigned in the classical sense. Both quantum and classical state spaces are convex, but the classical state space is a simplex, and the quantum state space is not.

https://www.quantiki.org/wiki/mixed-states
"The key features of quantum mechanics, as opposed to classical probability, can be discussed in terms of the convex structure of their state spaces, the prototypes being the Bloch sphere in the quantum case and a simplex in the classical case. The latter are characterized by the property that any two convex decompositions of the same state admit a common refinement, so that ultimately, every state has a unique finest convex decomposition into pure states. In contrast, points in the interior of the Bloch sphere have many different decompositions into surface points."

https://www.researchgate.net/publication/266435541_Geometry_of_Quantum_States_An_Introduction_to_Quantum_Entanglement

(p40) " From a purely mathematical point of view a probability distribution is simply a measure on a sample space, constrained so that the total measure is one. Whatever the point of view one takes on this, the space of states will turn into a convex set when we allow probabilistic mixtures of its pure states. "

(p205) "This is the Bloch ball discussed in Section 5.2. There is no particular difficulty in understanding a ball as a convex set. Physically however there is much to think about, because we now have two different ways of adding two pure states together. We can form a complex superposition—another pure state—and we can form a statistical mixture—a mixed state. In the other direction, any given point in the interior can be obtained as a mixture of pure statesin many different ways. We are confronted with an issue that does not arise in classical statistics at all: any mixed state can be expressed as a mixture of pure states in many different ways, indeed in as many ways as a point in a ball can be thought of as the ‘centre of mass’ of a mass distribution on the surface of the ball. Physically it is a basic tenet of quantum mechanics that there does not exist an operational procedure to distinguish different ensembles of pure states if they yield the same density matrix—otherwise quantum correlations between separated systems could be used for instantaneous signalling [426].

Quantum mechanics is a significant generalization of classical probability theory. When N = 2 there are two possible outcomes of any measurement described by a Hermitian operator, or put in another way the sample space belonging to a given observable consists of two points. They correspond to two orthogonal pure states, placed antipodally on the surface of the Bloch ball. Each pair of antipodal points on the surface corresponds to a new sample space coexisting with the original."

 

[EPR]

I think of an event as something that has a probability rather than something that is a probability.

It may look like something extended in space and time that has definite properties but QT treats it differently. These objects that look classical are not compatible with QM and are treated as a peculiar, special case. This gives rise to a multitude of attempts at interpretation of this special case. They may be compatible with Bohmian Mechanics though.

QT assigns probabilities of observing an event. Schrodinger famously said of the situation with the probability interpretation - "I don't like it and I am sorry I had anything to do with it". He wanted to show how ridiculous the implications were and devised the famous Cat thought experiment.

 

[Stephen Tashi]

The standard formalism does not distinguish between macroscopic properties and microscopic properties. Both "the spin-x of the electron is up" and "the cat is alive" are represented by projectors.

Yes and no. A macroscopic event is something that may "really happen" and to which a probability can be assigned in the classical sense.

Is it?

As I said above in post #10, it seems to me that observing a macroscopic event is realistically modeled as an incomplete observation performed on a system where a more complete observation is theoretically possible. In discussing cats, one may begin by declaring observations in one's QM model of a cat will have only two outcomes , Cat is alive, Cat is dead. However, is such a simple model necessarily consistent with a higher resolution model where observations on the system produce more detailed results? - and taking into account that an observation performed in the simple model must now be viewed as an incomplete observation performed in the higher resolution model?

 

[atyy]

Is it?

As I said above in post #10, it seems to me that observing a macroscopic event is realistically modeled as an incomplete observation performed on a system where a more complete observation is theoretically possible. In discussing cats, one may begin by declaring observations in one's QM model of a cat will have only two outcomes , Cat is alive, Cat is dead. However, is such a simple model necessarily consistent with a higher resolution model where observations on the system produce more detailed results? - and taking into account that an observation performed in the simple model must now be viewed as an incomplete observation performed in the higher resolution model?

No. The cat is irrelevant. The essential features can be seen by considering measurements on a spin 1/2 system.

