Complementarity (physics)

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https://en.wikipedia.org/wiki/Complementarity_(physics) 

In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that certain pairs of complementary properties cannot all be observed or measured simultaneously. For example, position and momentum, frequency and lifetime, or optical phase and photon number. In contemporary terms, complementarity encompasses both the uncertainty principle and wave-particle duality.

Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an electron, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.

History

Background

Complementarity as a physical model derives from Niels Bohr's 1927 lecture during the Como Conference in Italy, at a scientific celebration of the work of Alessandro Volta 100 years previous. Bohr's subject was complementarity, the idea that measurements of quantum events provide complementary information through seemingly contradictory results. While Bohr's presentation was not well received, it did crystallize the issues ultimately leading to the modern wave-particle duality concept. The contradictory results that triggered Bohr's ideas had been building up over the previous 20 years.

This contradictory evidence came both from light and from electrons. The wave theory of light, broadly successful for over a hundred years, had been challenged by Planck's 1901 model of blackbody radiation and Einstein's 1905 interpretation of the photoelectric effect. These theoretical models use discrete energy, a quantum, to describe the interaction of light with matter. Despite confirmation by various experimental observations, the photon theory (as it came to be called later) remained controversial until Arthur Compton performed a series of experiments from 1922 to 1924 demonstrating the momentum of light. The experimental evidence of particle-like momentum seemingly contradicted other experiments demonstrating the wave-like interference of light.

The contradictory evidence from electrons arrived in the opposite order. Many experiments by J. J. Thompson, Robert Millikan, and Charles Wilson, among others, had shown that free electrons had particle properties. However, in 1924, Louis de Broglie proposed that electrons had an associated wave and Schrödinger demonstrated that wave equations accurately account for electron properties in atoms. Again some experiments showed particle properties and others wave properties.

Bohr's resolution of these contradictions is to accept them. In his Como lecture he says: "our interpretation of the experimental material rests essentially upon the classical concepts." Direct observation being impossible, observations of quantum effects are necessarily classical. Whatever the nature of quantum events, our only information will arrive via classical results. If experiments sometimes produce wave results and sometimes particle results, that is the nature of light and of the ultimate constituents of matter.

Bohr's lectures

Niels Bohr apparently conceived of the principle of complementarity during a skiing vacation in Norway in February and March 1927, during which he received a letter from Werner Heisenberg regarding an as-yet-unpublished result, a thought experiment about a microscope using gamma rays. This thought experiment implied a tradeoff between uncertainties that would later be formalized as the uncertainty principle. To Bohr, Heisenberg's paper did not make clear the distinction between a position measurement merely disturbing the momentum value that a particle carried and the more radical idea that momentum was meaningless or undefinable in a context where position was measured instead. Upon returning from his vacation, by which time Heisenberg had already submitted his paper for publication, Bohr convinced Heisenberg that the uncertainty tradeoff was a manifestation of the deeper concept of complementarity. Heisenberg duly appended a note to this effect to his paper, before its publication, stating:

Bohr has brought to my attention [that] the uncertainty in our observation does not arise exclusively from the occurrence of discontinuities, but is tied directly to the demand that we ascribe equal validity to the quite different experiments which show up in the [particulate] theory on one hand, and in the wave theory on the other hand.

Bohr publicly introduced the principle of complementarity in a lecture he delivered on 16 September 1927 at the International Physics Congress held in Como, Italy, attended by most of the leading physicists of the era, with the notable exceptions of Einstein, Schrödinger, and Dirac. However, these three were in attendance one month later when Bohr again presented the principle at the Fifth Solvay Congress in Brussels, Belgium. The lecture was published in the proceedings of both of these conferences, and was republished the following year in Naturwissenschaften (in German) and in Nature (in English).

In his original lecture on the topic, Bohr pointed out that just as the finitude of the speed of light implies the impossibility of a sharp separation between space and time (relativity), the finitude of the quantum of action implies the impossibility of a sharp separation between the behavior of a system and its interaction with the measuring instruments and leads to the well-known difficulties with the concept of 'state' in quantum theory; the notion of complementarity is intended to capture this new situation in epistemology created by quantum theory. Physicists F.A.M. Frescura and Basil Hiley have summarized the reasons for the introduction of the principle of complementarity in physics as follows:

In the traditional view, it is assumed that there exists a reality in space-time and that this reality is a given thing, all of whose aspects can be viewed or articulated at any given moment. Bohr was the first to point out that quantum mechanics called this traditional outlook into question. To him the "indivisibility of the quantum of action" [...] implied that not all aspects of a system can be viewed simultaneously. By using one particular piece of apparatus only certain features could be made manifest at the expense of others, while with a different piece of apparatus another complementary aspect could be made manifest in such a way that the original set became non-manifest, that is, the original attributes were no longer well defined. For Bohr, this was an indication that the principle of complementarity, a principle that he had previously known to appear extensively in other intellectual disciplines but which did not appear in classical physics, should be adopted as a universal principle.

