For some weird reason many people seem to have trouble with the fact that the space of states of a quantum system has many (infinitely) more states than its classical limit. So it is necessary and sufficient to take all states of the quantum system into account, not only the states that remain upon taking the classical limit. What quantum mechanics also provides are unambiguous rules for computing transition probabilities between any pair of states. If one were to take into account only the states that remain upon taking the classical limit one would find inconsistencies; that, simply, reflect the fact of having made a mistake. Upon taking into account all the states of the quantum system, no inconsistency appears.

Many people implicitly assume that ``realist=classical''; and that's, just, wrong. ``Realist=quantum'' and the classical description is a limiting case of the quantum description, not the other way around.

One can't deduce quantum mechanics from classical mechanics; one can deduce classical mechanics from quantum mechanics.

Stam Nicolis More specifically, do you agree with the conservation of the 2-time correlation on Bob's side when only Alice performs measurements on her side and with that 2-time correlation's conservation to be a falsifiable prediction that noting happens on Bob's side when Alice performs a measurement on her side (contrary to Valentini, Scarani, Goldstein, Aspect, Hemmick, Towler, Bricmont... realist/objective interpretation of quantum state and quantum non-locality)?

I think the statement about the 2-time correlation being conserved is either right or wrong. Nothing else. The confusion, once more, lies in introducing additional assumptions, that are, simply, vague enough to make any calculation meaningless. Like all the ambiguous expressions that can lead to different results, depending on where one actually puts the parentheses to enforce priority.

So the bottom line is, No, it's not a falsifiable prediction of anything else than that vague statements lead nowhere.

Alice and Bob are in interaction, by definition, so it makes no sense pretending they're not and being surprised to find that they are.

The evolution operator of any quantum system is U = exp(itH/hbar), so, if H=0, there's no evolution at all. Therefore the statement that there can be any evolution with H=0 is wrong.

The space of states of the system is the space of all linear combinations of the states |Alice>|Bob>. If the Hamiltonian of the system H = HAlice x IBob + IAlice x HBob, where I is the corresponding identity operator, then there's no interaction between them, by definition. If H = HAlice x IBob + IAlice x HBob + V, where V is not block-diagonal, with respect to the subspaces of Alice and Bob, then-and only then-Alice and Bob interact. There's nothing more or less to it. Transition amplitudes can be unambiguously computed and, from them, transition probabilities, also, for anything in each subspace. Nothing more is needed. What each computes, in their own subspace, is computed, according to the usual rules of probability.

In non-relativistic quantum mechanics all calculations are performed in the space of states-phase space-not spacetime. Locality or non-locality just doesn't make sense in phase space in general. In particular locality in spacetime doesn't have anything to do with locality in phase space and it's not true that the fact that the space of states of a quantum system is infinitely larger than the space of states of its classical limit has anything to do with locality or lack thereof in spacetime.

It must be stressed that non-relativistic quantum mechanics knows as much about special relativity-namely nothing-as non-relativistic classical mechanics. For some reason people seemed to believe (and many, curiously, still believe) that non-relativistic quantum mechanics must be consistent with special relativity; they're mistaken. The statement itself that events in spacetime are separated by a space-like distance doesn't make sense in non-relativistic classical or quantum mechanics.

It's relativistic quantum mechanics that's consistent with special relativity, not non-relativistic quantum mechanics.

The way locality is enforced in relativistic quantum mechanics is by demanding that operators that act on subspaces of states that describe events that are separated-in spacetime-by a space-like distance, commute.

Further subtle issues are presented in these lecture notes: Article Notes from Sidney Coleman's Physics 253a

Stam Nicolis "The evolution operator of any quantum system is U = exp(itH/hbar), so, if H=0, there's no evolution at all. Therefore the statement that there can be any evolution with H=0 is wrong" and that's precisely the reason why I had added quotes around "evolution".

"I think the statement about the 2-time correlation being conserved is either right or wrong.... So the bottom line is, No, it's not a falsifiable prediction of anything else than that vague statements lead nowhere."

So, if I understand your point, you disagree with Aharonov, Popescu and Tollaksen statement (cf. each instant of time a new universe" https://arxiv.org/pdf/1305.1615) that, considering any system, provided its absence of quantum evolution (H = 0) and provided that no quantum measurement be performed between instants t1 and t2, the 2-time measurement O(t2) - O(t1) is achievable and vanishes whatever the observable O ?

Yes, that statement can't mean anything, but the triviality that O(t_2)=O(t_1), whatever t_2 and t_1 are, by construction.

