Quantum superposition

For broader coverage of this topic, see Superposition principle.

Quantum superposition of states and decoherence

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 $|0\rangle$ and $|1\rangle$:

$|\Psi \rangle =c_{0}|0\rangle +c_{1}|1\rangle ,$

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.

Background

Paul Dirac described the superposition principle as follows:

The general principle of superposition of quantum mechanics applies to the states [that are theoretically possible without mutual interference or contradiction] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely, any two or more states may be superposed to give a new state...

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

$|\alpha \rangle =\sum _{n}c_{n}|n\rangle ,$

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

$|\alpha \rangle =\int dx'|x'\rangle \langle x'|\alpha \rangle ,$

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

Example

Given the Schrödinger equation

${\hat {H}}|n\rangle =E_{n}|n\rangle ,$

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

${\hat {H}}{\big (}|n\rangle +|n'\rangle {\big )}=E_{n}|n\rangle +E_{n'}|n'\rangle ,$

where

$|\Psi \rangle =|n\rangle +|n'\rangle$

is a solution of the Schrödinger equation but is not generally an eigenstate because $E_{n}$ and $E_{n'}$ are not generally equal. We say that $|\Psi \rangle$ 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 ${\hat {z}}$ basis:

$|\Psi \rangle =c_{1}|{\uparrow }\rangle +c_{2}|{\downarrow }\rangle ,$

where $|{\uparrow }\rangle$ and $|{\downarrow }\rangle$ 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:

$P{\big (}|{\uparrow }\rangle {\big )}=|c_{1}|^{2},$

$P{\big (}|{\downarrow }\rangle {\big )}=|c_{2}|^{2},$

$P_{\text{total}}=P{\big (}|{\uparrow }\rangle {\big )}+P{\big (}|{\downarrow }\rangle {\big )}=|c_{1}|^{2}+|c_{2}|^{2}=1,$

where the probability of finding the particle with either spin up or down is normalized to 1. Notice that $c_{1}$ and $c_{2}$ are complex numbers, so that

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

is an example of an allowed state. We now get

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

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

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

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

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

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

Hamiltonian Evolution of General States

Any quantum state (a single eigenstate of the Hamiltonian or a superposition of eigenstates) evolve according to the time dependent Schrödinger equation. These states, as previously stated, are required to be normalized and live in the Hilbert space (either finite or infinite dimensional depending on the basis eigenstates used to write the general state). Under time evolution the state remains normalized but the amplitudes of each particular eigenstate can change with time:

$\sum _{n}|c(t=t_{0})_{n}|^{2}=\sum _{n}|c(t=t')_{n}|^{2}=1$

$c_{n}(t=t_{0})\neq c_{n}(t=t')$

where generally the $c_{n}$ coefficients at time $t=t_{0}$ and $t=t'$ are not the same. To evolve the original state to the new state the time evolution operator is introduced:

$|\alpha ,t_{0};t_{0}+dt\rangle ={\mathcal {U}}(t_{0}+dt,t_{0})|\alpha ,t_{0}\rangle$

where ${\mathcal {U}}$ is the infinitesimal time evolution operator that acts on a state $|\alpha \rangle$ at time $t=t_{0}$ and evolves it to the state $|\alpha ,t_{0};t_{0}+dt\rangle$. This notation means that the state started out as $|\alpha ,t_{0}\rangle$ and evolved into $|\alpha ,t_{0};t_{0}+dt\rangle$ after a time $dt$. The normalization condition on the state before and after time evolution means that ${\mathcal {U}}$ is a unitary operator that becomes identity as $dt\rightarrow 0$ and can be written as:

${\mathcal {U}}=1-i{\frac {H}{\hbar }}dt$

where $H$ is the Hamiltonian that defines the dynamics of the state. Additionally, the time evolution operator intuitively obeys a composition property:

${\mathcal {U}}(t+dt,t_{0})={\mathcal {U}}(t+dt,t){\mathcal {U}}(t,t_{0})$

which allows for arbitrary time evolution. Using the composition property the general, instead of infinitesimal, time evolution operator is derived:

${\mathcal {U}}(t+dt,t_{0})=(1-i{\frac {H}{\hbar }}dt){\mathcal {U}}(t,t_{0})$

${\mathcal {U}}(t+dt,t_{0})-{\mathcal {U}}(t,t_{0})=i{\frac {H}{\hbar }}{\mathcal {U}}(t,t_{0})dt$

$i\hbar {\frac {\partial {\mathcal {U(t,t_{0})}}}{\partial {t}}}=H{\mathcal {U(t,t_{0})}}$

results in the Schrödinger equation for the time evolution operator. Multiplying by the original state vector, $|\alpha ,t_{0}\rangle$ from the right:

$i\hbar {\frac {\partial {\mathcal {U(t,t_{0})}}}{\partial {t}}}|\alpha ,t_{0}\rangle =H{\mathcal {U(t,t_{0})}}|\alpha ,t_{0}\rangle$

$i\hbar {\frac {\partial {\mathcal {U(t,t_{0})}}}{\partial {t}}}|\alpha ,t_{0};t\rangle =H{\mathcal {U(t,t_{0})}}|\alpha ,t_{0};t\rangle$

gives the explicit time evolution of the new state and shows that this state must also satisfy the Schrödinger equation. However, we can work with just the time evolution operator depending on how the Hamiltonian evolves in time and if the Hamiltonians at different times commute with each other.