 

[Stephen Tashi]

No. The cat is irrelevant. You can consider measurements on a spin 1/2 system.

Just to make sure I understand, are you answering "No" to the question:

However, is such a simple model necessarily consistent with a higher resolution model where observations on the system produce more detailed results? - and taking into account that an observation performed in the simple model must now be viewed as an incomplete observation performed in the higher resolution model?

 

[atyy]

Just to make sure I understand, are you answering "No" to the question:

Yes, I don't think what you wrote is relevant at all, since most of the weirdness (except entanglement) can be obtained with a single spin 1/2 system, and almost all the weirdness (including entanglement) can be obtained with a system of two spin 1/2 particles.

Every experimental result is a macroscopic outcome.

In spin 1/2, if you choose to measure whether it is spin up or down (±z), then the possible experimental results are spin up or down (±z).

But if you choose to measure spin ±x then you get ±x as possible experimental results.

And if you choose to measure spin ±y then you get ±y as possible experimental results.

The states associated with ±x experimental results are superpositions of the states associated with ±z experimental results.

Now the Schroedinger cat problem just replaces ±z with dead or alive. Then the states associated with ±x or ±y are superpositions of dead and alive.

 

[Stephen Tashi]

Now the Schroedinger cat problem just replaces ±z with dead or alive. Then the states associated with ±x or ±y are superpositions of dead and alive.

Ok, but what I'm asking about isn't the Schroedinger cat scenario.

Yes, I don't think what you wrote is relevant at all

It isn't intended to be relevant to questions raised by Schroedinger's cat.

 

[atyy]

Ok, but what I'm asking about isn't the Schroedinger cat scenario.

The cat and the incomplete observations are simply not relevant, since we have macroscopic results even with a single spin 1/2.

 

[Stephen Tashi]

The cat and the incomplete observations are simply not relevant, since we have macroscopic results even with a single spin 1/2.

As far as I can see, the 1/2 spin system does not explicitly postulate that it is a consequence of a model that has greater resolution - nor that the measurement of spin corresponds to an incomplete observation made on outcomes in the higher resolution model.

Are you saying that it is self-evident that some higher resolution model underlies the 1/2 spin system?

 

[EPR]

There is no such thing as higher resolution observation, if I understand your post correctly. It is not necessary for all/100%/ of atoms of the cat to be observed directly to collapse them. Practically, an observation of some of the atoms of the cat, collapses all of the atoms of the cat. How all that takes place is interpretation-dependent.

 

[atyy]

As far as I can see, the 1/2 spin system does not explicitly postulate that it is a consequence of a model that has greater resolution - nor that the measurement of spin corresponds to an incomplete observation made on outcomes in the higher resolution model.

Are you saying that it is self-evident that some higher resolution model underlies the 1/2 spin system?

There is no higher resolution model underlying the spin 1/2 system. Even in such a simple system where greater resolution is impossible, the experimental results are macroscopic.

If one wants, one can use the indirect measurement formalism to discuss the measurement of the spin 1/2 system, but that will bring no further (basic) insight.

 

[Stephen Tashi]

There is no higher resolution model underlying the spin 1/2 system. Even in such a simple system where greater resolution is impossible, the experimental results are macroscopic.

I understand your offer an example where a model of a macroscopic event needs no higher resolution model underlying it. Are you also saying that all macroscopic events can be modeled in that manner?

There is no such thing as higher resolution observation, if I understand your post correctly.

I take that to mean that a higher resolution observation is not necessary to determine whether the cat is alive (not that no such observation can exist).

It is not necessary for all/100%/ of atoms of the cat to be observed directly to collapse them. Practically, an observation of some of the atoms of the cat, collapses all of the atoms of the cat. How all that takes place is interpretation-dependent.

That is a definite answer to post #10.

Is it interpretation dependent where the collapse of physical systems into definite outcomes stops? Does it extend beyond the cat?

 

[EPR]

That is a definite answer to post #10.

Is it interpretation dependent where the collapse of physical systems into definite outcomes stops? Does it extend beyond the cat?

