Well, quantum computing’s use of tensors does not involve the horror of multiple sums of multiple objects with both subscripts and superscripts, but for multiQubit systems (which is what the book is focused on) more notation is needed to keep things clear and short.
Here is the definition of tensor product again
The tensor product takes two vector spaces (V and W) of dimension n and m and combines them to form the tensor product (written V ® W) of dimension n*m. It is not your father’s product like +, *, / which takes two objects from a set and produces another object from the same set.
® : V x W –> V ® W
® : (|v>,|w>) –> |v>®|w> ; |v>,|w> are the notations for vectors used by quantum computing
Let V have an n dimensional basis { |a1>, .. |an> }
W have an m dimensional basis { |b1>, .. , |bm> }
By definition, the basis of V ® W is the n*m elements
{ |ai> ® |bj> } where i ranges from 1 to n, j ranges from 1 to m.
Given ßnm, a complex number, The definition of basis means that every element of V®W is expressible as a linear combination of n*m basis elements of the form
ß11 (|ai> ® |bj>) + . . + ß1m ( |a1> ® |bm>) +
ß21 (|a2> ® |bj>) + . . + ß2m ( |a2> ® |bm>) +
+
+
ßn1 (|an> ® |bj>) + . . + ßnm ( |an> ® |bm>)
Some ßij’s can be zero
This is all well and good and is generalized to fit any pair of finite dimensional vector spaces, but we’re going to be dealing, not with pairs of vector spaces, but with n vector spaces, all of whom are qubits who live in a 2 dimensional complex vector space. So we need 3 separate notation shifts to make things manageable.
So we’ll start with an n Qubit system, each member of which is in the standard basis {|0>,|1> } forming a 2^n dimensional complex vector space by tensoring the n Qubits together. Everything from here on refers to a n Qubit system with the standard basis {|0>,|1> }. Here is the basis, counting the qubits starting at 0 and ending at n-1.
|0>n-1 means that the value of the (n-1)th qubit is 0
It is a set with 2^n members
{ |0>n-1 ® . . . ® |0>1 ® |0>0 ,
|0>n-1 ® . . . ® |0>1 ® |1>0 ,
|0>n-1 ® . . . ® |1>1 ® |1>0 ,
. . .
|1>n-1 ® . . . ® |1>1 ® |1>0 }
First notation change
Bag the ®
|alpha> ® |beta> goes to |alpha>|beta>
and drop the n-1’s so that it is given by the position of what follows
The basis becomes
{ |0> . . . |0>|0>,
|0> . . . |0>|1>,
|0> . . . |1>|1>,
. . .
|1> . . . |1>|1> }
Second notation change
Bag the |’s and >’s
The basis becomes
{ |0 . . . 00>,
|0 . . . 01>,
|0 . . . 11>,
. . .
|1 . . . 11> }
Third notation — really an alternate
{ |0 . . . 00>, becomes |0>
|0 . . . 01>, becomes |1>
|0 . . . 11>, becomes |2>
. . .
|1 . . . 11> } becomes |n-1>
E.g. regard the basis elements as expression in numbers to the base 2 (binary notation).
{ |0>, |1>, |2>, , , , |n-1> }
Get used to this, as Reiffel uses it a lot.
In this notation the standard basis for a 2 qubit system is
{ |00>, |01>, |10>, |11> notation2
or
{ |0>, |1>, |2>, |3> } notation3
and for a 3 qubit system
{ |000>, |001>, |010>, |011>, |100>, |101>, |110>, |111> } notation2
or
{ |0>, |1>, |2>, |3>, |4>, |5>, |6>, |7> } notation3
Well that was dull wasn’t it? Here is the beginning of the payoff. If you’re interested in Quantum Mechanics at all, never mind computation you’ve heard of Bell’s theorem which shows that local variables are inconsistent with quantum mechanics and the experimental work that shows that quantum mechanical predictions on a two state system are correct, giving experimental proof that the local variable theory of reality is wrong. It starts with the
Bell basis for a 2 qubit system, which are a new and different set of linear combinations of the standard basis elements {|0>,|1> }. The 4 of them are also called Bell states.
| Phi+ > = ( 1/√2 ) (|00> + |11> )
| Phi- > = ( 1/√2 ) (|00> – |11> )
| Psi+ > = ( 1/√2 ) (|01> + |10> )
| Psi- > = ( 1/√2 ) (|01> – |10> )
The ( 1/√2 ) is to normalize them so that =1 etc.
It is worth unpacking |00> + |11> to understand why all 4 Bell states are entangled.
First |00 > is really |0> ® |0>, so if you wanted to untangle |00> + |11> into a tensor product of linear combinations of |0> and |1> you would have to have something like this
(a1 |0> + b1|1>) ® (a2 |0> + b2|1>)
Why not (a1 |0> + a2|0>) ® (b1 |1> + b2|1>) ?
Because you would only get |0>®|1> == |01> and not the ones you want |00> and |11>
So plow ahead and use
(a1 |0> + b1|1>) ® (a2 |0> + b2|1>)
giving a1a2 |00> + a1b2 |01> + a2b1 |10> + b1b2 |11>
to get rid of the a1b2 |01> either a1 = 0 or b2 = 0
a1=0 won’t work as it implies a1a2 = 0 getting rid of a term you want |00>
b2=0 won’t work either as it implies a2b2 = 0 getting rid of another term you want |11>
It’s a slightly different proof of entanglement than that given earlier, but it works for all 4 Bell states.
Now you are ready for Bell’s theorem (to be described later).
There are further subtlities of entanglement which I won’t go into here, but on p. 39 Reiffel notes that “entanglement depends on the particular decomposition into subsystems under considerations. States entangled with respect to single unit decomposition (such as the Bell states) may be untangled with respect to decomposition into subsystems (made of grouped qubits)”. A complicated example is given.
Previous post — https://luysii.wordpress.com/2024/09/05/survival-guide-for-quantum-computing-a-gentle-introduction-iv-multiqubit-systems-and-tensors/
