Download Report on “A system`s wave function is uniquely determined by its

Report on “A system’s wave function is uniquely
determined by its underlying physical state”
Roger Colbeck, Renato Renner
by Sergiu Irimie
1. Introduction
The article tries to answer a question that dates back to the beginning of quantum
theory, is the wave function the complete way of describing the reality or it is just a
component of a bigger reality? Before going into technical details, I would like to present
briefly what is wave function and why it is so important.
Since the beginning of modern science, physicist have tried to define and formalize
the world based on the observations they made. With each new experiment, new rules have
been described and combined with previous knowledge, giving new theories and
improving everyday life. One of the most notable discoveries are the Newton’s laws of
classical mechanics.
This evolutionary trend continued until the beginning of the 20th century, when
some experiments didn’t fit the basic physical assumptions seen do far. Physicist realized
in order to explain the new results, some previous knowledge should be rejected and they
had to come with something else. This new set of rules are at the core of quantum theory.
It was observed that classical laws did not apply to the world of particles. One of
the most notable result is the “double slit” experiment. To describe shortly the setup, there
is an electron gun, a wall with two slits and a back wall. When the electrons were launched,
two situations occurred, depending on weather the experiment was “watched” or not. When
the experiment was watched (i.e. measured), the pattern was as expected, two lines on the
back wall. The strange thing happened when nobody looked at the experiment, when the
electrons appeared as a set of straight lines, like an interference pattern. Somehow, the
electrons were either particles or waves.
The first to clearly formalize this behavior was Erwin Schroedinger with its famous
wave equation. Since then, the quantum state of any system of particles is described by the
wave function. While these discoveries have produced a shift in the way we see the world
with many philosophical implications, the aforementioned article, focuses on proving that
the wave function is indeed an objective way of describing the reality and it is determined
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by a complete description of a system’s physical state, under some assumptions that will
be discussed in the next paragraphs.
2. Motivation
The wave function has a probabilistic nature. There are two rules that govern its behavior,
the superposition principle and the measurement rule. The most common interpretation is that
the wave function describes all the possible states a quantum system can be in, which exist in
superposition. When a measurement is made, the physical state would collapse into only one
state, with a certain probability. Exactly this idea, pose questions on the real nature of the wave
function.
On one side, it is considered to be an objective description of a physical system. Since it
corresponds to a complete state, maximally informative, it is considered an element of reality
that exists before any measurement. On the other side, because of its probabilistic nature, it is
viewed as an incomplete knowledge of an underlying reality.
Therefore, this dual interpretation of wave function has been a long debate in the scientific
community since the early days of quantum theory. Einstein, Podolsky and Rosen (EPR) stated
that even with a complete description of the wave function, one will not be able to predict the
outcomes of the measurements with certainty. They believed that a more comprehensive theory
should exist where the wave function has a counterpart and that would represent a complete
description of reality. The objective nature was supported by Schroedinger and is also debated
here.
Since the way we view the wave function, either objective or subjective, changes the way
one see the world and influences future experiments, it is a hot topic in the field and a good
motivation for the article.
3. Related work
Going into more technical details, the article tries to answer the following question: given
a complete set of variables (i.e. complete means that nothing can be added to the system to
increase the outcomes of the measurements) would a wave function that describe this system
be unique? If so, then the wave function depends on the variables and since they represent
all the possible states of the system, it will be a complete, objective description of the system.
The article is based on some previous work of the same authors. They proved in [2] that
the wave function, denoted from now one by , is completely defined by set if and only if
is itself complete. In [4] they proved is complete in any framework where the notion of
free choice is considered and the measurements done are unitaries of an extended system.
The current work proves that a less restrictive framework can be considered, in order to
prove the uniqueness of . They get rid of the unitary assumption and maintain only the free
choice constraint.
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4. Main contributions
All the assumptions and results are formalized into a theorem, called “the uniqueness
theorem”. The proof is quite mathematical and is based on several lemmas that are shown in
the article’s appendices. I will present here only the main proof and I will state the required
results.
