# Quantum Computing, 2.2

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2021.01.21 Thursday ACHK

# The square root of the probability, 4.3

Eigenstates 3.4.3

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The indistinguishability of cases is where the quantum probability comes from.

— Me@2020-12-25 06:21:48 PM

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In the double slit experiment, there are 4 cases:

1. only the left slit is open

2. only the right slit is open

3. both slits are open and a measuring device is installed somewhere in the experiment setup so that we can know which slit each photon passes through

4. both slits are open but no measuring device is installed; so for each photon, we have no which-way information

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For simplicity, we rephrase the case-3 and case-4:

1. only the left slit is open

2. only the right slit is open

3. both slits are open, with which-way information

4. both slits are open, without which-way information

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Case-3 can be regarded as a classical overlapping of case-1 and case-2, because if you check the result of case-3, you will find that it is just an overlapping of result case-1 and result case-2.

However, case-4 cannot be regarded as a classical overlapping of case-1 and case-2. Instead, case-4 is a quantum superposition. A quantum superposition canNOT be regarded as a classical overlapping of possibilities/probabilities/worlds/universes.

Experimentally, no classical overlapping can explain the interference pattern, especially the destruction interference part. An addition of two non-zero probability values can never result in a zero.

Logically, case-4 is a quantum superposition of go-left and go-right. Case-4 is neither AND nor OR of the case-1 and case-2.

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We can discuss AND or OR only when there are really 2 distinguishable cases. Since there are not any kinds of measuring devices (for getting which-way information) installed anywhere in the case-4, go-left and go-right are actually indistinguishable cases. In other words, by defining case-4 as a no-measuring-device case, we have indirectly defined that go-left and go-right are actually indistinguishable cases, even in principle.

Note that saying “they are actually indistinguishable cases, even in principle” is equivalent to saying that “they are logically indistinguishable cases” or “they are logically the same case“. So discussing whether a photon has gone left or gone right is meaningless.

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If 2 cases are actually indistinguishable even in principle, then in a sense, there is actually only 1 case, the case of “both slits are open but without measuring device installed anywhere” (case-4). Mathematically, this case is expressed as the quantum superposition of go-left and go-right.

Since it is only 1 case, it is meaningless to discuss AND or OR. It is neither “go-left AND go-right” nor “go-left OR go-right“, because the phrases “go-left” and “go-right” are themselves meaningless in this case.

— Me@2020-12-19 10:38 AM

— Me@2020-12-26 11:02 AM

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It is a quantum superposition of go-left and go-right.

Quantum superposition is NOT an overlapping of worlds.

Quantum superposition is neither AND nor OR.

— Me@2020-12-26 09:07:22 AM

When the final states are distinguishable you add probabilities:

$\displaystyle{P_{dis} = P_1 + P_2 = \psi_1^*\psi_1 + \psi_2^*\psi_2}$

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When the final state are indistinguishable,[^2] you add amplitudes:

$\displaystyle{\Psi_{1,2} = \psi_1 + \psi_2}$

and

$\displaystyle{P_{ind} = \Psi_{1,2}^*\Psi_{1,2} = \psi_1^*\psi_1 + \psi_1^*\psi_2 + \psi_2^*\psi_1 + \psi_2^*\psi_2}$

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[^2]: This is not precise, the states need to be “coherent”, but you don’t want to hear about that today.

edited Mar 21 ’13 at 17:04
answered Mar 21 ’13 at 16:58

dmckee

— Physics Stack Exchange

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$\displaystyle{ P_{ind} = P_1 + P_2 + \psi_2^*\psi_1 + \psi_2^*\psi_2 }$

$\displaystyle{ P_{\text{indistinguishable}} = P_{\text{distinguishable}} + \text{interference terms} }$

— Me@2020-12-26 09:07:46 AM

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interference terms ~ indistinguishability effect

— Me@2020-12-26 01:22:36 PM

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# The square root of the probability, 4.2

Eigenstates 3.4.2

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The difference between quantum and classical is due to the indistinguishability of cases.