Stam Nicolis Let's take the discussion a little further (always in the case H=0 and no quantum measurement). When one performs the two measurements O(t1) and O(t2) and gets the same result O(t1) = O(t2), one gets an information on the system and one changes its state.

When one performs, on the contrary, a 2-time measurement O(t1)-O(t2) one finds 0. One gets no information on the system and one doesn't change its state.

Let us now consider the time "evolution" model proposed by Aharonov, Popescu and Tollaksen Article Each Instant of Time a New Universe

Let us assume, on the contrary, that Alice polarization measurement would cause a real (but up to day hidden) collapse of Bob's photon in the same polarization state than Alice's at a given time t on Bob's side (in violation of Relativistic causality at an interpretive level). Then O(t1) - O(t2) (with t1 < t < t2) could be found different of zero due to that alleged Bob’s photon polarization collapse.

So, in my view, the experimental verification of O(t1) - O(t2) = 0 on Bob's side, as long as only Alice performs measurements, proves than nothing happens on Bob's side when only Alice performs measurements on her side. It provides a proof that the interpretation of quantum measurement in EPRB experiments as violating the relativistic causality at an interpretive level is refuted.

Once more, the evolution operator of a quantum system is U(t)=exp(iHt/hbar).Operators evolve as O(t)=U(t)O(0)U(-t). If H=0, O(t)=O(0).

Furthermore: The reason that the statement ``each instant of time a new universe" is wrong is that quantum mechanics describes physical systems in the presence of fluctuations. Before one makes a measurement, one can compute the probability of finding the system in any of its states. After one has made the measurement, the system will be found in some state. The other states aren't available, anymore! Anymore than when one tosses a coin, there's a probability that it will be found in the state ``heads" or the state `` tails"; after the measurement it will be found in either the one state or the other, full stop.

However new states are NOT ``created'' during the evolution of the system! The space of states of the system remains the same during the evolution; the probability of finding the system in any particular state evolves with time.

When dealing with a qubit, the space of states is a 2-sphere: |ψ>(t)=a|0>+b|1>, with |a|^2+|b|^2=1. This is a 3-sphere, in fact, since a and b are complex numbers, but there's a relative phase that can't affect anything and all the states of the 3-sphere that belong to the same great circle are equivalent, hence the 2-sphere. The evolution operators are the operators that realize rotations on the 2-sphere, for example: U(t) = exp(iθ.σt/hbar), where σ are the three Pauli matrices and θ is a vector of three components, that labels points on the 3-sphere. So H is, definitely, not zero for a qubit, otherwise nothing happens.

The statement that O(t2)-O(t1) describes a measurement of anything is wrong; it doesn't. A measurement on a quantum system means the action of an operator, M, on the state of the system. So, with a system that evolves with time, a measurement at time t1 would be described by the operator MU(t1) acting on some initial state |ψ>, giving the state MU(t1)|ψ>. The evolution to the instant t2 is described by acting on this state by the operator U(t2-t1), giving the state U(t2-t1)MU(1)|ψ>. The probability of finding the system, at time t2, in some state |φ>, is given by the expression

||2. That's all there is.

If M commutes with U(t) for any t this means that the probability for finding the system in a state |φ>, at time t2, given that the system was, initially, in state |ψ>, is ||2=||2 and is independent of t1.

Of course it is possible to describe a measurement that is sensitive to the fact that it is carried out at the instant t1; this means that M=M(t1), so the transition probability becomes

||2

If H=0, then U=1 and the probability of finding the system in state |φ>, given that, at t=0, it was in state |ψ>, if a measurement was realized at time t1, is given by

||2

In all these calculations the states are assumed to be normalized, ||2=||2=1.

So, for Alice and Bob, their space of states is the space of all linear combinations of |Alice>|Bob>. They don't interact if the evolution operator for Alice is of the form UAlice x IBob and for Bob is of the form IAlice x UBob; otherwise they're interacting. I.e. they don't interact if the evolution operator of their system as a whole, that acts on linear combinations of |Alice>|Bob>, can be written as

U = UAlice x IBob + IAlice x UBob

At t=0, the initial state can be written as |ψ>Alice|ψ>Bob. This state can be expanded in the basis of the eigenstates of UAlice and UBob. And a similar calculation as above for the qubit can be done and leads to unambiguous expressions for whatever measurements Alice and Bob do.

If the evolution operator is given by the expression indicated, then Alice and Bob don't interact.

