User:Johnjbarton/sandbox/quantum entanglement concept

Concept

Definition

In classical systems, the overall state can be broken down into a product of states for each component. A quantum entangled system can be defined to be one whose state cannot be factored as a product of such states. The state of a composite quantum system is always expressible as a sum, or superposition, of products of states; it is entangled if this sum cannot be written as a single product term.

Quantum superposition involving more than one particle produces new effects not possible with classical superposition. In entanglement, one constituent cannot be fully described without considering the other(s); they are not individual particles but are parts of an inseparable whole. The quantum entangled state completely describes the combined system but information about the components may be limited. In the case of maximally entangled states, nothing at all will be known about the components.

On the other hand the number of observable properties of a quantum system increases with the square of the number of subsystems. These extra properties are quantum correlations. Measurements on entangled states result in nonlocal correlations that cannot be reproduced by classical theories.

The formal definition of entanglement negative – it is a quantum state that is not separable – and it is complex because the problem of determining when a state is separable is not solved. An alternative, positive definition focuses on behavior: entangled states are those whose correlations cannot be mimicked with classical systems.

Representations

Entanglement can be represented in an abstract qubit notation or any of a variety of notations for specific applications. In the qubit notation, two orthogonal states, $|0\rangle$ and $|1\rangle$ stand for two polarizations of light, two projections of spin, or the ground and excited states of an atom as examples. While a classical digital bit can be compared to classical objects like coin with heads or tails, qubits can represent a combination, a superposition, of two states $$|\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,{\text{where}}\ |\alpha |^{2}+|\beta |^{2}=1.$$ A measurement on a qubit results in only one of the two states. Entanglement involves two or more qubits, written as the sum of products of the single states: $$|\psi '\rangle =c_{1}|0_{a}\rangle |0_{b}\rangle +c_{2}|1_{a}\rangle |0_{b}\rangle +c_{3}|0_{a}\rangle |1_{b}\rangle +c_{4}|1_{a}\rangle |1_{b}\rangle ,$$ where the sum of the squares of the complex numbers $c_{i}$ equals 1. The subscripts on the individual states can be omitted: $$|\psi '\rangle =c_{1}|0\rangle |0\rangle +c_{2}|1\rangle |0\rangle +c_{3}|0\rangle |1\rangle +c_{4}|1\rangle |1\rangle .$$ and the individual products and be represented by ordered adjacent numbers: $$|\psi '\rangle =c_{1}|00\rangle +c_{2}|10\rangle +c_{3}|01\rangle +c_{4}|11\rangle .$$ A classical two-bit system like two coins has only 4 states, but a quantum system acts like a combination of the 4 states until a measurement is made, at which point one of the four classical states results; observation of one of the particles gives random values but the result for the other particle is alway correlated with the it.

Spin systems

Entanglement is often introduced using spin states. Descriptions often use two experimentalists, Alice and Bob, each with their own lab and spin-measurement equipment. Working independently, Alice and Bob might measure a spin in their labs as spin up or spin down, giving a total of four outcomes or degrees of freedom. A composite space of states describes systems that include the spins in the two labs together. Using the bra–ket notation, Alice measuring spin up while Bob measures spin down in the same experiment could be written as $|\uparrow \downarrow \rangle$. When the states in the two labs interact before measurement, new non-classical entangled states arise that cannot be described as two independent spins. The total quantum space has six degrees of freedom. The entangled singlet state: $${\frac {1}{\sqrt {2}}}\left(|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle \right)$$ is an example of a state of the composite system that has no analog among the states that represent results in the two labs independently. This singlet state is an example of a maximally entangled state. The entanglement means Alice and Bob will measure correlated spin values: if Bob measures spin up in his lab on Alpha Centauri, Alice in Palo Alto measures spin down. However, nothing happens to Alice's model immediately after Bob's measurement. Alice's measurements are correlated with Bob's but they have no knowledge of the correlation until they communicate with each other.

Polarization

Entanglement can be demonstrated with light polarization. A beam of polarized light incident on a polarizing film will transmit only if the axis of the light polarization matches the polarization axis of the film. A two photon state created by combining equal amounts of vertical and horizontal polarization will have entangled polarization. If a photon from this state passes a horizontal polarizing film, the other photon from the state will also pass through such a film. The two polarizing films can be far apart and their orientation can be fixed after emission of the photons.

