Mathematical definition of classical entanglement?

There is so much going on with quantum entanglement that it seems to completely obscure what non-quantum entanglement is. From the best I can piece together, the classical entanglement is just that in which two systems, say systems A and B are connected in such a way that knowledge of one system, say A, gives some knowledge of system B.

From what I am understanding about entanglement, loosely speaking, one can judge how much two systems are entangled by the measure of entropy between them, the more entropy the less joint information (if I have that correct).

Considering a classical propositional logical expression $X \oplus Y$. is it safe to say that, if the valuation of $X \oplus Y = 1$, then $X$ and $Y$ are entangled as if I know the valuation of $X$ - that is if $X = 1$ or $X = 0$ then I know the valuation of $Y$ so as to satisfy $X \oplus Y = 1$, that is the valuation of $Y$ is $\neg X$ always so as to make the expression true.

If this is true then would $X$ and $Y$ be maximally entangled? I should add on that if there is any fixed valuation for the system, say instead that $X \oplus Y = 0$ then to satisfy this expression, if we know the valuation of one variable then we know the other, in this case the valuations of $X = Y$ to make the whole system false. So what does this say about the "amount of entanglement"?

• I don't think that it's standard to refer to classical correlations as "entanglement" - that word is generally reserved for inherently quantum correlations. – tparker May 21 '17 at 2:42
• In classical mechanics, the state space of a pair of systems is the direct product of the individual systems, not the tensor product, so the usual notion of entanglement just doesn't apply. – WillO May 21 '17 at 2:45
• – user126422 May 22 '17 at 0:13

The term "classical entanglement" is unpopular in the quantum information community, because "entanglement" is usually associated with an essensial quantum property. A better term is perhaps "classical non-separability." It follows from the purely formal equivalence between the expression in, for instance, Dirac-notation of an entangled bipartite states (Bell state) $$|\psi^+\rangle_{QM} = \frac{1}{\sqrt{2}} (|1\rangle_A |0\rangle_B + |0\rangle_A |1\rangle_B) ,$$ and the expression of classical light using the same Dirac notation $$|\psi^+\rangle_{class} = \frac{1}{\sqrt{2}} (|1\rangle_{pol} |0\rangle_{OAM} + |0\rangle_{pol} |1\rangle_{OAM}) .$$ In the case of the quantum mechanical state, the two partites - $A$ and $B$ - can represent two different particles that may be located at different spatially separated locations. On the other hand, the `partites' of the classical field are different degrees of freedom, such as polarization ($pol$) and orbital angular momentum ($OAM$).

Due to the formal equivalence of the two expressions, any calculation of the amount of entanglement represented by the respective states would come out exactly the same. In other words, I can calculate the concurrence of the classical non-separable state and found it acts exactly as if it is "maximally entangled."

It is however important to note that in the classical case, one cannot separate the non-separable degrees of freedom to be located at spatially separated points in space. This is the essential difference between quantum entanglement and classical non-separability (and serves as the answer to your question - I hope).

This difference not withstanding, several practical implementations have to date been made to demonstrate that if the requirement for entanglement in certain quantum protocols does not include a requirement for it to be nonlocal, such quantum protocols can be implemented with the aid of classical non-separability. Examples include quantum walk (1), the Deutsch–Jozsa algorithm (2) and characterization of quantum channels (3). These are nontrivial examples, implying that classical non-separability does seem to share some essential feature with quantum entanglement.

(1) K Goyal, F S Roux, A Forbes and T Konrad, “Implementation of multidimensional quantum walks using linear optics and classical light,” Physical Review A, 92, 040302(R) (2015).

(2) B. Perez-Garcia, M. McLaren, S. K. Goyal, R. I. Hernandez-Aranda, A. Forbes, T. Konrad, "Quantum computation with classical light: Implementation of the Deutsch–Jozsa algorithm," Physics Letters A 380, 1925 (2016).

(3) B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad and A. Forbes, "Characterizing quantum channels with non-separable states of classical light," Nature Physics 13, 397 (2017).

Mathematical definition depends upon the phenomena being observed, and how that phenomena can be described mathematically.

Suppose the particles are (quantum) entangled in form of numerous pairs, one pair at a time. Such entanglement has two parts to it

1. Behavior of each particle of a single pair with respect to the other particle of the same pair.
2. Statistical correlation of measurements of numerous pairs. I think more fancy name for that is ensemble.

In other words,

1. Intera-pair instant behavior.
2. Inter-pair statistical behavior.

Part 1 can be a direct consequence of conservation laws. Meaning, you create mirror images (zero sum) which remain mirror images irrespective of distance between them. So all that "other end of universe" claim needs no magic.

Therefore, I will skip part 1 in classical context. Also, I admit that this part, in classical sense can only be demonstrated in terms of static hidden variables. In quantum sense, "conservation" is the hidden variable, and does not require any communication, let alone FTL communication.

I can give a very simple and crude classical example of part 2. and I am sure some mathematical model can describe the phenomena -

Example - Keep pouring dirt at one place on a flat surface. It will always take shape of a heap. i.e. you keep pouring the dirt for enough amount of time, it will take shape of a heap. This will happen every time you perform this experiment, guaranteed!

Even though, a mathematical/statistical model may be capable of describing the heap outcome, We know that the heap is in reality formed by gravity in order to keep things in balance. And that balancing is cumulative over the period of the experiment, without involving any instantaneous link. Because this correlation is cumulative, it does not require FTL communication, normal light speed communication, over the duration of experiment, should suffice.

It may take a different kind of shape if you do not pour the dirt on a flat surface. So, depending upon the type of surface, you will get different type of shape. "Different states of entanglement". But for one kind of surface, you will always get same type of shape, always.