If the evolution operator

U = UAlice x IBob + IAlice x UBob + UAB

contains a term of the form UAB that cannot be written as VAlice x WBob, then Alice and Bob are interacting: If one expands the initial state in the eigenstates of UAlice and UBob, the new term will produce correlations. Vice versa, if such correlations are found, they imply the presence of such a term, that can't be written as a direct product over the two subspaces. These are all standard exercises in quantum mechanics.

All these operations refer to the space of states, not spacetime. Where Alice and Bob are found, relative to one another doesn't enter the picture at all.

Stam Nicolis I have read your answers. I have no remark or question with regard to your answers. As you point out, these are all standard exercises in quantum mechanics. I nevertheless come back to this discussion because some of your answers prove, in my view, the need for providing more context about the Article Each Instant of Time a New Universe

First, I edited a little my post to specify my notations O(t1) and O(t2). They are intended to mean measurements' outcomes of observable O at instant t1 et t2 (not observable O at time t1 and t2 according to Heisenberg picture)

Now, the main question answered by that article could be summarized as:

is it possible to model the time "evolution" of the state of a quantum system and its successive states

|Ψ>1, |Ψ>2 ...|Ψ>N at N instants t1, t2... tN (initial state unknown)

by N systems in states |Ψ>1, |Ψ>2...|Ψ>N at the same instant?

(All the article relies on this time discretization. H=0 is not excluded, hence the "evolution" quotes. However, accounting for H not null is no big deal, see fig6)

In the standard one state-vector formalism, the answer is no:

• N systems in the same sate provide too much information. In case H=0 for instance, we can know this same state "by making different measurements on the different copies"
• O(|Ψ>i) is not necessarily equal to O(Ψ>j) (for instance measuring σx of a spin 1/2 in a proper initial state of σz)

In the two-state vectors formalism, it's possible. It's enough:

1/ to represent each instant of time (a little "brick" of time) by a two-state:

• a |ket> vector state at past boundary of the "brick" (called preselection)
• a and |1>, this maximal entanglement between 2 successive "bricks of time" k and k+1 writes:

  |0>k+1 k+1

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It's useful to have sharp statements; the sharper the statement, the easier it is to check whether (a) it makes sense or (b) it doesn't; and, once it's established that it makes sense, whether it's (a1) true or (a2) false.

The answer to the question, whether it is possible, by acting by the evolution operator on some state |ψ> to generate N states, by successive application, that can be equally well reached by the application of the evolution operator only once on N other states isn't different in quantum mechanics than it is in classical mechanics. So it's wrong to state that it requires special attention in quantum mechanics than it would in classical mechanics.

So let me deal with this issue, first. In classical mechanics, IF the system is integrable, the answer is No. A typical example is a particle in a time-independent potential in one spatial dimension. In that case each initial condition define a curve in phase space described by the equation E = (p2/(2m))+V(x)=(p02/(2m))+V(x0), where (x0,p0) denotes the initial condition. These curves don't cross-they can't since the uniqueness theorem of the equations of motion implies the can't-so any point on any one of these curves is equivalent to any other-but points that don't belong to the same curve are not.

Now in quantum mechanics the phase space for a particle isn't 2-dimensional, like its classical limit; it's infinite-dimensional, it's the space of the wavefunction, ψ(x,t),solution of the Schrödinger equation, ihbar(d/dt)ψ(x,t) = Hψ(x,t),where H is the Hamiltonian operator; for the particle we're discussing it would be given by -hbar2(d2/dx2)+V(x). Therefore the solution would be given by the expression

ψ(x,t) = exp(-iHt/hbar)ψ(x,0)

and, once more, the space of solutions is divided into sectors, since Schrödinger's equation, also, has conserved quantities, only they're a bit more complicated to describe. (Energy, for instance, isn't conserved, in the sense that the probability distribution of E = (p2/(2m))+V(x) is not a delta function about (x0,p0); it has a width, since position and momentum don't commute. What is conserved is probability, just like in classical mechanics, in fact, a statement that is known as Liouville's theorem: (dρ/dt)+(dJ/dx) = 0.)

However this subject (that goes under the name of ``ergodic behavior'', incidentally) doesn't have anything to do with EPR or with the topic of your original question. Nor does it have anything to do with relativity! Once more, non-relativistic quantum mechanics knows nothing about what space-like separations can mean-any more or less than rigid bodies do in non-relativistic classical mechanics.

The statements about measurements in spacetime only make sense within relativistic mechanics, whether classical or quantum. There, too, the causal structure is the same, whether in quantum mechanics or in classical mechanics, it differs from the non-relativistic approximation in both cases.