Debate following the lectures

Main article: Bohr–Einstein debates

Complementarity was a central feature of Bohr's reply to the EPR paradox, an attempt by Albert Einstein, Boris Podolsky and Nathan Rosen to argue that quantum particles must have position and momentum even without being measured and so quantum mechanics must be an incomplete theory. The thought experiment proposed by Einstein, Podolsky and Rosen involved producing two particles and sending them far apart. The experimenter could choose to measure either the position or the momentum of one particle. Given that result, they could in principle make a precise prediction of what the corresponding measurement on the other, faraway particle would find. To Einstein, Podolsky and Rosen, this implied that the faraway particle must have precise values of both quantities whether or not that particle is measured in any way. Bohr argued in response that the deduction of a position value could not be transferred over to the situation where a momentum value is measured, and vice versa.

Later expositions of complementarity by Bohr include a 1938 lecture in Warsaw and a 1949 article written for a festschrift honoring Albert Einstein. It was also covered in a 1953 essay by Bohr's collaborator Léon Rosenfeld.

Mathematical formalism

For Bohr, complementarity was the "ultimate reason" behind the uncertainty principle. All attempts to grapple with atomic phenomena using classical physics were eventually frustrated, he wrote, leading to the recognition that those phenomena have "complementary aspects". But classical physics can be generalized to address this, and with "astounding simplicity", by describing physical quantities using non-commutative algebra. This mathematical expression of complementarity builds on the work of Hermann Weyl and Julian Schwinger, starting with Hilbert spaces and unitary transformation, leading to the theorems of mutually unbiased bases.

In the mathematical formulation of quantum mechanics, physical quantities that classical mechanics had treated as real-valued variables become self-adjoint operators on a Hilbert space. These operators, called "observables", can fail to commute, in which case they are called "incompatible":$${\displaystyle \left[\hat{A},\hat{B}\right]:=\hat{A}\hat{B}-\hat{B}\hat{A}\ne \hat{0}.}$$ Incompatible observables cannot have a complete set of common eigenstates; there can be some simultaneous eigenstates of ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$, but not enough in number to constitute a complete basis. The canonical commutation relation $${\displaystyle \left[\hat{x},\hat{p}\right]=i\hslash }$$ implies that this applies to position and momentum. In a Bohrian view, this is a mathematical statement that position and momentum are complementary aspects. Likewise, an analogous relationship holds for any two of the spin observables defined by the Pauli matrices; measurements of spin along perpendicular axes are complementary. The Pauli spin observables are defined for a quantum system described by a two-dimensional Hilbert space; mutually unbiased bases generalize these observables to Hilbert spaces of arbitrary finite dimension. Two bases ${\displaystyle \{|a_j⟩\}}$ and ${\displaystyle \{|b_k⟩\}}$ for an ${\displaystyle N}$-dimensional Hilbert space are mutually unbiased when$${\displaystyle |⟨a_j|b_k⟩{|}^2=\frac{1}{N}\text{for all}j,k=1,...N-1.}$$

Here the basis vector ${\displaystyle a_1}$, for example, has the same overlap with every ${\displaystyle b_k}$; there is equal transition probability between a state in one basis and any state in the other basis. Each basis corresponds to an observable, and the observables for two mutually unbiased bases are complementary to each other. This leads to a definition of 'Principle of Complementarity' as:

For each degree of freedom the dynamical variables are a pair of complementary observables.

The concept of complementarity has also been applied to quantum measurements described by positive-operator-valued measures (POVMs).