Wavefunctions

Although the qubit notation emphasizes two-level component systems, any quantum system can be entangled. For example two non-interacting particles in potential wells can be entangled.

The wavefunction for can be written as the product of two one-particle wavefunctions: $$\Psi (\mathbf {r} _{1},\mathbf {r} _{2},t)=\Psi _{a}(\mathbf {r} _{1},t)\Psi _{b}(\mathbf {r} _{2},t)$$ In this case particle 1 can be said to be in state a and particle 2 in state b. In this state they are not entangled.

Other wavefunctions of two particles can be built up from linear combinations (also known as superpositions) of the single-particle wavefunctions. For example: $$\Psi '(\mathbf {r} _{1},\mathbf {r} _{2},t)={\frac {4}{5}}\Psi _{a}(\mathbf {r} _{1},t)\Psi _{b}(\mathbf {r} _{2},t)+{\frac {3}{5}}\Psi _{c}(\mathbf {r} _{1},t)\Psi _{d}(\mathbf {r} _{2},t)$$ where subscripts a and c represent different energy levels of one potential well and b and d states of the other well. Such states cannot be factored into product of single-particle states. These are entangled states, that is to say, they do not represent individual particles but an inseparable whole.

Measurements on such a system results in characteristics of only one of the terms in the sum. For the above state, the predicted energy measurements can be summarized as:

Predicted energy measurements on $\Psi '$

Probability	Measurement 1	Measurement 2
36%	$E_{a}$	$E_{b}$
64%	$E_{c}$	$E_{d}$

The rows represent different measurements. The first measurement gives one of the values randomly, in proportion to the square of its weight in the wavefunction. So 36% of the time the first measurement gives $E_{a}$. The second measurement is correlated with the first one. Reading across the rows, if the first measurement randomly gives $E_{a}$ the second one is not random: it results in $E_{b}$ with certainty. This is a general property of entanglement: measurements of properties become correlated.

Bell states

Bell states are four entangled basis states that describe a two particle system: $$|\psi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}\left(|0\rangle |1|\rangle \pm |1\rangle |0|\rangle \right)$$ $$|\phi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}\left(|0\rangle |0|\rangle \pm |1\rangle |1|\rangle \right)$$ Also called EPR states, these states are maximally entangled and a measurement on these states is equally likely to find a subsystem in $|0\rangle$ as to find $|1\rangle$. Bell states play an important role in quantum information and communication theory.

Multipartite systems

An entangled state of three particles called the Greenberger–Horne–Zeilinger state allows succinct, deterministic evidence against any local hidden-variable theory. This state can be written as a wavefunction: $$\psi =f(\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3}){\frac {|1,1,1\rangle +|-1,-1,-1\rangle }{\sqrt {2}}}$$ where the function f represents the physical separation of three measurements on the state and the numbers in angle brackets represent spin states identified by their eigenvalue along the z axis. At each location 1, 2, or 3, spin operators along other axes x and y have these effects: $$\sigma _{x}|1\rangle =|-1\rangle \ \ \ \sigma _{y}|1\rangle =i|-1\rangle$$ $$\sigma _{x}|-1\rangle =|1\rangle \ \ \ \sigma _{y}|-1\rangle =-i|-1\rangle$$ If measurements are along the y axis are applied at locations 2 and 3 and along x at location 1, the wavefunction is an eigenstate of the combined operators eigenvalue 1: $$\sigma _{1x}\sigma _{2y}\sigma _{3y}\psi =\psi$$ If however all three measurements are the x axis, the eigenvalue is -1: $$\sigma _{1x}\sigma _{2x}\sigma _{3x}\psi =-\psi .$$ A model for this state as independent particles with inherent spin properties would predict that the measurement at location 1 would depend only the the particle, not on the measurements at the other 2 locations. Such a model would predict a +1, exactly the opposite the observed value.

Mermin's three state thought experiment device

N. David Mermin demonstrated the non-classical quantum entanglement results from a three particle state with a thought experiment that used three identical detectors. Each detector flashed red or green depending on the eigenvalue (1, -1); each detector has a switch corresponding to the axis measured (x, y). A source in the middle emits particles into all three detectors. Mermin shows how the results that correspond to quantum mechanics cannot be predicted if you assume that particles from the source contain instructions on how to respond to the switch settings.

Multipartite states are useful in quantum teleportation and entanglement swapping, and are used in quantum computing systems.

