Quantum Measurement and the Measurement Problem
State Reduction, Probability, and Interpretation
Quantum Measurement and the Measurement Problem: Overview and Plan
In the previous section, we introduced quantum states as vectors in Hilbert space and observables as Hermitian operators. We also learned how the expansion of a state in the eigenbasis of an observable determines the probabilities for the possible outcomes of a measurement.
But this leaves an important question unanswered:
What actually happens to the quantum state when a measurement is made?
This question exposes one of the most unusual features of quantum mechanics.
Between measurements, the state evolves continuously and deterministically according to the Schrödinger equation. A measurement, however, generally produces one of several possible outcomes probabilistically, and the state used to describe the system after the measurement is different from the state before the measurement.
In the traditional language of quantum mechanics, the state is said to collapse onto an eigenstate of the measured observable.
Whether this collapse represents a real physical process, an update of our knowledge, or only an apparent effect arising from a more complete description is the subject of the quantum measurement problem. Different interpretations of quantum mechanics answer this question differently, while agreeing on most ordinary experimental predictions.
For the calculations in this course, we do not need to settle this interpretational question. We do, however, need to understand the operational consequences of measurement—most importantly, that a measurement generally changes the state and therefore affects subsequent measurements.
The specific goals for this chapter are:
- Review how measurement probabilities follow from the expansion of a state in an eigenbasis.
- Understand how an ideal quantum measurement changes the state.
- Distinguish deterministic Schrödinger evolution from the probabilistic measurement postulate.
- Introduce the quantum measurement problem.
- Compare several important interpretations and approaches to quantum measurement.
- Understand which parts of the measurement formalism are experimentally established and which depend on interpretation.
- Establish the operational measurement rules that we will use throughout this course.
- Prepare for the discussion of successive measurements and compatible observables in the next section.
1. Measurement and the Quantum State
1.1 Measurement probabilities
Consider an observable represented by the Hermitian operator \(\hat A\).
Its eigenstates satisfy
\[ \hat A\left\lvert a_n \right\rangle= a_n\left\lvert a_n \right\rangle. \]
For simplicity, we first assume that the eigenvalues are discrete and nondegenerate.
A general state can be expanded in the eigenbasis of \(\hat A\):
\[ \boxed{ \left\lvert \psi \right\rangle= \sum_n c_n\left\lvert a_n \right\rangle. } \]
Because the eigenstates form an orthonormal basis,
\[ c_n = \left\langle a_n \middle| \psi \right\rangle. \]
As discussed in the previous section on measurement probabilities, the Born rule states that a measurement of \(A\) produces the result \(a_n\) with probability
\[ \boxed{ P(a_n) = |c_n|^2 = \left| \left\langle a_n \middle| \psi \right\rangle\right|^2. } \]
Quantum mechanics therefore does not, in general, predict the result of an individual measurement.
Instead, it predicts the probabilities of the possible outcomes.
1.2 The state after a measurement
Suppose the initial state is
\[ \left\lvert \psi \right\rangle= \sum_n c_n\left\lvert a_n \right\rangle, \]
and a measurement of \(A\) yields the result \(a_k\).
For subsequent predictions, the state is then taken to be
\[ \boxed{ \left\lvert \psi \right\rangle\longrightarrow \left\lvert a_k \right\rangle. } \]
This change is traditionally called wave-function collapse or state reduction.
The second name is more general: the abstract state \(\left\lvert \psi \right\rangle\) changes, not merely its position-space wave function.
The measurement therefore does more than reveal one of the possible values of \(A\). It also changes the state used to describe the system.
A simple example
Suppose
\[ \left\lvert \psi \right\rangle= \frac{1}{\sqrt{2}} \left( \left\lvert a_1 \right\rangle+ \left\lvert a_2 \right\rangle\right). \]
A measurement of \(A\) gives
\[ P(a_1)=P(a_2)=\frac12. \]
Before the measurement, neither outcome is predicted with certainty.
If the measurement gives \(a_1\), however, the state afterward is
\[ \left\lvert \psi \right\rangle\longrightarrow \left\lvert a_1 \right\rangle. \]
An immediate repetition of the same ideal measurement then gives
\[ P(a_1)=1. \]
The first measurement has therefore changed the quantum state.
If the measured eigenvalue is degenerate, a measurement does not necessarily select one unique eigenvector.
Suppose several orthonormal states
\[ \left\lvert a,\mu \right\rangle\]
share the same eigenvalue \(a\):
\[ \hat A\left\lvert a,\mu \right\rangle= a\left\lvert a,\mu \right\rangle. \]
The projector onto this eigenspace is
\[ \hat P_a = \sum_\mu \left\lvert a,\mu \right\rangle\!\!\left\langle a,\mu \right\rvert. \]
For an ideal projective measurement yielding \(a\), the post-measurement state is
\[ \left\lvert \psi \right\rangle\longrightarrow \frac{ \hat P_a\left\lvert \psi \right\rangle}{ \sqrt{ \left\langle \psi \middle| \hat P_a \middle| \psi \right\rangle} }. \]
This is known as the Lüders rule.