In the MWI, everything is 'classical' all the time.
All events are particles.
In the CI, the collapse(the cut) can be placed anywhere(it's not specified). Since interacting quantum systems are always entangled, collapse of one part of the system, can in principle and in theory spread the collapse infinitely. One test confirmed that entangled separated particles can remain one system over a distance of over 1000 km. https://www.scientificamerican.com/...ance-rdquo-record-preps-for-quantum-internet/

 

[atyy]

I understand your offer an example where a model of a macroscopic event needs no higher resolution model underlying it. Are you also saying that all macroscopic events can be modeled in that manner?

Yes, if a macroscopic event is a measurement outcome, then by definition it can be modeled by a self-adjoint operator called an observable and the Born rule. That is the most fundamental sense of "macroscopic" in quantum mechanics. It is definitional, so other people may prefer the term "classical" or "definite outcome". So in the most basic sense, the measurement side is "macroscopic" and the quantum side is "microscopic".

A second idea of "macroscopic" is if we use the indirect measurement formalism and split the quantum side into two systems - one called the measurement ancilla and the other the quantum system. The measurement ancilla and its interaction with the quantum system can be modeled by a Hamiltonian. This specification alone does not give us the idea that the measurement ancilla consists of many particles. However, we also construct the Hamiltonian so that the measurement ancilla can be interpreted as consisting of many particles. Here the measurement ancilla would be "macroscopic" on the quantum side. We expect decoherence to occur, and the resulting density operator describing the quantum system to be almost diagonal, where the diagonals represent the measurement outcomes that would be made on the "classical" side (our first definition of "macroscopic").

So the first idea of macroscopic is the measurement outcome where the measurement is modeled with a self-adjoint operator and the Born rule (both the quantum state and the self-adjoint operator enter the Born rule). The second idea of macroscopic is the description of the measurement ancilla as part of the quantum state in the pre-measurement process (in the pre-measurement process the self-adjoint operator and the Born rule are not involved).

There is a third idea called the classical limit (not directly related to either of the two ideas above, but related in language by using the term "classical"), which is simply the limit of taking Planck's constant to zero.

 

[Stephen Tashi]

A second idea of "macroscopic" is if we use the indirect measurement formalism and split the quantum side into two systems - one called the measurement ancilla and the other the quantum system.

I understand either formalism as a model of a macroscopic event that occurs at an instant in time. I don't understand how such models apply to events that have a duration - the most obvious example is our (macroscopic) sensation that we are proceeding through life in a continuous series of "real" situations - i.e. encountering specific outcomes. For example, a macroscopic phenomena like "The cat is dead at 9:00 AM" is (with current technology) followed by the event "The cat is dead at 10 AM". What explains the persistence of macroscopic properties? Perhaps such questions also apply to microscopic events? - is this the famous Measurement Problem?

An example of a macroscopic event is a physicist preparing an experiment that puts a physical system S in a superposition of states. If the model for the entire situation includes S together with the physicist, the lab equipment etc. then how do we model the macroscopic outcome of sucessfuly putting S in a superposition? It seems (to me) that the occurrence of such a macrscopic outcome cannot not cause everything involved in the model to collapse to a specific outcome.

 

[PeterDonis]

What explains the persistence of macroscopic properties?

Decoherence.

An example of a macroscopic event is a physicist preparing an experiment that puts a physical system S in a superposition of states.

Superposition is basis dependent. There will always be some basis in which the system S is not in a superposition.

A better way to state what I assume you mean here is that the physicist can prepare a system S in a state that is not an eigenstate of a measurement operator that the physicist intends to apply to the state: for example, he can prepare a qubit in a state which is not an eigenstate of, say, the spin-z operator that he intends to apply with a Stern-Gerlach device oriented in the z direction. In the basis of eigenstates of that measurement operator, the prepared state of the system S will therefore be a superposition.

If the model for the entire situation includes S together with the physicist, the lab equipment etc. then how do we model the macroscopic outcome of sucessfuly putting S in a superposition?

