Quantum indeterminacy
Encyclopedia
Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system
, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that (a) a physical system had a determinate state
which uniquely determined all the values of its measurable properties, and conversely (b) the values of its measurable properties uniquely determined the state. Albert Einstein
may have been the first person to carefully point out the radical effect the new quantum physics would have on our notion of physical state.
Quantum indeterminacy can be quantitatively characterized by a probability distribution
on the set of outcomes of measurements
of an observable
. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.
Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors
in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.
and quantum measurement continues to be an active research area in both theoretical and experimental physics. Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann
. The kind of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of projection-valued measure
s for self-adjoint operator
s which had been recently developed (by von Neumann and independently by Marshall Stone) and the Hilbert space formulation of quantum mechanics
(attributed by von Neumann to Paul Dirac
).
In this formulation, the state of a physical system corresponds to a vector of length 1 in a Hilbert space
H over the complex number
s. An observable is represented by a self-adjoint (i.e. Hermitian) operator A on H. If H is finite dimensional, by the spectral theorem
, A has an orthonormal basis
of eigenvectors. If the system is in state ψ, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value λ will be the corresponding eigenvalue of the equation A e = λ e. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ψ. The probability is
where E(λ) is the projection onto the space of eigenvectors of A with eigenvalue λ.
(such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space C2, with each quantum state corresponding to a unit vector in C2 (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.
The Pauli spin matrices
are self-adjoint
and correspond to spin-measurements along the 3 coordinate axes.
The Pauli matrices all have the eigenvalues +1, −1.
Thus in the state
σ1 has the determinate value +1, while measurement of σ3 can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both σ1 and σ3 have determinate values.
There are various questions that can be asked about the above indeterminacy assertion.
Von Neumann formulated the question 1) and provided an argument why the answer had to be no, if one accepted the formalism he was proposing. However according to Bell, von Neumann's formal proof did not justify his informal conclusion. A definitive but partial negative answer to 1) has been established by experiment: because Bell's inequalities are violated, any such hidden variable(s) cannot be local (see Bell test experiments
).
The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the observer effect
, which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) which measures exclusively σ1 and (B) which measures only σ3 of a spin system in the
state ψ. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with equal probability.
The units involved in quantum uncertainty are on the order of Planck's constant (found experimentally to be 6.6 x 10−34 J·s).
, in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable properties (observable
s) has a determinate (sharp) value. The values of an observable will be obtained non-deterministically in accordance with a probability distribution which is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.
This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ1 as a filter which retains only those particles such that σ1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ.
However, Einstein did believe that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein, Boris Podolsky
and Nathan Rosen
did show that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas:
This failure of the classical view was one of the conclusions of the EPR thought experiment
in which two remotely located observers
, now commonly referred to as Alice and Bob
, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was a conclusion of EPR, using the formal apparatus of quantum theory, that once Alice measured spin in the x direction, Bob's measurement in the x direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. From this it follows that either value of spin in the x direction is not an element of reality or that the effect of Alice's measurement has infinite speed of propagation.
the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:
Let A be an observable of a quantum mechanical system. A is given by a densely
defined self-adjoint operator on H. The spectral measure of A is a projection-valued measure defined by the condition
for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:
This is a probability measure defined on the Borel subsets of R
which is the probability distribution obtained by measuring A in
S.
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that (a) a physical system had a determinate state
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
which uniquely determined all the values of its measurable properties, and conversely (b) the values of its measurable properties uniquely determined the state. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
may have been the first person to carefully point out the radical effect the new quantum physics would have on our notion of physical state.
Quantum indeterminacy can be quantitatively characterized by a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
on the set of outcomes of measurements
Measurement problem
The measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...
of an observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.
Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors
Observational error
Observational error is the difference between a measured value of quantity and its true value. In statistics, an error is not a "mistake". Variability is an inherent part of things being measured and of the measurement process.-Science and experiments:...
in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.
Measurement
An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and quantum measurement continues to be an active research area in both theoretical and experimental physics. Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
. The kind of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of projection-valued measure
Projection-valued measure
In mathematics, particularly functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space...
s for self-adjoint operator
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
s which had been recently developed (by von Neumann and independently by Marshall Stone) and the Hilbert space formulation of quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
(attributed by von Neumann to Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
).
In this formulation, the state of a physical system corresponds to a vector of length 1 in a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. An observable is represented by a self-adjoint (i.e. Hermitian) operator A on H. If H is finite dimensional, by the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
, A has an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of eigenvectors. If the system is in state ψ, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value λ will be the corresponding eigenvalue of the equation A e = λ e. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ψ. The probability is
where E(λ) is the projection onto the space of eigenvectors of A with eigenvalue λ.
Example
In this example, we consider a single spin 1/2 particleElementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
(such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space C2, with each quantum state corresponding to a unit vector in C2 (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.
The Pauli spin matrices
are self-adjoint
Self-adjoint
In mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
and correspond to spin-measurements along the 3 coordinate axes.