— Me@2020-12-26 01:25:03 PM

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Statistical effects of indistinguishability

The indistinguishability of particles has a profound effect on their statistical properties.

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The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled $\displaystyle{ |0 \rangle }$ and $\displaystyle{|1\rangle}$, which have the same energy.

The composite system can evolve in time, interacting with a noisy environment. Because the $\displaystyle{|0\rangle}$ and $\displaystyle{|1\rangle}$ states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on quantum entanglement.) After some time, the composite system will have an equal probability of occupying each of the states available to it. The particle states are then measured.

If A and B are distinguishable particles, then the composite system has four distinct states: $\displaystyle{|0\rangle |0\rangle}$, $\displaystyle{|1\rangle |1\rangle}$ , $\displaystyle{ |0\rangle |1\rangle}$, and $\displaystyle{|1\rangle |0\rangle }$. The probability of obtaining two particles in the $\displaystyle{|0\rangle}$ state is 0.25; the probability of obtaining two particles in the $\displaystyle{|1\rangle}$ state is 0.25; and the probability of obtaining one particle in the $\displaystyle{|0\rangle}$ state and the other in the $\displaystyle{|1\rangle}$ state is 0.5.

If A and B are identical bosons, then the composite system has only three distinct states: $\displaystyle{|0\rangle |0\rangle}$, $\displaystyle{ |1\rangle |1\rangle }$, and $\displaystyle{{\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle +|1\rangle |0\rangle)}$. When the experiment is performed, the probability of obtaining two particles in the $\displaystyle{|0\rangle}$ is now 0.33; the probability of obtaining two particles in the $\displaystyle{|1\rangle}$ state is 0.33; and the probability of obtaining one particle in the $\displaystyle{|0\rangle}$ state and the other in the $\displaystyle{|1\rangle}$ state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to “clump.”

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state $\displaystyle{{\frac {1}{\sqrt {2}}}(|0\rangle |1\rangle -|1\rangle |0\rangle)}$. When the experiment is performed, one particle is always in the $\displaystyle{|0\rangle}$ state and the other is in the $\displaystyle{|1\rangle}$ state.

The results are summarized in Table 1:

— Wikipedia on Identical particles

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# The square root of the probability, 4

Eigenstates 3.4

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quantum ~ classical with the indistinguishability of cases

— Me@2020-12-23 06:19:00 PM

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In statistical mechanics, a semi-classical derivation of the entropy that does not take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs who proposed this thought experiment in 1874‒1875. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the “mixing paradox”. If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, the paradox is averted.

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# Pointer state, 3

Eigenstates 3.3 | The square root of the probability, 3

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In calculation, if a quantum state is in a superposition, that superposition is a superposition of eigenstates.

However, real superposition does not just include eigenstates that make macroscopic senses.

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That is the major mistake of the many-worlds interpretation of quantum mechanics.

— Me@2017-12-30 10:24 AM

— Me@2018-07-03 07:24 PM

— Me@2020-12-18 06:12 PM

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Mathematically, a quantum superposition is a superposition of eigenstates. An eigenstate is a quantum state that is corresponding to a macroscopic state. A superposition state is a quantum state that has no classical correspondence.

The macroscopic states are the only observable states. An observable state is one that can be measured directly or indirectly. For an unobservable state, we write it as a superposition of eigenstates. We always write a superposition state as a superposition of observable states; so in this sense, before measurement, we can almost say that the system is in a superposition of different (possible) classical macroscopic universes.

However, conceptually, especially when thinking in terms of Feynman’s summing over histories picture, a quantum state is more than a superposition of classical states. In other words, a system can have a quantum state which is a superposition of not only normal classical states, but also bizarre classical states and eigen-but-classically-impossible states.

A bizarre classical state is a state that follows classical physical laws, but is highly improbable that, in daily life language, we label such a state “impossible”, such as a human with five arms.

An eigen-but-classically-impossible state is a state that violates classical physical laws, such as a castle floating in the sky.