This isn't a "macroscopic outcome" by your definition, because the system S has not yet been measured.

If you're viewing the preparation process itself as a measurement, then you have to add more information to your scenario: what was the state of the system S before the preparation? In the example I gave above, where was the physicist getting the qubit from that he prepares, and in what state did he get it? Adding that information should make it obvious how you model the process: the same way you model any other measurement.

 

[EPR]

I understand either formalism as a model of a macroscopic event that occurs at an instant in time. I don't understand how such models apply to events that have a duration - the most obvious example is our (macroscopic) sensation that we are proceeding through life in a continuous series of "real" situations - i.e. encountering specific outcomes. For example, a macroscopic phenomena like "The cat is dead at 9:00 AM" is (with current technology) followed by the event "The cat is dead at 10 AM". What explains the persistence of macroscopic properties? Perhaps such questions also apply to microscopic events? - is this the famous Measurement Problem?

Yes. It's the measurement problem. Some interpretations do not have it but they trade it for an infinite number of worlds. Or hidden variables.

 

[atyy]

I understand either formalism as a model of a macroscopic event that occurs at an instant in time. I don't understand how such models apply to events that have a duration - the most obvious example is our (macroscopic) sensation that we are proceeding through life in a continuous series of "real" situations - i.e. encountering specific outcomes. For example, a macroscopic phenomena like "The cat is dead at 9:00 AM" is (with current technology) followed by the event "The cat is dead at 10 AM". What explains the persistence of macroscopic properties? Perhaps such questions also apply to microscopic events? - is this the famous Measurement Problem?

Yes, a measurement only produces an event at a specific time.

For repeated measurements, a simple scenario that produces the same result for two successive measurements is that the first measurement produces the result the "cat" is "dead", and the quantum state collapses from a superposition of "dead and alive" into a state of "dead". For an appropriate Hamiltonian, the "cat" will remain in the "dead" state, so that a second measurement on that state will produce the outcome "dead" with certainty. Here by "cat" I mean a spin 1/2 system. Obviously, one could get other outcomes for other Hamiltonians.

For many repeated measurements or for continuous measurements, one can measure frequently or even continuously, taking into account that measurement perturbs the system (by collapsing the state).
https://arxiv.org/abs/quant-ph/0306072
https://arxiv.org/abs/2010.07575

Decoherence can also form part of the answer here. However, decoherence on its own produces no macroscopic events or measurement outcomes, and needs to be supplemented with additional criteria or variables that are beyond the basic postulates of quantum mechanics.
An example of an additional criterion is the predictability sieve:
https://arxiv.org/abs/quant-ph/0105127
An example of additional variables is Bohmian mechanics:
https://arxiv.org/abs/1206.1084 (see VII.2 and Fig 11 about the measurement process and collapse)

An example of a macroscopic event is a physicist preparing an experiment that puts a physical system S in a superposition of states. If the model for the entire situation includes S together with the physicist, the lab equipment etc. then how do we model the macroscopic outcome of sucessfuly putting S in a superposition? It seems (to me) that the occurrence of such a macrscopic outcome cannot not cause everything involved in the model to collapse to a specific outcome.

In orthodox quantum mechanics, there is no way to put everything into the quantum state. One must always have the measurement apparatus outside the quantum state (if the measurement apparatus is in the quantum state, one needs another measurement apparatus to measure the measurement apparatus) in order to have definite measurement outcomes and probabilities for the outcomes. This is the measurement problem.

Approaches to solving the measurement problem (being able to include the measurement apparatus in the quantum state) include Bohmian mechanics and the Many-Worlds Interpretation(s).

 

[Stephen Tashi]

Superposition is basis dependent. There will always be some basis in which the system S is not in a superposition.

Is the model of a measurement as a linear operator also basis independent? As I understand it, the Born rule gives the probabilities for eigenvectors of the operator being outcomes of the measurement. So the eigenvectors are, in a manner of speaking, the mutually exclusive categories that the person making the measurement wishes to use in classifying the world. If the probability of the result being eigenvector v is p(v) and in some basis v = w + x, is there a way to interpret a measurement of v = w+x in terms of w and x individually?