The Pauli matrices all have the eigenvalues +1, −1.
- For σ1, these eigenvalues correspond to the eigenvectors
-
- For σ3, they correspond to the eigenvectors
Thus in the state
σ1 has the determinate value +1, while measurement of σ3 can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both σ1 and σ3 have determinate values.
There are various questions that can be asked about the above indeterminacy assertion.
- Can the apparent indeterminacy be construed as in fact deterministic, but dependent upon quantities not modeled in the current theory, which would therefore be incomplete? More precisely, are there hidden variables that could account for the statistical indeterminacy in a completely classical way?
- Can the indeterminacy be understood as a disturbance of the system being measured?
Von Neumann formulated the question 1) and provided an argument why the answer had to be no, if one accepted the formalism he was proposing. However according to Bell, von Neumann's formal proof did not justify his informal conclusion. A definitive but partial negative answer to 1) has been established by experiment: because Bell's inequalities are violated, any such hidden variable(s) cannot be local (see Bell test experiments
Bell test experiments
The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...
).
The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the observer effect
Observer effect
Observer effect may refer to:* Observer effect , the impact of observing a process while it is running* Observer effect , the impact of observing a physical system...
, which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) which measures exclusively σ1 and (B) which measures only σ3 of a spin system in the
state ψ. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with equal probability.
Other examples of indeterminacy
Quantum indeterminacy can also be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified. This quantum uncertainty principle can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy.The units involved in quantum uncertainty are on the order of Planck's constant (found experimentally to be 6.6 x 10−34 J·s).
Indeterminacy and incompleteness
Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed, according to the Kochen-Specker theoremKochen-Specker theorem
In quantum mechanics, the Kochen–Specker theorem is a "no go" theorem proved by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden variable theories which try to explain the apparent randomness of quantum mechanics as a deterministic model...
, in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable properties (observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s) has a determinate (sharp) value. The values of an observable will be obtained non-deterministically in accordance with a probability distribution which is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.
This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ1 as a filter which retains only those particles such that σ1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ.
However, Einstein did believe that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein, Boris Podolsky
Boris Podolsky
Boris Yakovlevich Podolsky , was an American physicist of Russian Jewish descent.-Education:In 1896, Boris Podolsky was born into a poor Jewish family in Taganrog, in what was then the Russian Empire, and he moved to the United States in 1913...
and Nathan Rosen
Nathan Rosen
Nathan Rosen was an American-Israeli physicist noted for his study on the structure of the hydrogen molecule and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox.-Background:Nathan Rosen was born into a Jewish family in Brooklyn, New York...
did show that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas:
- A measurable property of a physical system whose value can be predicted with certainty is actually an element of reality (this was the terminology used by EPREPR paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
). - Effects of local actions have a finite propagation speed.
This failure of the classical view was one of the conclusions of the EPR thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
in which two remotely located observers
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...
, now commonly referred to as Alice and Bob
Alice and Bob
The names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party...
, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was a conclusion of EPR, using the formal apparatus of quantum theory, that once Alice measured spin in the x direction, Bob's measurement in the x direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. From this it follows that either value of spin in the x direction is not an element of reality or that the effect of Alice's measurement has infinite speed of propagation.
Indeterminacy for mixed states
We have described indeterminacy for a quantum system which is in a pure state. Mixed states are a more general kind of state obtained by a statistical mixture of pure states. For mixed statesthe "quantum recipe" for determining the probability distribution of a measurement is determined as follows:
Let A be an observable of a quantum mechanical system. A is given by a densely
defined self-adjoint operator on H. The spectral measure of A is a projection-valued measure defined by the condition
for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:
This is a probability measure defined on the Borel subsets of R
which is the probability distribution obtained by measuring A in
S.
See also
- Uncertainty principleUncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
- Quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
- Quantum entanglementQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
- Complementarity (physics)Complementarity (physics)In physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...
- Interpretation of quantum mechanicsInterpretation of quantum mechanicsAn interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...
- Quantum measurement
- Counterfactual definitenessCounterfactual definitenessIn some interpretations of quantum mechanics, counterfactual definiteness is the ability to speak with meaning of the definiteness of the results of measurements that have not been performed...
- EPR paradoxEPR paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
Other references
- A. Aspect, Bell's inequality test: more ideal than ever, Nature 398 189 (1999). http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf
- G. Bergmann, The Logic of Quanta, American Journal of Physics, 1947. Reprinted in Readings in the Philosophy of Science, Ed. H. Feigl and M. Brodbeck, Appleton-Century-Crofts, 1953. Discusses measurement, accuracy and determinism.
- J.S. Bell, On the Einstein-Poldolsky-Rosen paradox, Physics 1 195 (1964).
- A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777 (1935). http://prola.aps.org/abstract/PR/v47/i10/p777_1
- G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004).
- J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. Reprinted in paperback form. Originally published in German in 1932.
- R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999.
External links
- Common Misconceptions Regarding Quantum Mechanics See especially part III "Misconceptions regarding measurement".