I Is Entanglement Properly Defined in Bohr's Idea of Complementarity?

[bhobba]

What am I missing?

I think you may have picked up a logical issue with the paper - at least a badly expressed idea. It says 'An interesting finding is that quantum theory is the only non-classical probability theory that can exhibit entanglement without conflicting one or more axioms.'

It has not ruled out entanglement in classical probability theory - but for the life of me I can't think of how classical probability theory can have entanglement. It has me beat. In fact the following paper states it outright:
http://staff.utia.cas.cz/swart/lecture_notes/qua17_04_05.pdf
' In classical probability, entangled states do not exist'

Thanks for your careful reading - I should have spotted it - but didn't.

Someone else may be able to disentangle (pun intended) the intent - but I can't see it at the moment.

Thanks
Bill

 

[RUTA]

I think you may have picked up a logical issue with the paper - at least a badly expressed idea. It says 'An interesting finding is that quantum theory is the only non-classical probability theory that can exhibit entanglement without conflicting one or more axioms.'

It has not ruled out entanglement in classical probability theory - but for the life of me I can't think of how classical probability theory can have entanglement. It has me beat. In fact the following paper states it outright:
http://staff.utia.cas.cz/swart/lecture_notes/qua17_04_05.pdf
' In classical probability, entangled states do not exist'

Thanks for your careful reading - I should have spotted it - but didn't.

Someone else may be able to disentangle (pun intended) the intent - but I can't see it at the moment.

Thanks
Bill

I just emailed Professor Dakic, perhaps someone in his group will clarify their use of the terminology in that paper. I'll let you know what they say :-)

 

[RUTA]

I just emailed Professor Dakic, perhaps someone in his group will clarify their use of the terminology in that paper. I'll let you know what they say :-)

Here is Dakic's reply:

the sentence in question: "As we will see later, taking the latter choice, it will follow from axiom 1 alone that the state space must contain entangled states." refers to the previous: ...also called “generalized bit”. So, the logic is that any 1 bit-system obeying Axiom 1 is either: a) discrete (and in this case only classical bit is a consistent solution) or b) continuous (such as real, quantum or some general bit). In the latter case, entanglement is the necessary feature. The only counter-example to this is the classical bit and consequently classical probability theory. They do not exhibit entanglement.

Here is the entire paragraph, so that his reply makes more sense:

In logical terms axiom 1 means the following. We can think of two basis states as two binary propositions about an individual system, such as (1) “The outcome of measurement A is +1” and (2) “The outcome of measurement A is -1”. An alternative choice for the pair of propositions can be propositions about joint properties of two systems, such as (1’) “The outcomes of measurement A on the first system and of B on the second system are correlated” (i.e. either both +1 or both -1) and (2’) “The outcomes of measurement A on the first system and of B on the second system are anticorrelated”. The two choices for the pair of propositions correspond to two choices of basis states which each can be used to span the full state space of an abstract elementary system (also called “generalized bit”). As we will see later, taking the latter choice, it will follow from axiom 1 alone that the state space must contain entangled states.

So, yes, trivially the discrete (classical bit) option is an "entangled state" in the sense of the "alternative choice for the pair of propositions about joint probabilities," i.e., the outcomes are either correlated or anti-correlated. But, since there is no continuous mapping between such basis states in the classical case, there is no entanglement proper, as is well known. Thus, strictly speaking, I would say axiom 1 and axiom 3 are both required for entanglement. Maybe I'm missing something. What to the mathemagicians say?