This isn't a "macroscopic outcome" by your definition, because the system S has not yet been measured.

Just to clarify, I didn't define a macroscopic event to be a situation where everything has been measured. My thought is that a macroscopic event is a situation that can be theoretically defined as a set of outcomes where everything has been assigned a value , but can be reliably determined to be in the set by an incomplete measurement. For example, "There is a cat in the picture" can theoretically be defined for a 1024x1024 gray scale pixel image by listing all 1024x1024 arrays of pixels that contain a picture of a cat. But if "There is a cat in the picture" is a macroscopic event (by my definition) then there exists a measurement procedure that employs many-to-one functions of pixels and detects a cat based on the values of such functions.

In orthodox quantum mechanics, there is no way to put everything into the quantum state. One must always have the measurement apparatus outside the quantum state (if the measurement apparatus is in the quantum state, one needs another measurement apparatus to measure the measurement apparatus) in order to have definite measurement outcomes and probabilities for the outcomes. This is the measurement problem.

Is " not in a quantum state" defined relative to some assumed basis of the state space? Or is "not in a quantum state" a description of something entirely different than vectors in the state space - for example, a linear operator on vector space is a distinct concept from a vector in that space.

 

[PeterDonis]

Is the model of a measurement as a linear operator also basis independent?

The operator itself is. The operator's matrix elements in a particular matrix representation are not.

If the probability of the result being eigenvector v is p(v) and in some basis v = w + x, is there a way to interpret a measurement of v = w+x in terms of w and x individually?

There might be in particular cases, but not in general. The obvious counterexample is measurements of spin about different axes; the vector spin-z up (v) is a sum of spin-x up (w) and spin-x down (x), but you cannot interpret a measurement of spin-z that gives an "up" result in terms of spin-x measurement results.

 

[PeterDonis]

I didn't define a macroscopic event to be a situation where everything has been measured.

I didn't say you did. The issue I raised was not that "everything" had not been measured, but that system S had not been measured.

 

[atyy]

Is " not in a quantum state" defined relative to some assumed basis of the state space? Or is "not in a quantum state" a description of something entirely different than vectors in the state space - for example, a linear operator on vector space is a distinct concept from a vector in that space.

"Not in a quantum state" is a basis independent statement.
The quantum state is a vector in the vector space (strictly speaking it is a ray or unit vector).
The measurement apparatus is represent by a self-adjoint operator on the vector space. Thus the measurement apparatus is not represented in the quantum state.
Orthodox quantum mechanics requires us to decide which part of "reality" to put in the quantum state, and which part of reality (such as the measurement apparatus) stays outside the quantum state. While we are pretty sure that measurement results are real (or classical or macroscopic), what exactly the quantum state alone represents is a puzzle.

 

[martinbn]

"Not in a quantum state" is a basis independent statement.
The quantum state is a vector in the vector space (strictly speaking it is a ray or unit vector).
The measurement apparatus is represent by a self-adjoint operator on the vector space. Thus the measurement apparatus is not represented in the quantum state.
Orthodox quantum mechanics requires us to decide which part of "reality" to put in the quantum state, and which part of reality (such as the measurement apparatus) stays outside the quantum state. While we are pretty sure that measurement results are real (or classical or macroscopic), what exactly the quantum state alone represents is a puzzle.

The observalbes are represented by self-adjoint operators, not the measuring device.

 

[Morbert]

It's probably useful to distinguish between a measurement outcome–a property of the measurement apparatus–and a measurement result–a property of the measured system. The logical equivalence between them is why the formalism is so flexible re/ what must be included in the state space.

 

[AlexCaledin]

Does QM have to describe events specially? It describes all the properties and probabilities of the possible decoherent histories - which include everything to happen in our very brains - and then we observe some events and can calculate what we really need - knowing that every measurable property is formed by (means of) the universe quantum state reduction (= the actual history choice) and therefore the textbook QM is FAPP-applicable.