For a superposition, if we allow only normal classical states as the component eigenstates, a lot of the quantum phenomena, such as quantum tunnelling, cannot be explained.

If you want multiple universes, you have to include not only normal universes, but also the bizarre ones.

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Actually, even for the double-slit experiment, “superposition of classical states” is not able to explain the existence of the interference patterns.

The superposition of the electron-go-left universe and the electron-go-right universe does not form this universe, where the interference patterns exist.

— Me@2020-12-16 05:18:03 PM

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One of the reasons is that a quantum superposition is not a superposition of different possibilities/probabilities/worlds/universes, but a superposition of quantum eigenstates, which, in a sense, are square roots of probabilities.

— Me@2020-12-18 06:07:22 PM

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# Logical arrow of time, 6.4.2

Logical arrow of time, 6.1.2

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The source of the macroscopic time asymmetry, aka the second law of thermodynamics, is the difference between prediction and retrodiction.

In a prediction, the deduction direction is the same as the physical/observer time direction.

In a retrodiction, the deduction direction is opposite to the physical/observer time direction.

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— guess —

If a retrodiction is done by a time-opposite observer, he will see the entropy increasing. For him, he is really making a prediction.

— guess —

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— Me@2013-10-25 3:33 AM

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A difference between deduction and observation is that in observation, the probability is updated in real time.

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each update time interval ~ infinitesimal

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In other words, when you observe a system, you get new information about that system in real time.

Since you gain new knowledge of the system in real time, the probability assigned to that system is also updated in real time.

— Me@2020-10-13 11:27:59 AM

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# Consistent histories, 8

Relationship with other interpretations

The only group of interpretations of quantum mechanics with which RQM is almost completely incompatible is that of hidden variables theories. RQM shares some deep similarities with other views, but differs from them all to the extent to which the other interpretations do not accord with the “relational world” put forward by RQM.

Copenhagen interpretation

RQM is, in essence, quite similar to the Copenhagen interpretation, but with an important difference. In the Copenhagen interpretation, the macroscopic world is assumed to be intrinsically classical in nature, and wave function collapse occurs when a quantum system interacts with macroscopic apparatus. In RQM, any interaction, be it micro or macroscopic, causes the linearity of Schrödinger evolution to break down. RQM could recover a Copenhagen-like view of the world by assigning a privileged status (not dissimilar to a preferred frame in relativity) to the classical world. However, by doing this one would lose sight of the key features that RQM brings to our view of the quantum world.

Hidden variables theories

Bohm’s interpretation of QM does not sit well with RQM. One of the explicit hypotheses in the construction of RQM is that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, the Bohmian view seems to imply an underlying, “absolute” set of states of all systems, which is also ruled out as a consequence of RQM.

We find a similar incompatibility between RQM and suggestions such as that of Penrose, which postulate that some processes (in Penrose’s case, gravitational effects) violate the linear evolution of the Schrödinger equation for the system.

Relative-state formulation

The many-worlds family of interpretations (MWI) shares an important feature with RQM, that is, the relational nature of all value assignments (that is, properties). Everett, however, maintains that the universal wavefunction gives a complete description of the entire universe, while Rovelli argues that this is problematic, both because this description is not tied to a specific observer (and hence is “meaningless” in RQM), and because RQM maintains that there is no single, absolute description of the universe as a whole, but rather a net of inter-related partial descriptions.

Consistent histories approach

In the consistent histories approach to QM, instead of assigning probabilities to single values for a given system, the emphasis is given to sequences of values, in such a way as to exclude (as physically impossible) all value assignments which result in inconsistent probabilities being attributed to observed states of the system. This is done by means of ascribing values to “frameworks”, and all values are hence framework-dependent.

RQM accords perfectly well with this view. However, the consistent histories approach does not give a full description of the physical meaning of framework-dependent value (that is it does not account for how there can be “facts” if the value of any property depends on the framework chosen). By incorporating the relational view into this approach, the problem is solved: RQM provides the means by which the observer-independent, framework-dependent probabilities of various histories are reconciled with observer-dependent descriptions of the world.

