Also read – Creating Operators and Eigenvectors

What are Product States?

Product States are states of entangled particles – or particles that have interacted in some manner (some energy exchange). They are ‘correlated’ in this sense.

They are written as the product of state 1 and state 2 – but given that this is quantum mechanics, we have to consider the superposition of all such products.

So – | 1> |2> + |2> |1>   – would be the superposition of states 1 and 2.

How are these different from classically correlated states?

If you know the combined (correlated state) in classical physics, you also know  everything about the INDIVIDUAL component states.

Here is an example (borrowed from Susskind’s Quantum Mechanics):

If Joe hands out two dice – one to Alice and one to Bob – these dice are correlated – in that they came from the same source. Before handing them out, Joe peeks at the entire state – i.e. he sees what is showing on DICE 1 and what is showing on DICE 2.

Now, no matter how far apart Alice and Bob take their individual die, Joe will always know what  the individual reading on each die was.  So – knowledge of the ORIGINAL COMBINED state – also leads to DIRECT KNOWLEDGE of the individual states.

This is JUST NOT TRUE in Quantum Mechanics.

 

Why is it difficult to visualize these product states?

Even the simplest SINGLE particle state  (in QM) is a 2 state system. So – a single particle is itself a multi-state system. Now, think about more than one particle – and the number of particle states. Instead of individual particle states, there is now a ‘combined state’ of the multiple particles.

What does the product state operator act on? Individual states or a product state?

Are measurements the same as Operating on a state?

In general, No. Only when the state is actually in an eigenstate of the operator, can we say that ‘measuring’ the state is the same as operating on a state vector.

Prepared states vs. Unknown states

This is at the heart of a lot of QM confusion when speaking about Operators and States. States  can be KNOWN in advance of a measurement (i.e. we PREPARE the state to be a specific state). This can be PREPARED For either single particle or MULTI particle states.