Entanglement (physics)

There are three interrelated meanings of the word entanglement in physics. They are listed below and then discussed, both separately and in relation to each other.

• A combination of empirical facts, observed or only hypothetical, incompatible with the conjunction of three fundamental assumptions about nature, called "counterfactual definiteness", "relativistic local causality" and "no-conspiracy" (see below), but compatible with the conjunction of the last two of them ("relativistic local causality" and "no-conspiracy"). Such a combination will be called "empirical entanglement" (which is not a standard terminology).
• A prediction of the quantum theory stating that the empirical entanglement must occur in appropriate physical experiments (below called "quantum entanglement").
• In quantum theory there is a technical notion of "entangled state".

Empirical entanglement

Some people understand it easily, others find it difficult and confusing.

It is easy, since no physical or mathematical prerequisites are needed. Nothing like Newton laws, Schrödinger equation, conservation laws, nor even particles or waves. Nothing like differentiation or integration, nor even linear equations.

It is difficult and confusing for the very same reason! It is highly abstract. Many people feel uncomfortable in such a vacuum of concepts and rush to return to the particles and waves.

The framework, and local causality

The following concepts are essential here.

• A physical apparatus that has a switch and several lights. The switch can be set to one of several possible positions. A little after that the apparatus flashes one of its lights.
• "Local causality": widely separated apparata are incapable of signaling to each other.

Otherwise the apparata are not restricted; they may use all kinds of physical phenomena. In particular, they may receive any kind of information that reaches them. We treat each apparatus as a black box: the switch position is its input, the light flashed is its output; we need not ask about its internal structure.

However, not knowing what is inside the black boxes, can we know that they do not signal to each other? There are two approaches, non-relativistic ("loose") and relativistic ("strict").

The loose approach: we open the black boxes, look, see nothing like mobile phones and rely on our knowledge and intuition.

The strict approach: we do not open the black boxes. Rather, we place them, say, 360,000 km apart (the least Earth-Moon distance) and restrict the experiment to a time interval of, say, 1 sec. Relativity theory states that they cannot signal to each other, for a good reason: a faster-than-light communication in one inertial reference frame would be a backwards-in-time communication in another inertial reference frame!

Falsifiabilty, and no-conspiracy assumption

A claim is called falsifiable (or refutable) if it has observable implications. If some of these implications contradict some observed facts then the claim is falsified (refuted). Otherwise it is corroborated.

The relativistic local causality was never falsified; that is, a faster-than-light signaling was never observed. Does it mean that local causality is corroborated? This question is more intricate than it may seem.

Let A, B be two widely separated apparata, x_A the input (the switch position) of A, and y_B the output (the light flashed) of B. (For now we do not need y_A and x_B.) Local causality claims that x_A has no influence on y_B.

An experiment consisting of n trials is described by x_A(i), y_B(i) for i = 1,2,...,n. Imagine that n = 4 and

x_A(1) = 1,   x_A(2) = 2,   x_A(3) = 1,   x_A(4) = 2,  

y_B(1) = 1,   y_B(2) = 2,   y_B(3) = 1,   y_B(4) = 2.

The data suggest that x_A influences y_B, but do not prove it. Two alternative explanations are possible:

• the apparatus B chooses y_B at random (say, tossing a coin); the four observed equalities y_B(i) = x_A(i) are just a coincidence (of probability 1/16);
• the apparatus B alternates 1 and 2, that is, y_B(i) = 1 for all odd i but y_B(i) = 2 for all even i.

Consider a more thorough experiment: n = 1000, and x_A(i) are chosen at random, say, tossing a coin. Imagine that y_B(i) = x_A(i) for all i = 1,2,...,n. The influence of x_A on y_B is shown very convincingly! But still, an alternative explanation is possible.

For choosing x_A, the coin must be tossed within the time interval scheduled for the trial, since otherwise a slower-than-light signal can transmit the result to the apparatus B before the end of the trial. However, is the result really unpredictable in principle (not just in practice)? Not necessarily so. Moreover, according to classical mechanics, the future is uniquely determined by the past! In particular, the result of the coin tossing exists in the past as a complicated function of a huge number of coordinates and momenta of micro particles.

It is logically possible, but quite unbelievable that the future result of coin tossing is somehow spontaneously singled out in the microscopic chaos and transmitted to the apparatus B in order to influence y_B. The no-conspiracy assumption claims that such exotic scenarios may be safely neglected.

