Survival Guide for Quantum Computing a Gentle Introduction IV: MultiQubit systems and Tensors
Reiffel starts off her Chapter on Multi-Qubit systems (pp. 31 – 46) on the wrong foot by saying “Unlike classical systems, the state space of a quantum grows exponentially with the number of particles”. Even a single qubit has an uncountably infinite number of states — https://luysii.wordpress.com/2024/09/04/survival-guide-for-quantum-computing-a-gentle-introduction-iii-state-representations/. What she means is that the dimension of a multiparticle quantum system grows exponentially with the number of particles.
Just as the number of states of a classical computer register grows exponentially with the number of bits (a byte has 8 bits and 256 states, two bytes have 2^16 65536 states etc. etc.) The dimension of the complex vector space representing a quantum system with n qubits is 2^n.
I still do not understand her statement “often the state of a system cannot be described in terms of the states of its component pieces. Such states are called entangled states.” This, although I think I understand what entanglement is at least mathematically (see below) and hopefully you will too.
Mathematicians can define mathematical object any way they like and tensor products (don’t panic) were defined before quantum mechanics was a gleam in the eye of any physicist. If you’ve ever looked at tensors you seen horrible expressions with multiple summations, subscripts and superscripts galore. Relax, the tensors used by quantum computation don’t use them.
Quantum computation looks at tensors as a way of combining vector spaces to build bigger and better ones. The awful expressions are about changing the basis of the underlying vector spaces (which doesn’t happen for the discussion to follow).
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) n*m. It’s 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 ßij a complex number. The definition of basis means that every element of V®W is expressible as a linear combination of 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 0.
I don’t think Reiffel explained entanglement very well. It all comes from the 3 following abstract relations on ®, all put in by mathematicians long before quantum mechanics existed.
|v1>, |v2>, |v> are vectors in vector space V
|w1>, |w2>, |w> are vectors in vector space W
a is a complex number
1. (|v1> + ||v2>) ® |w> = |v1> ® |w> + |v2> ® |w>
2. |v > ® (|w1> ® |w2>) = |v> ® |w1> + |v> ® |w2>
1 and 2 are called multilinearity
3. a|v> ® |w> = |v> ® a|w> = a (|v> + ||w>)
Next, consider (|v1> + ||v2>) ® (|w1> + |w2>)
Multilinearity gives you all 4 possible tensor products. Here they are
|v1> ® |w1> +
|v1> ® |w2> +
|v2> ® |w1> +
|v2> ® |w2>
Now consider
|v1> ® |w2> +
|v2> ® |w1>
It is not the tensor product of (|v1> + ||v2>) and (|w1> + |w2>) because it lacks two of the 4 terms of the product. In fact it satisfies the definition of an entangled
state of V ® W ::= It can’t can’t be written as the tensor product of a vector in V (which (|v1> + ||v2>) is) with a vector in W (which (|w1> + |w2>) is).
I guess this is what she means by “”often the state of a system cannot be described in terms of the states of its component pieces”.
If you are wondering where the qubits went, the above article was about generic vector spaces, of which the 2 dimensional complex vector space of the qubit is just a single example.
Why such a mathematial contraption should model quantum mechanics and quantum computation is not known, but that is the way it is. Deal with it.
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