Bottom line: Alice and Bob can do experiments and calculations and both can be described consistently and unambiguously within classical mechanics and quantum mechanics as both are presented in all the standard textbooks. Confusion results when introducing additional assumptions that don't have anything to do with either classical or quantum mechanics. It's not a question of differing interpretations! It's a question of doing the calculations consistently, or not. If the calculations are inconsistent, the results are, just, meaningless.

Once more:If Alice and Bob interact, what they're measuring is correlated between them. If one performs the calculation of a quantity under the assumption that they aren't interacting, when they are, the calculation will show that. The assumption that a calculation made within the framework of non-relativistic mechanics, whether classical or quantum, must, nonetheless, be consistent with relativity, is inconsistent and all the confusion arises because this isn't recognized.

Regarding spins: The interaction Hamiltonian presented just doesn't make sense. The way a spin interacts with the external world is known-it's through a magnetic field and nothing else-and doesn't have anything to do with anything called a pointer, or its momentum.

The Hamiltonian, that describes the interaction of a spin 1/2 with an external magnetic field is given by the expression

Hint = g σ.B

and that's it, where g is aa constant called the so-called gyromagnetic ratio and σ=(σx , σy , σz )

is the vector of matrices in the 2-dimensional representation of SU(2); these are called the Pauli matrices and it's possible to choose labels so that σz=diag(1,-1); then the two others are not diagonal.

If the magnetic field is constant in time, in direction, it's always possible to choose this direction to be the z-axis, so

Hint = gσzBz= diag(Bz,-Bz)

where Bz may depend on time.

If the magnetic field is not constant in time, in direction, then, in general, the Hamiltonian will have three terms,

Hint = g(σxBx(t)+σyBy(t)+σzBz(t))

It will still be a Hermitian 2x2 matrix, but with time-dependent coefficients. That's not a problem, the Schrödinger equation,

i hbar (d/dt)(ψ1,ψ2)T = HInt(ψ1,ψ2)T

can, still, be solved for its evolution.

The coefficients ψ1,2(t) describe the spin's evolution. Its state at any time is defined by the equation

|ψ>(t) = ψ!(t)|0> + ψ2(t)|1>

where σz|0>=|0> and σz|1> = -|1>.

The external world affects the spin only through the functions B(t).

This is all that's needed to describe the evolution of the spin, along with the initial conditions for the coefficients ψ!(0) and ψ2(0).

So Alice and Bob exist, as far as the spin is concerned, only through the functions B(t). One, particularly interesting, case is when Bz(t)=0, Alice controls Bx(t) and Bob controls By(t).

If they both control the same component of the magnetic field, it's always possible to take this to be Bz(t), so they don't interact. They will only be interacting if they control two different components (one can then be taken to be Bz but the other can't). And, from then on, everything each measures is unambiguous. And by studying each the time series of their individual measurements, they will discover whether or not they are in interaction.

This is a standard exercise in quantum mechanics.

If the magnetic field is of the form B(t)=Bz(t)z, where z is the unit vector along the z-direction, then the spin states (that can be identified, through the conservation of probability, with the points described by the equation

|ψ1|2+|ψ2|2=1, with the points of the unit 3-sphere, (Re ψ!)2+(Im ψ1)2+(Re ψ2)2+(Im ψ2)2=1, but, since it's possible to eliminate a phase, all points that belong to the same great circle on this sphere can be shown to describe the same state) define a circle (not necessarily a great circle) on the unit 2-sphere. So in this case, any initial configuration will define a circle on the unit 2-sphere that's the intersection of a plane perpendicular to the z axis and the sphere.

If the magnetic field is not of constant direction in time, then it's an interesting exercise to find what portion of the surface of the unit 2-sphere is covered during the evolution.

A measurement at t=t1 means the insertion of a projection operator at that instant, that describes this measurement. For instance to measure whether the spin is along the +z direction means to insert the operator M(t1)=δ(t-t1)(I+σz)/2 at the instant t=t1. So the state of the spin at a time after t=t1 will be |ψ>(t)=exp(-iHint (t-t1)/hbar)M(t1)exp(-iHint t/hbar)|ψ>(0). Either Alice or Bob may have done this measurement. And similarly for a measurement at t=t2.

Stam Nicolis I have no question or remark concerning a spin ½ evolution in a magnetic field and the Hamiltonian of its evolution. Neither have I questions or remarks about its state-space, that’s to say the special unitary group SU(2), this group being diffeomorphic with the unit radius sphere S3 and a double-cover of the rotation group SO(3), nor with the use of the 3 Pauli matrices and the Bloch sphere being enough to model qubits due to phase invariance (be they spin ½ or any 2-state quantum systems). As you point out, these are all matters of standard exercises in quantum mechanics.