Quantum teleportation and entanglement swapping

If Alice and Bob share a Bell state, Alice can tell Bob over a telephone call how to reproduce a quantum state, $|\Psi \rangle$, she has in her lab. Alice performs measurements on the $|\Psi \rangle$ quantum state using the Bell state and tells Bob the results. Bob operates on his Bell state to create a state that matches Alice's $|\Psi \rangle$. The $|\Psi \rangle$ is said to be "quantum teleported" through the Bell state quantum channel with the understanding that classical communications limited to the speed of light are required.

Bob's quantum state is not an independent copy or "clone" of Alice's state. Alice's Bell state measurement does not reveal the state of the individual photons and she can't learn anything more about state. After his operations Bob has a quantum state equivalent to the one Alice started with. Thus, in addition to being limited to the speed of light, the teleportation process does not defy quantum no-cloning theorem.

Entanglement of states from independent sources can be swapped through Bell state measurement.

Entanglement swapping is a form of quantum teleportation. Four particles are involved, arriving at locations labeled Alice, Bob, Clare, and David. Two independent and separate sources of entanglement each send two particle states with one particle from each source, for Bob and Clare, entering a Bell state analysis. Together Bob and Clare project onto a Bell state basis and tell Alice and Bob the results. Alice and Bob they can operate on their states to find that their particles from independent sources are entangled.

Collapse and relativity

Wavefunction collapse or state-reduction, selection of a single term of a quantum superposition as a result of a measurement, is an postulate added to explain quantum measurement. This concept applies only to scenarios involving one observer. Scenarios like quantum teleportation involve multiple observers. If, for example, Alice and Bob are moving apart there is no unique way to determine which collapse occurred first. In all cases however the observers all agree with the measured results and those results agree with the quantum theory.

Nonlocality

Correlations between measurements of a common state at two separate locations signals quantum nonlocality.

Locality is the idea that a measurement in one location only depends upon the past characteristics of the system being measured and not on other, distant measurements. In quantum mechanics, measurements are probabilistic and consequently locality is expressed as a probability. Alice (A), making a measurement on a system with a history labeled $\lambda$ gets a result x with probability $p(A|x,\lambda )$. Far away Bob (B), making a measurement on a system with the same history, gets result y with $p(B|y\lambda )$. If the system obeys the principle of locality in the sense of Bell's theorem, then the two results will be uncorrelated and their joint probability factors into a simple product: $$p(AB|xy,\lambda )=p(A|x,\lambda )p(B|y\lambda )$$ In certain experiments with entangled particles, this locality condition is not observed. Alice and Bob find that their results have correlations that cannot be explained by any theory based on the history labeled by $\lambda$. This result is known as Bell's theorem. When the locality condition fails, meaning non-local correlations are observed, the measured system is entangled. All pure states that are entangled are nonlocal. However, not all mixed entangled systems show nonlocal correlations.

Practical complications

Simple descriptions of entanglement use pure states. A general theoretical analysis also involves

• mixed states described by a density matrix,
• exchange symmetry required by indistinguishable particles, and
• consideration of the effects of relativity

Paradox

Two quantum paradoxes relate to entanglement. These arise from a misuse of quantum concepts.

The first paradox is known as the EPR paradox. A singlet state, as discussed in the spin example but with the component states marked A for Alice and B for Bob, $${\frac {1}{\sqrt {2}}}\left(|\uparrow _{A}\downarrow _{B}\rangle -|\downarrow _{A}\uparrow _{B}\rangle \right)$$ shows that if Alice measures spin up, Bob will measure spin down: their results are correlated. Since the particles Alice and Bob measure came from the same source, Einstein, Podolsky, and Rosen argued that origin of the correlation lies inside those particles. John Stewart Bell showed that the EPR reasoning meant that the particles would have simultaneous properties of spin along other axes, properties that cannot be known according to quantum mechanics. Experiments measure correlations between Alice's and Bob's measurements of spin along these other axes, correlations that the EPR reasoning do not predict. The intuitive EPR reasoning is not correct. Another way to say this is that the entangled singlet state completely specifies the composite system but contains no information about the components. There are no hidden variables to explain the correlations. The component information only appears after a measurement.

