WEBVTT

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Why can an implication measurement become a constant?

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A numeric implication C M between bra X and ket Y gives the formula X implies Y.

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A logical matrix also contains formulas in its cells. We will trace how the bra and ket match the

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expressions defining that logical matrix, why the matched
result is constant, and what changes with a different ket.

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The four cells are X implies Y, X implies not Y,

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not X implies Y, and not X implies not Y.

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Bra X is X, not X; ket Y is Y above not Y.

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X and Y mean the same expressions everywhere in this calculation.

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Use both entries in the first column.

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Pair X with X implies Y, and not X with not X implies Y.

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AND each pair, then exclusive-or the two products. They

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AND each pair, then exclusive-or the two products. They simplify to Y.

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The whole first column participates in this contraction.

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Now use both entries in the second column. Pair X with X implies not Y,

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and not X with not X implies not Y.

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and not X with not X implies not Y. AND each pair, then exclusive-or the two products. They

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simplify to not Y, leaving the complete intermediate row [Y, not Y].

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Contract that row with ket Y. Y AND Y

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Contract that row with ket Y. Y AND Y is Y; not Y AND not Y

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is not Y. Their exclusive-or is

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is not Y. Their exclusive-or is always one.

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We have not asserted that every implication expression is true.

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This constant follows from the matched symbolic LM measurement we just computed.

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Keep the same bra and logical matrix, but use ket not Y.

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The intermediate row is unchanged. It now contracts with not Y, Y.

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Both products contain a value and its negation, so

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Both products contain a value and its negation, so both vanish. The

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Both products contain a value and its negation, so both vanish. The result is zero.

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The four choices of X or not X on the left and Y or not Y on the right return

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the four numeric entries of implication. This is a property of the matched symbolic basis.

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It does not license substituting an unrelated expression without recalculation.

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Replace the right ket with ket Z, keeping the same X bra
and the same implication logical matrix built from X and Y.

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Our intermediate row remains Y, not Y. What expression results when it meets Z, not Z?

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The result is

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The result is Y and Z exclusive-or

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The result is Y and Z exclusive-or not Y and not Z.

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Those terms are disjoint, and together they say that Y
and Z agree. The measurement is now an expression,

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Y equivalent to Z, rather than the constant one from the matched case.