The conjunction of the two assumptions, relativistic local causality and no-conspiracy, is falsifiable, but was never falsified; thus, both assumptions are corroborated.

Below, the no-conspiracy is always assumed (unless explicitly stated otherwise).

Counterfactual definiteness

In this section a single apparatus is considered.

A trial is described by a pair (x,y) where x is the input (the switch position) and y is the output (the light flashed). Is y a function of x? We may repeat the trial with the same x and get a different y (especially if the apparatus tosses a coin). We can set the switch to x again, but we cannot set all molecules to the same microstate. Still, we may try to imagine the past changed, asking a counterfactual question:

• Which outcome the experimenter would have received (in the same trial) if he/she did set the switch to another position?

It is meant that only the input x is changed in the past, nothing else. The question may seem futile, since an answer cannot be verified empirically. Strangely enough, the question will appear to be very useful in the next section.

Classical physics can interpret the question as a change of external forces acting on a mechanical system of a large number of microscopic particles. It is unfeasible to calculate the answer, but anyway, the question makes sense, and the answer exists in principle:

$y=f(x)$

for some function $f:X\to Y,$ where X is the finite set of all possible inputs, and Y is the finite set of all possible outputs. Existence of this function f is called "counterfactual definiteness".

Repeating the experiment we get

$y(i)=f_{i}(x(i))$

for $i=1,2,\dots$ Each time a new function f_i appears; thus x(i)=x(j) does not imply y(i)=y(j). In the case of a single apparatus, counterfactual definiteness is not falsifiable, that is, has no observable implications. Surprisingly, for two (and more) apparata the situation changes dramatically.

Local causality and counterfactual definiteness

For two apparata, A and B, an experiment is described by two pairs, (x_A,y_A) and (x_B,y_B) or, equivalently, by a combined pair ((x_A,x_B), (y_A,y_B)). Counterfactual definiteness alone (without local causality) takes the form

$(y_{\rm {A}},y_{\rm {B}})=f_{\rm {A,B}}(x_{\rm {A}},x_{\rm {B}})$

or, equivalently,

$y_{\rm {A}}=f_{\rm {A}}(x_{\rm {A}},x_{\rm {B}}),$   $y_{\rm {B}}=f_{\rm {B}}(x_{\rm {A}},x_{\rm {B}}).$

Assume in addition that A and B are widely separated and the local causality applies. Then x_A cannot influence y_B, and x_B cannot influence y_A, therefore

$y_{\rm {A}}=f_{\rm {A}}(x_{\rm {A}}),$   $y_{\rm {B}}=f_{\rm {B}}(x_{\rm {B}}).$

An alternative language is logically equivalent, but makes the presentation more vivid. Imagine an experimenter, Alice, near the apparatus A, and another experimenter, Bob, near the apparatus B. Alice is given some input x_A and must provide an output y_A. The same holds for Bob, x_B and y_B. Once the inputs are received, no communication is permitted between Alice and Bob until the outputs are provided. The input x_A is an element of a prescribed finite set X_A (not necessarily a number); the same holds for y_A and Y_A, x_B and X_B, y_B and Y_B.

It may seem that the apparata A, B are of no use for Alice and Bob. Significantly, this is an illusion.

Example

The simplest example of empirical entanglement is presented here. First, its idea is explained informally.

Alice and Bob pretend that they know a 2x2 matrix

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

consisting of numbers 0 and 1 only, satisfying four conditions:

$a=b,\quad c=d,\quad a=c,\quad b\neq d.$

Surely they lie; these four conditions are evidently incompatible. Nevertheless Alice commits herself to show on request any row of the matrix, and Bob commits himself to show on request any column. We expect the lie to manifest itself on the intersection of the row and the column (not always but sometimes). However, Alice and Bob promise to always agree on the intersection!