Firstly, the authors use the bra-ket notation of the quantum state. Since the wave function
can take various states with certain probabilities, it will be modeled by a random variable,
denoted also by . It takes as values unit vectors in a Hilbert space H.
The physical setup is presented in image below. A source emits a particle that decays
into two, each newly created particle being measured by a different device. The measurements
depend by parameters A, B and give the outcomes X, Y. U is an isometry (i.e. a distance
preserving injective map between metric spaces) from H to HA HB that represents the decay.
An important observation, the quantum state is pure, and hence defined by a wave function,
and not by a mixed distribution of states.
The law which gives the probability that a measurement would yield a given result is
expressed by the Born law. In our case, the joint probability of X and Y given relevant
parameters is: PXY|ABΨ(x,y|a,b,ψ) =
with
families of projectors
on HB and similar for HA.
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.
The physical state would be described by an additional random variable which does not
need to have any particular structure (i.e. it can be a list of values). Having these mentioned,
we will define the notions of free choice and completeness.
In order to understand these ideas, one must specify the experiments take place in a
chronological structure where we define the preorder relation
. In this case, A
B means
that X lies in the future of A (i.e. A occurs earlier in time than the generation of X). In our
setting we have:
. So, a random variable is
free of choice, if it is correlated only with variables it may cause. Formally, P A|A* = PA where
A* is the set of all random variables Z such that A Z.
The completeness is defined in a similar way. A variable is complete if
where
is the set of random variables Z such that
and
for
. On short,
completeness implies that predictions about future experiments cannot be improved by taking
into account information from the past.
Now, we can formulate the uniqueness theorem.
Theorem Let Λ and Ψ be random variables and assume that the support of Ψ contains
two wave functions, ψ and ψ’, with
. If for any isometry U and measurements
and
parametrized by a ϵ 𝒜 and b ϵ ℬ, there exist random variables A, B, X and Y such
that
1. PXY|ABΨ satisfies the Born rule
2. A and B are free choices from 𝒜 and ℬ w.r.t chronological order
3. Λ is complete w.r.t chronological order
then there exists a subset ℒ of the range 𝛬 such that PΛ|Ψ (ℒ|𝜓) = 1 and PΛ|Ψ (ℒ|𝜓’) = 0
The theorem states that the wave function Ψ is a function of Λ, since for different values
of Ψ the values taken by Λ are different.
The proof uses specific wave functions and a couple of results that would be stated next.
4.1.
Additional results
4.1.1. Lemma 1 For any 0 ≤ 𝛼 < 1 there exists k, d 𝜖 ℕ with k < d and 𝜉 𝜖[0,1] such
that the vectors 𝜙 𝑎𝑛𝑑 𝜙 ′ defined by (1) and (2) have overlap
(1)
(2)
where
is an orthonormal basis of Hilbert space H.
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4.1.2. Definition 1 The projective measurements are defined by (3) and (4)
(3)
(4)
for any n 𝜖 ℕ where
is the generalized Pauli operator,
a 𝜖 𝒜 n = {0,2,4, …,2n-2}, b 𝜖 ℬ n = {1,3,5, …,2n-1}, x, y 𝜖 {0,…,d-1} with
outcomes in 𝒳 d = {0,…,d-1}.
Note:
4.1.3. Definition 2 The correlation between outcomes X and Y will be correlated by
the linear function (5)
(5)
which is upper bounded by
.
(6)
4.1.4. Lemma 2 Let PXYABΛ be a distribution that satisfies PXΛ|AB = PXΛ|A, PYΛ|AB = PYΛ|B
and PABΛ = PA x PB x PΛ with supp(PA) ⊇ 𝒜 n and supp(PB) ⊇ ℬ n . Then
(7)
4.1.5. Lemma 3 For any unit vectors 𝜓, 𝜓 ′ 𝜖 H1 and 𝜙, 𝜙 ′ 𝜖 𝐻2 where dimH1 ≤
𝑑𝑖𝑚H2 and
, there exists an isometry U: H1 → H2 such that
U 𝜓 = 𝜙 and U 𝜓′ = 𝜙′.
