Lesson 16 of 23 · 2:54
Measure with a logical matrix
Compute the intermediate vectors; distinguish matched constants from general formulas.
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In this lesson
- 0:00 · Why can an implication measurement become a constant?
- 0:25 · Define the implication LM explicitly
- 0:44 · Expand the first intermediate entry
- 1:02 · Expand the second intermediate entry
- 1:19 · Finish the matched contraction
- 1:39 · Change the ket to its matched opposite
- 1:55 · Read all four matched relations
- 2:11 · Your turn: use a different expression
- 2:34 · Answer: an equivalence relation
Complete transcript
This transcript includes the answers to prediction prompts.
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Why can an implication measurement become a constant? · 0:00
Why can an implication measurement become a constant? A numeric implication C M between bra X and ket Y gives the formula X implies Y. A logical matrix also contains formulas in its cells. We will trace how the bra and ket match the expressions defining that logical matrix, why the matched result is constant, and what changes with a different ket.
Define the implication LM explicitly · 0:25
The four cells are X implies Y, X implies not Y, not X implies Y, and not X implies not Y. Bra X is X, not X; ket Y is Y above not Y. X and Y mean the same expressions everywhere in this calculation.
Expand the first intermediate entry · 0:44
Use both entries in the first column. Pair X with X implies Y, and not X with not X implies Y. AND each pair, then exclusive-or the two products. They simplify to Y. The whole first column participates in this contraction.
Expand the second intermediate entry · 1:02
Now use both entries in the second column. Pair X with X implies not Y, and not X with not X implies not Y. AND each pair, then exclusive-or the two products. They simplify to not Y, leaving the complete intermediate row [Y, not Y].
Finish the matched contraction · 1:19
Contract that row with ket Y. Y AND Y is Y; not Y AND not Y is not Y. Their exclusive-or is always one. We have not asserted that every implication expression is true. This constant follows from the matched symbolic LM measurement we just computed.
Change the ket to its matched opposite · 1:39
Keep the same bra and logical matrix, but use ket not Y. The intermediate row is unchanged. It now contracts with not Y, Y. Both products contain a value and its negation, so both vanish. The result is zero.
Read all four matched relations · 1:55
The four choices of X or not X on the left and Y or not Y on the right return the four numeric entries of implication. This is a property of the matched symbolic basis. It does not license substituting an unrelated expression without recalculation.
Your turn: use a different expression · 2:11
Replace the right ket with ket Z, keeping the same X bra and the same implication logical matrix built from X and Y. Our intermediate row remains Y, not Y. What expression results when it meets Z, not Z?
Prediction pause: 10 seconds.
Answer: an equivalence relation · 2:34
The result is Y and Z exclusive-or not Y and not Z. Those terms are disjoint, and together they say that Y and Z agree. The measurement is now an expression, Y equivalent to Z, rather than the constant one from the matched case.