— Wikipedia on Relational quantum mechanics

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2020.09.27 Sunday ACHK

# Space and Causality

When we say that A and B are at different places at the same time, that implies that A and B have no causal relation[, at least for that moment of time].

— Me@2011.08.25

— Me@2020-09-06

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# Consistent histories, 6.3

observer ~ an entity that can get an consistent history

— Me@2017-07-28 08:20:23 PM

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# Consistent histories, 6.2

observer ~ a consistent description

— Me@2017-08-03 07:58:50 AM

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# Consistent histories, 7

In quantum mechanics, the consistent histories (also referred to as decoherent histories) approach is intended to give a modern interpretation of quantum mechanics, …

— Wikipedia on Consistent histories

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It needs to be decoherent in order to be consistent.

— Me@2017-08-08 01:25:54 PM

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decoherent ~ no quantum superposition

consistent ~ classical logic can apply

— Me@2020-05-30 03:52:15 PM

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# Logical arrow of time, 7.2

Microscopically, there is no time arrow.

— Me@2011.06.23

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No. There is weak force.

— Me@2011.07.22

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Myth: The arrow of time is a consequence of CP-symmetry violation.

The weak nuclear interactions violate the CP symmetry which is equivalent to saying that they violate the T symmetry. Is it the reason why eggs don’t unbreak? Of course not. There are two basic ways to see why. First, the weak interactions much like all other interactions preserve the CPT symmetry – there is extensive theoretical as well as experimental evidence supporting this assertion. And the CPT symmetry would be enough to show that eggs break as often as unbreak. More precisely, eggs break as often as mirror anti-eggs unbreak. ;-)

— Myths about the arrow of time

— Lubos Motl

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Weak force’s T-symmetry-violation has nothing to do with the time arrow.

In other words, microscopic time arrow has nothing to do with the macroscopic time arrow.

— Me@2020-03-21 07:56:01 PM

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About T-violation and the arrow of time: the simple answer is that the weak interactions are perfectly unitary, even if they are not T-invariant. They don’t affect the entropy in any way, so they don’t help with the arrow of time.

A bit more carefully: if you did want to explain the arrow of time using microscopic dynamics, you would have to argue that there exist more solutions to the equations of motion in which entropy grows than solutions in which entropy decreases. But CPT invariance is enough to guarantee that that’s not true. For any trajectory (or ensemble of trajectories, or evolution of a distribution function) in which the entropy changes in one way, there is another trajectory (or set…) in which the entropy changes in precisely the opposite way: the CPT conjugate. Such laws of physics do not in and of themselves pick out what we think of as the arrow of time.

People talk about the “arrow of time of the weak interactions,” but ask yourself: in which direction does it point? There just isn’t any direct relationship to entropy.

— Sean Carroll

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# Superposition always exists, 2

Decoherence means that the different components in the superposition do not interact with each other, but it does not mean that the components separate to form different macroscopic realities.

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Just like when a 100-soldier army’s marching gets interrupted, the decoherent soldiers do not form a single army anymore, because their actions become out of sync.

However, they do not become 100 armies either.

Instead, they form a group of 100 random people in the street.

Although now they are out of sync with each other, all original soldiers still exist, forming the (new) average result; all or most of them have become part of the environment.

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But it is an analogy only. It has an important distinction.

In quantum superposition, we discuss the relationships between different component states of the superposition. Those states exist not in physical space, but in a mathematical space.

In the army analogy, we discuss the relationships between the actions between different material items (solders in this case). Those material items exist in physical space.

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The unselected eigenstates do not cooperate with other particles to form macroscopic realities.

Although the spirit of the statement is correct, the statement itself is incorrect in multiple senses.

First, an eigenstate is a quantum state. It interferes with other eigenstates, not other particles.

Second, although the “unselected” eigenstates seem to disappear, they actually still exist; they entangles with the environment, which includes the apparatus and measurement devices of that experiment.