For a nondegenerate eigenvalue,
\[ \hat P_a=\left\lvert a \right\rangle\!\!\left\langle a \right\rvert, \]
and this reduces to the familiar statement that the state collapses onto \(\left\lvert a \right\rangle\).
1.3 Deterministic evolution and probabilistic measurement
The Schrödinger equation determines the time evolution of a state:
\[ i\hbar \frac{d}{dt} \left\lvert \psi(t) \right\rangle= \hat H\left\lvert \psi(t) \right\rangle. \]
If the state is known at some initial time, this equation determines the state at all later times.
The evolution is therefore deterministic.
A measurement is different.
From
\[ \left\lvert \psi \right\rangle= \sum_n c_n\left\lvert a_n \right\rangle, \]
quantum mechanics predicts the probabilities
\[ P(a_n)=|c_n|^2, \]
but the standard measurement postulate does not predict which particular \(a_n\) will occur in an individual measurement.
Within the usual operational formulation of quantum mechanics, the individual outcome is therefore probabilistic.
This gives us two apparently very different rules for how a quantum state changes:
\[ \boxed{ \begin{array}{lll} \text{Schrödinger evolution:} & \left\lvert \psi(t_0) \right\rangle\longrightarrow \left\lvert \psi(t) \right\rangle& \text{continuous and deterministic}, \\[1em] \text{measurement:} & \displaystyle \sum_n c_n\left\lvert a_n \right\rangle\longrightarrow \left\lvert a_k \right\rangle& \text{probabilistic}. \end{array} } \]
The coexistence of these two kinds of evolution lies at the heart of the quantum measurement problem.
2. The Quantum Measurement Problem
2.1 Where does collapse come from?
To see the measurement problem more clearly, let us try to describe both the quantum system and the measuring apparatus using quantum mechanics.
Suppose we want to measure an observable \(\hat A\) with eigenstates
\[ \hat A\left\lvert a_n \right\rangle= a_n\left\lvert a_n \right\rangle. \]
Let
\[ \left\lvert M_0 \right\rangle\]
represent the initial “ready” state of the measuring apparatus, and let
\[ \left\lvert M_n \right\rangle\]
represent a state of the apparatus in which it records the result \(a_n\).
For an ideal measurement, we want the measurement interaction to correlate each eigenstate of the system with the corresponding state of the apparatus:
\[ \boxed{ \left\lvert a_n \right\rangle\left\lvert M_0 \right\rangle\longrightarrow \left\lvert a_n \right\rangle\left\lvert M_n \right\rangle. } \]
For example,
\[ \left\lvert a_1 \right\rangle\left\lvert M_0 \right\rangle\longrightarrow \left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle\]
and
\[ \left\lvert a_2 \right\rangle\left\lvert M_0 \right\rangle\longrightarrow \left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle. \]
So far, there is no difficulty. If the system begins in an eigenstate of the measured observable, the apparatus simply becomes correlated with the corresponding result.
What if the system begins in a superposition?
Now suppose the system is initially in the superposition
\[ \left\lvert \psi \right\rangle= c_1\left\lvert a_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle. \]
The combined initial state of system and apparatus is therefore
\[ \left( c_1\left\lvert a_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle\right) \left\lvert M_0 \right\rangle. \]
The crucial point is that the Schrödinger equation is linear.
If
\[ \left\lvert a_1 \right\rangle\left\lvert M_0 \right\rangle\longrightarrow \left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle\]
and
\[ \left\lvert a_2 \right\rangle\left\lvert M_0 \right\rangle\longrightarrow \left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle, \]
then linearity requires
\[ \boxed{ \left( c_1\left\lvert a_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle\right) \left\lvert M_0 \right\rangle\longrightarrow c_1\left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle. } \]
The system and apparatus have become entangled.
But notice what has not happened.
Ordinary unitary time evolution has not produced either
\[ \left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle\]
or
\[ \left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle. \]
Instead, it has produced a superposition containing both alternatives:
\[ c_1\left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle. \]
Yet when we actually read the measuring apparatus, we observe one definite result: the apparatus says \(a_1\), or it says \(a_2\).
We do not ordinarily experience a measuring device as being in a coherent superposition of macroscopically different readings.