Continuous complementarity

Main article: Wave–particle duality relation

While the concept of complementarity can be discussed via two experimental extremes, continuous tradeoff is also possible. In 1979 Wooters and Zurek introduced an information-theoretic treatment of the double-slit experiment providing on approach to a quantiative model of complementarity. The wave-particle relation, introduced by Daniel Greenberger and Allaine Yasin in 1988, and since then refined by others, quantifies the trade-off between measuring particle path distinguishability, ${\displaystyle D}$, and wave interference fringe visibility, ${\displaystyle V}$: $${\displaystyle D^2+V^2\le 1}$$ The values of ${\displaystyle D}$ and ${\displaystyle V}$ can vary between 0 and 1 individually, but any experiment that combines particle and wave detection will limit one or the other, or both. The detailed definition of the two terms vary among applications, but the relation expresses the verified constraint that efforts to detect particle paths will result in less visible wave interference.

Modern role

While many of the early discussions of complementarity discussed hypothetical experiments, advances in technology have allowed advanced tests of this concept. Experiments like the quantum eraser verify the key ideas in complementarity; modern exploration of quantum entanglement builds directly on complementarity:

The most sensible position, according to quantum mechanics, is to assume that no such waves preexist before any measurement.

— Anton Zeilinger

In his Nobel lecture, physicist Julian Schwinger linked complementarity to quantum field theory:

Indeed, relativistic quantum mechanics—the union of the complementarity principle of Bohr with the relativity principle of Einstein—is quantum field theory.

— Julian Schwinger

The Consistent histories interpretation of quantum mechanics takes a generalized form of complementarity as a key defining postulate.

Quantum superposition

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_superposition

Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.

An example is a qubit used in quantum information processing. A qubit state is most generally a superposition of the basis states ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$:

${\displaystyle |\mathrm{\Psi }⟩=c_0|0⟩+c_1|1⟩,}$

where ${\displaystyle |\mathrm{\Psi }⟩}$ is the quantum state of the qubit, and ${\displaystyle |0⟩}$, ${\displaystyle |1⟩}$ denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes ${\displaystyle c_0}$ and ${\displaystyle c_1}$ that both are complex numbers. Here ${\displaystyle |0⟩}$ corresponds to the classical 0 bit, and ${\displaystyle |1⟩}$ to the classical 1 bit. The probabilities of measuring the system in the ${\displaystyle |0⟩}$ or ${\displaystyle |1⟩}$ state are given by ${\displaystyle |c_0{|}^2}$ and ${\displaystyle |c_1{|}^2}$ respectively (see the Born rule). Before the measurement occurs the qubit is in a superposition of both states.

The interference fringes in the double-slit experiment provide another example of the superposition principle.

Wave postulate

The theory of quantum mechanics postulates that a wave equation completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be linear and homogeneous. These conditions mean that for any two solutions of the wave equation, ${\displaystyle {\mathrm{\Psi }}_1}$ and ${\displaystyle {\mathrm{\Psi }}_2}$, a linear combination of those solutions also solve the wave equation: $${\displaystyle \mathrm{\Psi }=c_1{\mathrm{\Psi }}_1+c_2{\mathrm{\Psi }}_2}$$ for arbitrary complex coefficients ${\displaystyle c_1}$ and ${\displaystyle c_2}$. If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.

Transformation

The quantum wave equation can be solved using functions of position, ${\displaystyle \mathrm{\Psi }(\overrightarrow{r})}$, or using functions of momentum, ${\displaystyle \mathrm{\Phi }(\overrightarrow{p})}$ and consequently the superposition of momentum functions are also solutions: $${\displaystyle \mathrm{\Phi }(\overrightarrow{p})=d_1{\mathrm{\Phi }}_1(\overrightarrow{p})+d_2{\mathrm{\Phi }}_2(\overrightarrow{p})}$$ The position and momentum solutions are related by a linear transformation, a Fourier transformation. This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.

Generalization to basis states

Other transformations express a quantum solution as a superposition of eigenvectors, each corresponding to a possible result of a measurement on the quantum system. An eigenvector ${\displaystyle {\psi }_i}$ for a mathematical operator, ${\displaystyle \hat{A}}$, has the equation $${\displaystyle \hat{A}{\psi }_i={\lambda }_i{\psi }_i}$$ where ${\displaystyle {\lambda }_i}$ is one possible measured quantum value for the observable ${\displaystyle A}$. A superposition of these eigenvectors can represent any solution: $${\displaystyle \mathrm{\Psi }=\sum _n{a}_i{\psi }_i.}$$ The states like ${\displaystyle {\psi }_i}$ are called basis states.