— Me@2013.01.01

— Me@2020-02-26 06:49:46 AM

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In “decoherence means that the different components do not interact with each other”, the meaning of “interact” is not defined yet.

The word should probably be “interfere”, instead of “interact”.

— Me@2020-02-25 10:44:23 PM

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interference ~ superposition with pattern

Decoherence means that the phase differences between different components in a superposition are not constants anymore. It does not mean that there is no superposition anymore.

Superposition is always there.

What disappears is the interference pattern, not the superposition.

— Me@2019-09-20 06:48:55 AM

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# Black hole information paradox, 4

So we seem to have a direct contradiction between [QM and unitarity] and [GR and causality]. Both of these principles, unitarity and causality, cannot be exactly correct because a contradiction arises from their explosive mixture.

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As most quantum mechanicians have known from the very beginning, it is the unitarity, a principle of quantum mechanics, that wins in the battle and remains universally valid.

On the other hand, causality becomes an approximate principle that is only valid in the limit of infinitely large causal diamonds. In the presence of black holes, the internal causal structure is modified by quantum phenomena and the information can “tunnel” out of the black hole.

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It shouldn’t be so surprising that unitarity survives completely while causality doesn’t. After all, the basic postulates of quantum mechanics, including unitarity, the probabilistic interpretation of the amplitudes, and the linearity of the operators representing observables, seem to be universally necessary to describe physics of any system that agrees with the basic insights of the quantum revolution.

On the other hand, geometry has been downgraded into an effective, approximate, emergent aspect of reality. The metric tensor is just one among many fields in our effective field theories including gravity. In string theory, there are, in some sense, infinitely many such fields besides the metric tensor – the whole “stringy tower”. The metric tensor doesn’t have to exist as a good degree of freedom at the Planck scale or in other extreme conditions. We know many other fields that are only good enough at low energies – e.g. the pion field.

— Black hole information puzzle

— Lubos Motl

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2020.02.09 Sunday ACHK

# Spacetime interval

Two contrasting viewpoints on time divide prominent philosophers.

One view is that time is part of the fundamental structure of the universe – a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is sometimes referred to as Newtonian time.

The opposing view is that time does not refer to any kind of “container” that events and objects “move through”, nor to any entity that “flows”, but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events.

This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, and thus is not itself measurable nor can it be travelled.

— Wikipedia on Time

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Special relativity declares a similar law for all motion: the combined speed of any object’s motion through space and its motion through time is always precisely equal to the speed of light.

— The Fabric of the Cosmos: Space, Time, and the Texture of Reality

— Brian Greene

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Space is relative, in the sense that the space interval, $\Delta d$, (aka distance) between two events can have different values for different observers.

\displaystyle{ \begin{aligned} \Delta {d} &= \sqrt{{\left(\Delta {x}\right)}^{2}+{\left(\Delta {y}\right)}^{2}+{\left(\Delta {z}\right)}^{2}} \\ \end{aligned} }

Time is relative, in the sense that the time interval, $\Delta t$, (aka duration) between two events can have different values for different observers.

Spacetime is absolute, in the sense that the spacetime interval, $(\Delta s)^2$, between two events cannot have different values for different observers.

\displaystyle{ \begin{aligned} (\Delta s)^{2} &= (\Delta ct)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2} \\ &= (\Delta ct)^{2}-(\Delta d)^{2} \\ \end{aligned} }

— paraphrasing The Fabric of the Cosmos

— Me@2020-01-26 12:46:41 AM

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# Frequency probability and Bayesian probability, 2.2

The probability frequentist vs Bayesian debate can be transcended by realizing that an observer cannot be separated from the observed.

— Me@2011.07.23

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# Compare results

Within a universe, any two observers can, at least in principle, compare results.

When they compare, they will have consistent results for any observables/measurables.

