It turns out that few people appreciate the relation between Schrödinger’s cat and entanglement.  When we hear entanglement, the first paper that comes to mind is Einstein Podolsky and Rosen’s “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”  EPR were the first to point out a strange prediction of quantum mechanics (which we now call entanglement),  but the   term entanglement (or its German equivalent) was coined by Schrödinger in a  paper  inspired by EPR. In the same paper Schrödinger describes an experiment involving a cat interacting with a  “small flask of hydrocyanic acid” (and a Geiger counter etc.) such that, at some point the best quantum description of the cat-flask system is an entangled state. If one then ignores the state of the flask, the cat is in a mixture of being dead and alive (and not in a superposition as some wold have you believe). Schrödinger noted that this is a peculiar situation,  the cat’s state has large uncertainty (it is maximally mixed) while the cat+flask etc. are in a well defined state, that is, the uncertainty is at the minimum allowed by the theory. We call such a state a pure state.

Schrödinger coined the term entanglement in the context of pure quantum states. A pure quantum state describing two subsystems is entangled  if (and only if) the state of each subsystem is mixed, i.e (within the context of the relevant operators) there is no (rank 1) measurement that yields a definite outcome¹.  But in reality the states we encounter are mixed and Schrödinger’s definition cannot be applied in a straightforward way.

A mixed quantum state is similar to a composite color such as pink, brown or white which have no specific wavelength.  Any composite color can be made by mixing elements from a set of primary colors such as red green and blue (RBG) but one can choose different conventions to produce the same color². Similarly, a mixed quantum state does not have a unique decomposition in terms of pure quantum states.  The cat in the box is in a mixture of being dead and alive, yet it is also in a mixture of being in various superpositions of dead and alive.  It turns out that this creates a serious problem when we try to define mixed state entanglement.

The standard way to define entanglement is to look for a decomposition into pure non-entangled states. So, if we can find some way to describe the mixed quantum state as a mixture of non-entangled (i.e separable) states, then the state is separable.  This is a convenient mathematical definition ³ but it is not consistent with the physical manifestation of entanglement.

What is the physical manifestation of entanglement? 

One way  to think of entanglement is as a resource for some physical tasks such as teleportation or quantum communication.  Ideally one would want to make a claim such as “If you gave me enough copies of an entangled state I could perform perfect teleportation”.  Indeed this would be the case if the states were pure, but in the case of mixed states there are counterexamples to this statement for practically any physical task (except channel discrimination).

Another way to think about entanglement is as a way to quantify complexity.  The intuition comes from the fact that a good enough description⁴  of an entangled state usually requires a very large memory.  If a system is in a pure quantum state and it is not entangled, we can fully describe it by specifying each part.  If it is entangled, we must also specify some global properties. Roughly speaking, these global properties describe the relations between the subsystems, and the number of parameters we need to keep track of grows exponentially with the number of subsystems.  However, as it turns out, some highly entangled states can be described in a very concise way.  When it comes to mixed states, the situation is different and it is unclear if we can give a concise description of a separable system.

The bottom line is,  the physical manifestation of entanglement is not trivial, especially when we consider mixed states. As a result, there is no obvious one-size-fits-all way to extend various ideas about entanglement to mixed states.

Quantum correlations and discord

So, while there is no unique way to generalize entanglement to mixed states,  one particular method (entangled = not separable) has become canonical. Other ways of generalizing entanglement from pure states must be given a different name. Many of these fall into the broad category of quantum correlations (or discord).  These quantities are equivalent to entanglement in pure states, but don’t correspond to non-separable in the case of mixed states.

Ok, but why should we care?

Entanglement is one of the central features of quantum theory, and there is good reason to suspect that it plays a crucial role in many physical scenarios, from many body physics to black holes and of course quantum information processing.  Unfortunately, it is not trivial to extend our mathematical treatment of entanglement beyond the two party, pure state case.   There are many examples  where separable mixed states or ensembles of separable pure states,  behave in a way that resembles pure entangled states.  Apart from the obvious joy of playing around with the mathematical structure of quantum states, there are many things we can learn by trying to understand this rich structure beyond the usual separable vs non-separable states.  Discord is one, and there are others, most notably Bell non-locality.

And if you want to know more, check out my paper with Danny Terno , arXiv:1608.01920

Footnotes

1.  The caveats here are simply to ensure that the measurement is not trivial in some sense. For example if the states are entangled in spin, asking about their position is not relevant, similarly making a trivial measurement (one that has outcome 1 if the spin is up and the same outcome 1 if the spin is down) is not interesting.
2. Actually,  the situation with colors is far more complex than I described, but as far as the human eye is concerned the statement is more or less correct.  Spectroscopy would reveal a unique decomposition to any color.  Quantum states on the other hand have no unique decomposition, in fact, if they did we would be in big trouble with relativistic causality (i.e we would be able to send information faster than the speed of light). As a side note: Schrödinger was interested in our perception of colors and made some interesting contributions to the field.
3. Given the complete description of a (mixed) quantum state, it can be very difficult (computationally) to decide if it is entangled or separable.
4. Think of trying to keep a description of the state in the memory of a computer for the purpose of simulating the evolution and finally reproducing some measurement statistics.