Case 1: Hamiltonian is Not Time Independent

If the Hamiltonian, $H$, is time independent we can solve the previous differential equation to obtain:

${\mathcal {U(t,t_{0})}}=\exp(-i{\frac {H(t-t_{0})}{\hbar }})$

We can also obtain this expression by compounding an infinite number of infinitesimal transformations to generate a finite transformation:

${\mathcal {U(t,t_{0})}}=\exp(-i{\frac {H(t-t_{0})}{\hbar }})=\lim _{N\to \infty }(1-i{\frac {H(t-t_{0})/\hbar }{N}})^{N}$

Case 2: Hamiltonian is Time Dependent and Commutes at All Times

If the Hamiltonian has explicit time dependence but the Hamiltonians at different times all commute with each other we can again solve the Schrödinger equation for ${\mathcal {U}}$ to obtain:

${\mathcal {U(t,t_{0})}}=\exp(-i{\frac {\int _{t_{0}}^{t}H(t')dt'}{\hbar }})$

an example of a Hamiltonian with this property is a magnetic field pointing in a direction whose magnitude changes but direction remains fixed.

Case 3: Hamiltonian is Time Dependent and Not Commuting at All Times

The final case is for a Hamiltonian that has explicit time dependence but does not commute with the Hamiltonian at a later time. In this case the Dyson Series is used and the final form the time evolution operator is:

${\mathcal {U(t,t_{0})}}=1+\sum _{n=1}^{\infty }({\frac {-i}{\hbar }})^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}...\int _{t_{0}}^{t_{n}-1}dt_{n}H(t_{1})H(t_{2})...H(t_{n})$

this may be familiar from time dependent perturbation theory and finds applications in scattering problems. An example of a Hamiltonian this property is one with a magnetic field whose magnitude remains the same but the direction switches from pointing along ${\hat {x}}$ to ${\hat {y}}$.

Experiments and applications

Experiments Realizing Superposition

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

• A "cat state" has been achieved with photons.
• 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
• 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.

Proposed Experiments

• An experiment involving a flu virus has been proposed.

• An experiment has been proposed, with a bacterial cell cooled to 10 mK, using an electromechanical oscillator. At that temperature, all metabolism would be stopped, and the cell might behave virtually as a definite chemical species. For detection of interference, it would be necessary that the cells be supplied in large numbers as pure samples of identical and detectably recognizable virtual chemical species.

Applications of Superposition

• Quantum computation: quantum computers seek to increase computing speed by using qubits to parallelize computations.

Physical Interpretation

Any measurement executed on a particle will collapse its wave function into an eigenstate of the Hamiltonian governing its time evolution. This means that a particle prepared into a general superposition of different eigenstates will, after the measurement, be in a definite eigenstate corresponding to the eigenvalue of that state read out from the measurement. For example, a valence electron in an atom that is in an equal superposition of its ground state and first excited state when measured will be in either the ground or first excited state. If an ensemble of atoms are prepared and the valence electron of each atom is placed in an equal superposition of its ground and first excited state then a measurement executed on all atoms will return approximately half the electrons in the ground and half in the first excited state. If this measurement is executed an infinite number of times on a single atom or once on an infinite ensemble of atoms then exactly half of the electrons will be in the first excited state and half will be in the ground state. This is analogous to having an infinite number of coins balanced on their edge on a table and covered by a box. When the box is removed the table shakes slightly and all of the coins collapse into either heads or tails with equal probability due to the act of measurement. As they are equally likely to land on either heads or tails exactly half should be heads and half should be tails after all have been counted.

The reason why macroscopic objects almost never exhibit quantum superposition is an active field of study and a proposed mechanism for the emergence of classical from quantum mechanics is due to quantum decoherence. Another proposed class of theories is that the fundamental time evolution equation is incomplete, and requires the addition of a 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.

See also

• Eigenstates – Mathematical entity to describe the probability of each possible measurement on a systemPages displaying short descriptions of redirect targets
• Goodhart's law – Adage about statistical measures — when an attempt is made to use a statistical measure for purposes of control (directing), its statistical validity breaks down
• Mach–Zehnder interferometer – Device to determine relative phase shift
• Penrose interpretation – Interpretation of quantum mechanics
• Pure qubit state – Basic unit of quantum informationPages displaying short descriptions of redirect targets
• Quantum computation – Technology that uses quantum mechanicsPages displaying short descriptions of redirect targets
• Schrödinger's cat – Thought experiment in quantum mechanics
• Superposition principle – Fundamental physics principle stating that physical solutions of linear systems are linear
• Uncertainty principle – Foundational principle in quantum physics — Certain pairs of quantum measurements cannot both give precise results.
• Wave packet

References

Bibliography of cited references

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• Dirac, P. A. M. (1930/1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press.
• Einstein, A. (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in Schilpp, P. A. editor (1949), Albert Einstein: Philosopher-Scientist, volume II, Open Court, La Salle IL.
• Feynman, R. P., Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics, volume 3, Addison-Wesley, Reading, MA.
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• Nielsen, Michael A.; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0521632358. OCLC 43641333.
• Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.
• Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum computing for computer scientists. Cambridge University Press. ISBN 978-0-521-87996-5.