Factor 122√|00⟩−122√|01⟩+3√22√|10⟩−3√22√|11⟩ into (a|0⟩+b|1⟩)(c|0⟩+d|1⟩), where |a|2+|b|2=1 and |c|2+|d|2=1. What is the value of |a|?

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a=1/2

To factor the given expression, we need to express it in the form (a|0⟩+b|1⟩)(c|0⟩+d|1⟩) and find the values of a, b, c, and d.

Let's start by expanding the expression:

122√|00⟩ - 122√|01⟩ + 3√2√|10⟩ - 3√2√|11⟩

Now we can rewrite each term of the expression using the basis states:

(12√√|0⟩ + 3√√2√|1⟩)(2√√|0⟩ - 3√√2√|1⟩)

We can see that a = 12√√, b = 3√√2√, c = 2√√, and d = -3√√2√.

To find the value of |a|, we need to calculate |a| = √(a^2 + b^2):

|a| = √((12√√)^2 + (3√√2√)^2)
= √(144 + 18)
= √162
= 9√2

So, the value of |a| is 9√2.