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.

For broader coverage of this topic, see Superposition principle.

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

where $|\Psi \rangle$ is the quantum state of the qubit, and $|0\rangle$ , $|1\rangle$ denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes $c_{0}$ and $c_{1}$ that both are complex numbers. Here $|0\rangle$ corresponds to the classical 0 bit, and $|1\rangle$ to the classical 1 bit. The probabilities of measuring the system in the $|0\rangle$ or $|1\rangle$ state are given by $|c_{0}|^{2}$ and $|c_{1}|^{2}$ 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.

Theory[edit]

General formalism[edit]

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

A "" has been achieved with photons.^[4]

cat state

A ion has been trapped in a superposed state.^[5]

beryllium

A has been performed with molecules as large as buckyballs and functionalized oligoporphyrins with up to 2000 atoms.^[6]^[7]

double slit experiment

A 2013 experiment superposed molecules containing 15,000 each of protons, neutrons and electrons. The molecules were of compounds selected for their good thermal stability, and were evaporated into a beam at a temperature of 600 K. The beam was prepared from highly purified chemical substances, but still contained a mixture of different molecular species. Each species of molecule interfered only with itself, as verified by mass spectrometry.

[8]

An experiment involving a ("SQUID") has been linked to the theme of the "cat state" thought experiment.^[9]

superconducting quantum interference device

Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.^[3]

In quantum computing the phrase "cat state" often refers to the GHZ state,^[18] the special entangled state of qubits wherein the qubits are in an equal superposition of all being 0 and all being 1; i.e.,

Physical interpretation[edit]

It is natural to ask why ordinary everyday objects and events do not seem to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded as "mysterious", for instance by Richard Feynman.^[21] In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted this dissonance between quantum mechanics and classical physics. One modern view is that this mystery is explained by quantum decoherence. A macroscopic system (such as a cat) may evolve over time into a superposition of classically distinct quantum states (such as "alive" and "dead"). The mechanism that achieves this is a subject of significant research. One mechanism suggests that the state of the cat is entangled with the state of its environment (for instance, the molecules in the atmosphere surrounding it). When averaged over the possible quantum states of the environment (a physically reasonable procedure unless the quantum state of the environment can be controlled or measured precisely), the resulting mixed quantum state for the cat is very close to a classical probabilistic state where the cat has some definite probability to be dead or alive, just as a classical observer would expect in this situation. Another proposed class of theories is that the fundamental time evolution equation is incomplete, and requires the addition of some type of fundamental Lindbladian, the reason for this addition and the form of the additional term varies from theory to theory. A popular theory is continuous spontaneous localization, where the Lindblad term is proportional to the spatial separation of the states. This too results in a quasi-classical probabilistic state.

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