WEBVTT

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How do a row and a column select one answer?

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Put the exclusive-or C M between bra zero and ket one.

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The paper calls this row-matrix-column calculation a logical measurement,

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and its final result is one Boolean value. We will calculate the intermediate

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vectors with AND and exclusive-or, then check that both legal groupings agree.

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Bra zero is the row zero, one.

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The exclusive-or C M is zero, one above one, zero. Ket one is the column one above zero.

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The dimensions fit: one-by-two, then two-by-two, then two-by-one.

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The final result has one entry.

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We use the shape of matrix multiplication, but with Boolean operations.

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AND takes the place of multiplication. Exclusive-or combines the products in each sum.

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This is not ordinary integer addition, and it is not OR: one exclusive-or one is zero.

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Start by multiplying the bra into the C M. For the first output entry,

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pair the bra with the first matrix column. Zero and zero

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pair the bra with the first matrix column. Zero and zero gives zero. One and one

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pair the bra with the first matrix column. Zero and zero gives zero. One and one gives one.

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Exclusive-or those two terms:

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Exclusive-or those two terms: zero exclusive-or one is one.

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Now use the second matrix column. Zero and one

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gives zero. One and zero

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gives zero. One and zero also gives zero. Their

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gives zero. One and zero also gives zero. Their exclusive-or is zero.

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Together the two column calculations have produced the intermediate row one, zero.

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We have completed the bra-times-matrix part. The intermediate row is one, zero,

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and the ket on the right is still one above zero. The
intermediate row is not yet the measurement result.

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One final row-by-column operation remains.

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Pair the first entries: one and one

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Pair the first entries: one and one is one. Pair the second entries:

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zero and zero

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zero and zero is zero. Exclusive-or those products to

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zero and zero is zero. Exclusive-or those products to get one.

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In this left-first calculation, the final terms appear as one exclusive-or zero.

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We can also keep the same operands and first apply
the C M to the ket. Its

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We can also keep the same operands and first apply
the C M to the ket. Its top row produces zero, and its

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bottom row produces one. The intermediate object is now the

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bottom row produces one. The intermediate object is now the column zero above one.

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This changes the grouping, not the input order.

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Bra zero is still zero, one.

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Pair it with that new column. The products are

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Pair it with that new column. The products are zero and zero, then

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Pair it with that new column. The products are zero and zero, then one and one.

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Their exclusive-or is zero exclusive-or one,

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Their exclusive-or is zero exclusive-or one, giving one.

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Both legal groupings agree on the measurement result.

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Keep bra zero and the same exclusive-or C M, but replace ket one with ket zero.

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Ket zero is zero above one.

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Use the intermediate row from the left-first calculation. What single bit do you get now?

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The intermediate row is still one, zero. Pair it with zero above one.

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Both AND products are zero, so their exclusive-or is zero.

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That agrees with the meaning of exclusive-or: the two input states are now both zero.

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You can now calculate a logical measurement instead of only pointing at a cell.

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Next we will use expression-valued state vectors with a numeric implication C M.

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The same arithmetic will recover the whole implication expression from the paper.