— Me@2018-02-06 08:48:24 PM

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being entangled ~ being consistent, with respect to any two observers, when they compare the results

— Me@2018-02-05 10:11:45 PM

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# Classical physics

Quantum Mechanics 6

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As Gene and Sidney Coleman have pointed out, the term “interpretation of quantum mechanics” is a misnomer encouraging its users to generate logical fallacies. Why? It’s because we should always use a theory, or a more accurate, complete, and universal theory, to interpret its special cases, to interpret its approximations, to interpret the limits, and to interpret the phenomena it explains.

However, there’s no language “deeper than quantum mechanics” that could be used to interpret quantum mechanics. Unfortunately, what the “interpretation of quantum mechanics” ends up with is an attempt to find a hypothetical “deeper classical description” underneath the basic wheels and gears of quantum mechanics. But there’s demonstrably none. Instead, what makes sense is an “interpretation of classical physics” in terms of quantum mechanics. And that’s exactly what I am going to focus in this text.

Plan of this blog entry

After a very short summary of the rules of quantum mechanics, I present the widely taught “mathematical limit” based on the smallness of Planck’s constant. However, that doesn’t really fully explain why the world seems classical to us. I will discuss two somewhat different situations which however cover almost every example of a classical logic emerging from the quantum starting point:

1. Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)

2. Decoherence which makes us interpret absorbed particles as point-like objects and which makes generic superpositions of macroscopic objects unfit for well-defined questions about classical facts

— How classical fields, particles emerge from quantum theory

— Lubos Motl

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There is no interpretation problem for quantum mechanics. Instead, if there is a problem, it should be the interpretation of classical mechanics problem.

— Lubos Motl

— paraphrased

— Me@2011.07.28

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# Pointer state, 2

Eigenstates 3.2

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Microscopically, a state can be definite or indefinite. Even if it is indefinite, the overlapping of superpositions of states of a lot of particles, or the superposition of a lot of system-microstates gives a definite macrostate.

If a state is definite, it is corresponding to one single system-macrostate directly.

I am referring to the physical definition, not the mathematical definition.

— Me@2012-12-31 09:28:08 AM

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If a microstate is definite, it is called an “eigenstate”. It is corresponding to one single system-macrostate directly.

However, the microstate is NOT the macrostate. The microstate is just corresponding to that macrostate.

— Me@2019-09-20 07:02:10 AM

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In quantum Darwinism and similar theories, pointer states are quantum states, sometimes of a measuring apparatus, if present, that are less perturbed by decoherence than other states, and are the quantum equivalents of the classical states of the system after decoherence has occurred through interaction with the environment. ‘Pointer’ refers to the reading of a recording or measuring device, which in old analog versions would often have a gauge or pointer display.

— Wikipedia on Pointer state

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In quantum mechanics, einselections, short for environment-induced superselection, is a name coined by Wojciech H. Zurek for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions.

In this approach, classicality is described as an emergent property induced in open quantum systems by their environments. Due to the interaction with the environment, the vast majority of states in the Hilbert space of a quantum open system become highly unstable due to entangling interaction with the environment, which in effect monitors selected observables of the system.

After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into an uncertain [in the sense of classical probability] state which can be decomposed into a mixture of simple pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of pure superpositions of pointer states. These ‘pointer states’ are stable despite environmental interaction. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition.

Advocates of this approach argue that since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers). However, the basic program has been criticized as relying on a circular argument (e.g. R. E. Kastner). So the question of whether the ‘einselection’ account can really explain the phenomenon of wave function collapse remains unsettled.

— Wikipedia on Einselection

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Here I simply review the basic approach to ‘deriving’ einselection via decoherence, and point to a key step in the derivation that makes it a circular one.

— Ruth E. Kastner

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We should not derive einselection via decoherence. Instead, they should be regarded as different parts or different presentations of the same theory.

In other words, “einselection” and “decoherence” are synonyms.

— Me@2019-09-21 05:53:53 PM

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There has been significant work on correctly identifying the pointer states in the case of a massive particle decohered by collisions with a fluid environment, often known as collisional decoherence. In particular, Busse and Hornberger have identified certain solitonic wavepackets as being unusually stable in the presence of such decoherence.

— Wikipedia on Einselection

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