Without local hidden variables to explain the correlation, the alternative would seem to be "spooky action at a distance". In this model, Alice's measurement seems to causes action that instantaneously alters the state even if Bob is far away. However, the quantum mechanics model does not account for the principle of relativity and this instantaneous change is an illusion. The results on Alice's end have no effect on Bob's knowledge. Alice's results are not known to Bob. Nothing at his distant location has changed. His only options are to measure the state himself or wait to hear from Alice using communications at the speed of light.

Emergent time

Main article: Problem of time

There is a fundamental conflict, referred to as the problem of time, between the way the concept of time is used in quantum mechanics, and the role it plays in general relativity. In standard quantum theories time acts as an independent background through which states evolve, while general relativity treats time as a dynamical variable which relates directly with matter. Part of the effort to reconcile these approaches to time results in the Wheeler–DeWitt equation, which predicts the state of the universe is timeless or static, contrary to ordinary experience. Work started by Don Page and William Wootters suggests that the universe appears to evolve for observers on the inside because of energy entanglement between an evolving system and a clock system, both within the universe. In this way the overall system can remain timeless while parts experience time via entanglement. The issue remains an open question closely related to attempts at theories of quantum gravity.

Emergent gravity

In general relativity gravity arises from the curvature of spacetime and that curvature derives from the distribution of matter. However, matter is governed by quantum mechanics. Integration of these two theories faces many problems. In an (unrealistic) model space called the anti-de Sitter space, the AdS/CFT correspondence allows a quantum gravitational system to be related to a quantum field theory without gravity. Using this correspondence, Mark Van Raamsdonk suggested that spacetime arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the spacetime.

Quantum-mechanical framework

The following subsections use the formalism and theoretical framework developed in the articles bra–ket notation and mathematical formulation of quantum mechanics.

Pure states

Consider two arbitrary quantum systems A and B, with respective Hilbert spaces H_A and H_B. The Hilbert space of the composite system is the tensor product

$H_{A}\otimes H_{B}.$

If the first system is in state $|\psi \rangle _{A}$ and the second in state $|\phi \rangle _{B}$, the state of the composite system is

$|\psi \rangle _{A}\otimes |\phi \rangle _{B}.$

States of the composite system that can be represented in this form are called separable states, or product states.

Not all states are separable states (and thus product states). Fix a basis $\{|i\rangle _{A}\}$ for H_A and a basis $\{|j\rangle _{B}\}$ for H_B. The most general state in H_A ⊗ H_B is of the form

$|\psi \rangle _{AB}=\sum _{i,j}c_{ij}|i\rangle _{A}\otimes |j\rangle _{B}$.

This state is separable if there exist vectors $[c_{i}^{A}],[c_{j}^{B}]$ so that $c_{ij}=c_{i}^{A}c_{j}^{B},$ yielding ${\textstyle |\psi \rangle _{A}=\sum _{i}c_{i}^{A}|i\rangle _{A}}$ and ${\textstyle |\phi \rangle _{B}=\sum _{j}c_{j}^{B}|j\rangle _{B}.}$ It is inseparable if for any vectors $[c_{i}^{A}],[c_{j}^{B}]$ at least for one pair of coordinates $c_{i}^{A},c_{j}^{B}$ we have $c_{ij}\neq c_{i}^{A}c_{j}^{B}.$ If a state is inseparable, it is called an 'entangled state'.

For example, given two basis vectors $\{|0\rangle _{A},|1\rangle _{A}\}$ of H_A and two basis vectors $\{|0\rangle _{B},|1\rangle _{B}\}$ of H_B, the following is an entangled state:

${\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right).$

If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure state. Another way to say this is that while the von Neumann entropy of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry. The above example is one of four Bell states, which are (maximally) entangled pure states (pure states of the H_A ⊗ H_B space, but which cannot be separated into pure states of each H_A and H_B).

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If in the entangled state given above Alice makes a measurement in the $\{|0\rangle ,|1\rangle \}$ eigenbasis of A, there are two possible outcomes, occurring with equal probability:

1. If Alice measures 0, she views the state as $|0\rangle _{A}|1\rangle _{B}$ and predicts with certainty that Bob will measure 1 in the same basis.
2. Alice measures 1, she views the state as $|1\rangle _{A}|0\rangle _{B}$ and predicts with certainty that Bob will measure 0 in the same basis.

These results make it seem that B has been altered by Alice performing a local measurement on system A. This remains true even if the systems A and B are spatially separated. This is the foundation of the EPR paradox.

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see no-communication theorem.