4.2.
Main proof
Given the premises of Lemma 1 and Lemma 3, there exists an isometry U that
corresponds to the particle decay. Besides that, consider n 𝜖 ℕ and let A, B, X, Y be random
variables that satisfy the three conditions of the theorem.
According to Born rule, the distribution
corresponds to the one in (5).
In order to apply Lemma 2, one needs to fulfill its conditions. By freedom of choice
we have
,
and
Thus, using (6) in (7) we get:
(8)
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By considering only the kth term of (8) (k < d) and n close to ∞ we have:
(9)
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Let ℒ be the set of all elements λ ϵ Λ for which
= 𝑑. Equality (9) is
true only for λ ϵ ℒ , such that we consider Λ = ℒ. Since ℒ is a sample space, we infer the
first claim of the theorem,
which is true because the probability of a
sample space is always 1.
Next, we want to make the transition to the second wave function 𝜓′and to prove that
. The second probability of equation (9) is conditioned by 𝜓, which is an
element of the past for Λ. By completeness condition, all probabilities (based on Λ) for
elements in the future of Λ are not influenced by past events of Λ. Therefore the following
equality holds true, since both wave functions are in the past of Λ.
(10)
Now we write the definition of a continuous random variable (11), using free of choice
condition (i.e.
)
(11)
Using relation (11) in our case and
, we get:
(12)
Let
be an indicator function defined by (13).
(λ) = {
1 𝑖𝑓 𝜆 ∈ ℒ
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(13)
Now we use (10) and (13) with (12) and we obtain (14).
(14)
Inequality (14) is true since the indicator function is 0 for 𝜆 ∉ ℒ.
According to Born rule, probability in relation (14) will have the following form:
𝑃𝑋|𝐴Ψ (𝑘|0, 𝜓′ ) = ⟨𝜓 ′ |𝑈 𝑡 Π𝑘0 𝑈|𝜓′⟩ = ⟨𝜓′ |𝑈 𝑡 |𝑘⟩⟨𝑘|𝑈|𝜓′⟩
= ⟨𝜙′|𝑘⟩⟨𝑘|𝜙′⟩ according to Lemma 3
(15)
Since k < d, the dot product in (15) is 0 and so, using (14) we get to the conclusion:
∎
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5. Conclusion
The article proves that the quantum wave function is a an element of reality of a system
considering two assumptions, the correctness and the freedom of choice of the measurement
setup. Technical considerations shows that values taken by any complete variable Λ
representing the physical state, are different for different forms of the wave function Ψ. This
can happen only when there is a relation between the two.
All three assumptions are needed to prove the result. Without Born rule, the wave function
could be taken independently of the outcomes X or Y. The completeness assumption assure
the link with reality (otherwise the wave function would encode only a part of the physical
state). Last, but not least, without the freedom of choice, measurements A and B could
be pre-determined by the list Λ, when Ψ would not be unique.
This article answered a question that exists from the beginning of the quantum theory. Even
so, the authors made a couple of assumption that can narrow its generality. An interesting fact
would be to obtain the same result using a setting with a single measurement. Also, the wave
functions have predefined forms and depend on some parameters chosen subjectively.
In my opinion, the choices made by the authors can be seen as a subjective influence over
the proof. Even if the setup tries to be as objective as possible this small constraints puts a
question mark over the generality of the result.
6. References
[1] R. Colbeck and R. Renner, A system’s wave function is uniquely determined by its
underlying physical state, arXiv:1312.7353 (2013).
[2] R. Colbeck and R. Renner, Is a system's wave function in one-to-one correspondence
with its elements of reality?, Phys. Rev. Lett. 108, 150402 (2012).
[3] R. Colbeck and R. Renner, A short note on the concept of free choice, arXiv:1302.4446
(2013).
[4] R. Colbeck and R. Renner, No extension of quantum theory can have improved predictive
power, Nat. Commun. 2, 411 (2011).
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