This is the central tension:
\[ \boxed{ \text{unitary Schrödinger evolution} \quad\longrightarrow\quad \text{superposition of outcomes}, } \]
whereas
\[ \boxed{ \text{measurement} \quad\longrightarrow\quad \text{one observed outcome}. } \]
The Schrödinger equation by itself does not contain a rule that selects one particular term,
\[ \left\lvert a_k \right\rangle\left\lvert M_k \right\rangle, \]
from the superposition.
This is the essence of the quantum measurement problem:
How does a quantum superposition of possible outcomes lead to the single definite outcome that we observe?
This is where the interpretations differ
Different interpretations of quantum mechanics agree on the unitary evolution above but disagree about what happens—or what it means—when a definite outcome is observed.
In Copenhagen-type interpretations, measurement is treated as a special process. Once an outcome \(a_k\) is observed, the state is reduced to the corresponding state,
\[ \left\lvert \psi \right\rangle\longrightarrow \left\lvert a_k \right\rangle. \]
Collapse is therefore included as part of the measurement postulate rather than derived from the Schrödinger equation.
In Everettian or Many-Worlds interpretations, there is no fundamental collapse. The complete state
\[ c_1\left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle\]
continues to exist and evolve unitarily. The different terms correspond to effectively distinct branches in which different outcomes are recorded.
In Bohmian mechanics, the wave function may retain the different branches, but the system and apparatus possess an actual configuration. That configuration lies in one branch, producing one definite observed outcome. Collapse is therefore effective rather than fundamental.
In objective-collapse theories, the unitary Schrödinger evolution is not exact. The dynamics are modified so that sufficiently different alternatives can undergo a real stochastic collapse into one outcome.
Decoherence shows how interaction with the environment rapidly suppresses observable interference between different macroscopic outcomes. If environmental states \(\left\lvert E_1 \right\rangle\) and \(\left\lvert E_2 \right\rangle\) become correlated with the alternatives,
\[ c_1\left\lvert a_1 \right\rangle\left\lvert M_1 \right\rangle\left\lvert E_1 \right\rangle+ c_2\left\lvert a_2 \right\rangle\left\lvert M_2 \right\rangle\left\lvert E_2 \right\rangle, \]
the two branches can become effectively unable to interfere.
Decoherence therefore explains why macroscopic alternatives behave approximately like classical alternatives. However, the total state is still a superposition, so decoherence by itself does not necessarily explain why one unique outcome is experienced.
Thus, the different interpretations do not disagree about the basic calculation that produces the entangled state. They disagree about how that state should be understood and how definite measurement outcomes arise from it.
Many interpretations of standard quantum mechanics are deliberately constructed to reproduce the same experimental probabilities.
For example, Copenhagen-type interpretations, Everettian quantum mechanics, and Bohmian mechanics in quantum equilibrium generally reproduce the ordinary predictions of nonrelativistic quantum mechanics.
Experiments that merely confirm the usual Born-rule probabilities therefore do not distinguish among them.
This does not mean that every proposal concerning quantum measurement is experimentally indistinguishable.
Objective-collapse models, for example, modify the ordinary quantum dynamics and can in principle predict small deviations from standard quantum mechanics. Such models are therefore subject to experimental tests.
Decoherence is somewhat different again: it is not simply an interpretation but a physical process arising from entanglement with uncontrolled environmental degrees of freedom, and its effects are experimentally observable.
The philosophical interpretations may differ, but the operational rules used for the calculations in this course are the same.
2.2 What we will use
For the purposes of this course, we will use the standard operational measurement rules:
- An observable \(A\) is represented by a Hermitian operator \(\hat A\).
- The possible measurement outcomes are its eigenvalues.
- The probability for a particular outcome follows from the Born rule.
- After an ideal measurement yielding \(a_n\), the state used for subsequent predictions lies in the corresponding eigenspace.
We do not need to choose a particular interpretation of quantum mechanics in order to use these rules.
What matters for our subsequent calculations is that measurement changes the state.
3. From One Measurement to the Next
Suppose we prepare a system in some state \(\left\lvert \psi \right\rangle\) and measure an observable \(\hat A\).
If the result is a nondegenerate eigenvalue \(a_n\), the state used for subsequent predictions becomes
\[ \left\lvert a_n \right\rangle. \]
What happens if we now measure a different observable \(\hat B\)?
Will the second measurement leave the value of \(A\) undisturbed?
If we then measure \(A\) again, are we guaranteed to recover the original result \(a_n\)?
The answer depends on the relationship between the eigenstates of \(\hat A\) and \(\hat B\).
For some pairs of observables, both quantities can possess definite values simultaneously. For others, measuring one generally destroys the definite value established by a previous measurement of the other.
This distinction leads to the concepts of
- compatible and incompatible observables,
- commutators, and
- ultimately the Heisenberg uncertainty principle.
We turn to these questions in the next section.