Survival Guide for Quantum Computing a Gentle Introduction VII: Qubit states and their redundant vector space representations

Reiffel spends a lot of time distinguishing between qubit states  (and by inference all quantum states) and their representations as elements of complex vector spaces.

First a word about infinity.  From a completely meatball perspective there are only two types.  The first is the countable type of infinity, whose elements can be put into a one to one correspondence with the positive integers.  The second is the uncountable infinity, whose members are so numerous that they can’t be put into such a correspondence.  Example: the real numbers.  The man who first ordered these two infinities (Georg Cantor) went nuts trying to figure out if there was another infinity between the integers and the real numbers (continuum hypothesis).  Twentieth century logicians (Kurt Godel and Paul Cohen) showed that it didn’t matter, set theory was consistent with or without the continuum hypothesis being true.

To cut to the chase.  Each qubit has an uncountable infinity of representations in the complex vector space of dimension 2 (C^2) where it lives, and the vector space of different qubits has uncountably many distinct members.

This is not mathematical diddle, as it is crucial for correctly measuring qubits, on which all of quantum computation rests.

Given

l. a, b in C ; e.g. a and b are complex numbers

2. Qubit state space (written QSS) ::=

{ a |0> + b |1> such that |a|^2 + |b|^2 = 1

recall that { |0>, |1> } is the standard basis for qubits and that it does not change

3. c1, c2 in C such that |ci|^2 = 1, e.g. ci has modulus 1

#ci = ∞ ; infinity (uncountable in this case)

4. |v1>, |v2>, |v3> in QSS

5.   |v1 > = a|0> + b|1> , |v2 > = c1|v1> = c1(a|0> + b|1>) = c1a|0> + c1b|1>

It is an empirical fact that |v1> and |v2> give the same results when measured making them the ‘same’ or equivalent quantum states.

Since #c1 = ∞, |v1> has an ∞ of vector representations in C^2 which are notated differently but which are actually the same (equivalent in the sense that they all measure the same).

def:  Global phase factor ::= c1 in the equation    |v2 > = c1|v1>

Are there elements of the Qubit State Space that are not the same?

Yes use the contrapositive.  Recall that the contrapositive of

a implies b (written a ==> b)

is

not b implies not a (written ~b ==> ~a

Proving one implies the other

Statement1  |v1>,|v2> in QSS are the same

==>  ; implies

there is a c in C such that |c| =1 &&  |v2 >= c|v1 >

Contrapositive of statement1

|v1>,|v2> in QSS are not the same

<==

there is no c in C such that such that |c| =1 &&  |v2 >= c|v1 >

So prove the contrapositive by constructing an example

l. Fix a and b in C    ; recall that |0> and  |1> are fixed by definition

2. Choose d in C such that |d| =1  ;  There are infinitely many such d’s, e.g.

#d = ∞

3.  Let |v1 > = a|0> + b|1> and |v3 > = a|0> + db|1>

There is no c in C such that |v3 > = c |v1 >  as

c|v1> = ca|0> + cb|1> and ca|0>  <> c|o>

So | v1 > is different from | v3 >

The number of |v3>’s different from |v1> is ∞ as #d = ∞

Qubits  like |v1> and |v3> are physically (as well as mathematically) distinct in that they give different results when measured.

They differ in relative phase, which can be crudely defined as

|coefficient of |0> |/ |coefficient of |1> |

| some sort of number | means the absolute value of that number

Actually all complex numbers can be said to have a phase — for details see argand diagram site:wikipedia.org

Qubits differing in relative phase are physically distinct.