Compact notation for superpositions

Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the Dirac bra-ket notation: $${\displaystyle |v⟩=d_1|1⟩+d_2|2⟩}$$ This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics.

Consequences

Paul Dirac described the superposition principle as follows:

The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.

Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:

"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.

Theory

General formalism

Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:

${\displaystyle |\alpha ⟩=\sum _n{c}_n|n⟩,}$

where ${\displaystyle |n⟩}$ are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, ${\displaystyle |x⟩}$:

${\displaystyle |\alpha ⟩=\int d{x}^{\prime }|x^{\prime }⟩⟨x^{\prime }|\alpha ⟩,}$

where ${\displaystyle {\varphi }_{\alpha }(x)=⟨x|\alpha ⟩}$ is the projection of the state into the ${\displaystyle |x⟩}$ basis and is called the wave function of the particle. In both instances we notice that ${\displaystyle |\alpha ⟩}$ can be expanded as a superposition of an infinite number of basis states.

Example

Given the Schrödinger equation

${\displaystyle \hat{H}|n⟩=E_n|n⟩,}$

where ${\displaystyle |n⟩}$ indexes the set of eigenstates of the Hamiltonian with energy eigenvalues ${\displaystyle E_n,}$ we see immediately that

${\displaystyle \hat{H}(|n⟩+|n^{\prime }⟩)=E_n|n⟩+E_{n^{\prime }}|n^{\prime }⟩,}$

where

${\displaystyle |\mathrm{\Psi }⟩=|n⟩+|n^{\prime }⟩}$

is a solution of the Schrödinger equation but is not generally an eigenstate because ${\displaystyle E_n}$ and ${\displaystyle E_{n^{\prime }}}$ are not generally equal. We say that ${\displaystyle |\mathrm{\Psi }⟩}$ is made up of a superposition of energy eigenstates. Now consider the more concrete case of an electron that has either spin up or down. We now index the eigenstates with the spinors in the ${\displaystyle \hat{z}}$ basis:

${\displaystyle |\mathrm{\Psi }⟩=c_1|\uparrow ⟩+c_2|\downarrow ⟩,}$

where ${\displaystyle |\uparrow ⟩}$ and ${\displaystyle |\downarrow ⟩}$ denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:

${\displaystyle P(|\uparrow ⟩)=|c_1{|}^2,}$

${\displaystyle P(|\downarrow ⟩)=|c_2{|}^2,}$

${\displaystyle P_{\text{total}}=P(|\uparrow ⟩)+P(|\downarrow ⟩)=|c_1{|}^2+|c_2{|}^2=1,}$

where the probability of finding the particle with either spin up or down is normalized to 1. Notice that ${\displaystyle c_1}$ and ${\displaystyle c_2}$ are complex numbers, so that

${\displaystyle |\mathrm{\Psi }⟩=\frac{3}{5}i|\uparrow ⟩+\frac{4}{5}|\downarrow ⟩.}$

is an example of an allowed state. We now get

${\displaystyle P(|\uparrow ⟩)={\left|\frac{3i}{5}\right|}^2=\frac{9}{25},}$

${\displaystyle P(|\downarrow ⟩)={\left|\frac{4}{5}\right|}^2=\frac{16}{25},}$

${\displaystyle P_{\text{total}}=P(|\uparrow ⟩)+P(|\downarrow ⟩)=\frac{9}{25}+\frac{16}{25}=1.}$

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

${\displaystyle \mathrm{\Psi }={\psi }_{+}(x)\otimes |\uparrow ⟩+{\psi }_{-}(x)\otimes |\downarrow ⟩,}$

where we have a general state ${\displaystyle \mathrm{\Psi }}$ is the sum of the tensor products of the position space wave functions and spinors.

Experiments

Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.

• A beryllium ion has been trapped in a superposed state.
• A double slit experiment has been performed with molecules as large as buckyballs and functionalized oligoporphyrins with up to 2000 atoms.
• Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed, and metal clusters with masses over 170,000 Da and containing more than 7,000 atoms have also been demonstrated in quantum superposition.
• Very sensitive magnetometers have been realized using superconducting quantum interference devices (SQUIDS) that operate using quantum interference effects in superconducting circuits.

• A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.
• Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.

In quantum computers

In quantum computers, a qubit is the analog of the classical information bit, but rather than having one of two distinct values, qubits are a superposition of two values. Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interaction are needed for computing with qubits. Qubit systems like nuclear spins with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.