More formally, x_A=1 requests from Alice the first row, x_A=2 the second; in every case y_A must be either ${\scriptstyle (}{\begin{smallmatrix}0&0\end{smallmatrix}}\scriptstyle {)}$ or ${\scriptstyle (}{\begin{smallmatrix}1&1\end{smallmatrix}}{\scriptstyle )}$. From Bob, x_B=1 requests the first column, in which case y_B must be $({\begin{smallmatrix}0\\0\end{smallmatrix}})$ or $({\begin{smallmatrix}1\\1\end{smallmatrix}})$; and x_B=2 requests the second column, in which case y_B must be $({\begin{smallmatrix}0\\1\end{smallmatrix}})$ or $({\begin{smallmatrix}1\\0\end{smallmatrix}})$. The agreement on the intersection means that, for example, if x_A=2 and x_B=1 then the first element of the row y_A must be equal to the second element of the column y_B.

Without special apparata (A and B), Alice and Bob surely cannot fulfill their promise. Can the apparata help? This crucial question is postponed to the section "Quantum entanglement". Here we consider a different question: is it logically possible, under given assumptions, that Alice and Bob fulfill their promise?

Under all the three assumptions (counterfactual definiteness, local causality and no-conspiracy) we have y_A = f_A(x_A) and y_B = f_B(x_B) for some functions f_A, f_B. (These functions may change from one trial to another.) Specifically, f_A(1) and f_A(2), being two rows, form a 2x2 matrix satisfying the conditions a=b, c=d. Also f_B(1) and f_B(2), being two columns, form a 2x2 matrix satisfying the conditions a=c, b≠d. These two matrices necessarily differ at least in one of the four elements (since the four conditions are incompatible). Therefore it can happen that Alice and Bob disagree on the intersection, and moreover, it happens with the probability at least 0.25. In the long run, Alice and Bob cannot fulfill their promise.

Waiving the counterfactual definiteness (but retaining local causality and no-conspiracy) we get the opposite result: Alice and Bob can fulfill their promise. Here is how.

Given x_A and x_B, there ate two allowed y_A and two allowed y_B, thus, 4 allowed combinations (y_A, y_B). Two of them agree on the intersection of the row and the column; the other two disagree. Imagine that the apparata A, B choose at random (with equal probabilities 0.5, 0.5) one of the two combinations (y_A, y_B) that agree on the intersection. For example, given x_A=2 and x_B=1, we get either $y_{\rm {A}}={\scriptstyle (}{\begin{smallmatrix}0&0\end{smallmatrix}}\scriptstyle {)}$ and $y_{\rm {B}}=({\begin{smallmatrix}0\\0\end{smallmatrix}})$, or $y_{\rm {A}}={\scriptstyle (}{\begin{smallmatrix}1&1\end{smallmatrix}}\scriptstyle {)}$ and $y_{\rm {B}}=({\begin{smallmatrix}1\\1\end{smallmatrix}})$.

This situation is compatible with local causality, since y_B gives no information about x_A; also y_A gives no information about x_B. For example, given x_A=2 and x_B=1, we get either $y_{\rm {B}}=({\begin{smallmatrix}0\\0\end{smallmatrix}})$ or $y_{\rm {B}}=({\begin{smallmatrix}1\\1\end{smallmatrix}})$, with probabilities 0.5, 0.5; exactly the same holds given x_A=1 and x_B=1.

Thus, empirical entanglement is logically possible. The question of its existence in the nature is addressed in the next section.

Quantum entanglement

Classical bounds and quantum bounds

Classical physics obeys the counterfactual definiteness and therefore negates entanglement. Classical apparata A, B cannot help Alice and Bob to always win (that is, agree on the intersection). What about quantum apparata? The answer is quite unexpected.

First, quantum apparata cannot ensure that Alice and Bob win always. Moreover, the winning probability does not exceed

${\frac {2+{\sqrt {2}}}{4}}=0.853\dots$

no matter which quantum apparata are used.

Second, there exist quantum apparata that ensure a winning probability higher than 3/4 = 0.75. This is a manifestation of entanglement, since under the three classical assumptions (counterfactual definiteness, local causality and no-conspiracy) the winning probability cannot exceed 3/4 (the classical bound). But moreover, ideal quantum apparata can reach the winning probability $(2+{\sqrt {2}})/4$ (the quantum bound), and non-ideal quantum apparata can get arbitrarily close to this bound.

Third, a modification of the game, called "magic square game", makes it possible to win always. To this end we replace 2x2 matrices with 3x3 matrices, still of numbers 0 and 1 only, with the following conditions:

• the parity of each row is even,
• the parity of each column is odd.

The classical bound is equal to 8/9; the quantum bound is equal to 1.