Now, let’s get to the topic of the discussion. First let’s remind the main points of the formalism used in Article Each Instant of Time a New Universe

• a model of the time evolution of a spin 1/2 in the time-symmetric two-state vector formalism.
• A time discretization of a spin evolution in time as N two-state vectors “bricks” of time.
• A maximal entanglement of each time ‘brick” state-vector at its past boundary (eq. (14) and (15)).
• Probabilities’ calculation in this time evolution formalism (§ Probabilities).
• The conservation of 2-time correlations, i.e. a null two-time outcome of a 2-time measurement, when no single measurement is performed. This interaction with a measurement apparatus is modelled by a von Neumann quantum measurement interaction Hamiltonian defined in eq (5). The physical meaning of that Hamiltonian and the meaning of the standard notion of pointer observable in this framework are detailed later.
• The loss of that 2-time correlation between a time “brick” preceding a quantum measurement and the time “brick” following that quantum measurement.
• The effect, on pairs of spin 1/2 in singlet state, of a quantum measurement on Alice’s side and no quantum measurement on Bob’s side. It gives rise to the loss of 2-time correlation on Alice’s side between times t1 and t2 before and after a quantum measurement on her side and, on the contrary, the conservation of the 2-time correlation conservation on Bob’s side (cf. §Measurement on EPR states and fig 7).

Conclusion:

The proposed time evolution model, based on the two-state vectors formalism Article The Two-State Vector Formalism

Moreover, the conservation of the 2-time correlation on Bob’s side provides, in my view and that of the authors of Article Each Instant of Time a New Universe

This discards, in my view, interpretations of some physicists, assuming a possible objective action-at-a-distance of Alice’s measurement causing an objective (though hidden) change of Bob’s spin state.

I skip the details generalizing this two-state vector time-symmetric evolution formalism to any quantum system and to a non-zero Hamiltonian evolution. They don’t change the stressed conclusion. Moreover, these details are no big deal to account for. See eq. (16), (17) and (18).

The 2-time quantum measurement interaction Hamiltonian eq. (5)

That interaction Hamiltonian is a model of a 2-time quantum measurement “written in the standard von Neumann’s measuring formalism, with a measuring device whose pointer position is denoted by q and its canonical conjugate momentum p”.

The standard definition of a pointer variable, as defined by R. Balian for instance, is: “pointer variable: a part of the apparatus which undergoes a change that can be read off or registered.” Whatever the measured quantity and the physical phenomenon creating the interaction. cf. “Understanding quantum measurement from the solution of dynamical models”

https://arxiv.org/pdf/1107.2138 §1.3.2, and

“The von Neumann Model of Measurement in Quantum Mechanics”

https://arxiv.org/pdf/1311.7649 §2.1.1 eq. (14.a) (14.b) and (15).

The reason for the presence of Dirac distributions in time in eq (5) of the quantum measurement interaction Hamiltonian is because of an assumed very short interaction duration of the apparatus with the spin 1/2.

Calculation detail of the 2-time measurement pointer variable outcome

Let's calculate the outcome of the pointer observable q, after time t2, by a two-time (t1, t2) measurement applied to the set of N time “bricks” (modelling a time evolution of a spin ½ particle under a H=0 “evolution"). In the case of a σx measurement, this 2-time measurement is modeled thanks to the von Neumann quantum measurement interaction Hamiltonian:

Hint = −δ(t − t1) pσx + δ(t − t2) pσx. (5)

The evolution of the pointer position q is provided by Heisenberg equation of motion (ℏ taken as 1) dq/dt = i [Hint, q] = (δ(t − t2) − δ(t − t1)) σx(t). (6)

Hence, integrating (6) we find at a time t >t2: q(t2) –q(t1) = σx(t2) −σx(t1) (7)

Once more: The only way a magnetic moment interacts with an external magnetic field is through the interaction Hamiltonian

Hint = gs.B

where s is the magnetic moment (which is the spin in the magnetic moment's rest frame) and B is the external magnetic field. There's nothing else and there can't be anything else.

You should not forget the third answer: Measurements of entganglement do not prove entanglement because of a fundamental missinterpretation of the results.

The most simple local "Entanglement" calculation:

Code Alice-and-Bob Excel file: Entanglement without action at a d...

A detection rate of > 82.8 % has never been realized.

Preprint Bell-Test and CHSH-Test for a local Alice-und-Bob-model