Lesson 3 of 23 · 3:39
Calculate a CM measurement
Compute both groupings of ⟨0|[⇕]|1⟩ using AND and XOR.
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In this lesson
- 0:00 · How do a row and a column select one answer?
- 0:26 · Replace every symbol by its array
- 0:43 · AND the pairs; XOR the terms
- 1:02 · First intermediate entry
- 1:20 · Second intermediate entry
- 1:36 · Keep the remaining ket visible
- 1:51 · Finish the contraction
- 2:07 · You can calculate the right side first
- 2:25 · The other grouping gives the same answer
- 2:43 · Your turn: change the right input
- 3:08 · Answer: equal inputs give zero
- 3:24 · From a numeric answer to a symbolic rule
Complete transcript
This transcript includes the answers to prediction prompts.
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How do a row and a column select one answer? · 0:00
How do a row and a column select one answer? Put the exclusive-or C M between bra zero and ket one. The paper calls this row-matrix-column calculation a logical measurement, and its final result is one Boolean value. We will calculate the intermediate vectors with AND and exclusive-or, then check that both legal groupings agree.
Replace every symbol by its array · 0:26
Bra zero is the row zero, one. The exclusive-or C M is zero, one above one, zero. Ket one is the column one above zero. The dimensions fit: one-by-two, then two-by-two, then two-by-one. The final result has one entry.
AND the pairs; XOR the terms · 0:43
We use the shape of matrix multiplication, but with Boolean operations. AND takes the place of multiplication. Exclusive-or combines the products in each sum. This is not ordinary integer addition, and it is not OR: one exclusive-or one is zero.
First intermediate entry · 1:02
Start by multiplying the bra into the C M. For the first output entry, pair the bra with the first matrix column. Zero and zero gives zero. One and one gives one. Exclusive-or those two terms: zero exclusive-or one is one.
Second intermediate entry · 1:20
Now use the second matrix column. Zero and one gives zero. One and zero also gives zero. Their exclusive-or is zero. Together the two column calculations have produced the intermediate row one, zero.
Keep the remaining ket visible · 1:36
We have completed the bra-times-matrix part. The intermediate row is one, zero, and the ket on the right is still one above zero. The intermediate row is not yet the measurement result. One final row-by-column operation remains.
Finish the contraction · 1:51
Pair the first entries: one and one is one. Pair the second entries: zero and zero is zero. Exclusive-or those products to get one. In this left-first calculation, the final terms appear as one exclusive-or zero.
You can calculate the right side first · 2:07
We can also keep the same operands and first apply the C M to the ket. Its top row produces zero, and its bottom row produces one. The intermediate object is now the column zero above one. This changes the grouping, not the input order.
The other grouping gives the same answer · 2:25
Bra zero is still zero, one. Pair it with that new column. The products are zero and zero, then one and one. Their exclusive-or is zero exclusive-or one, giving one. Both legal groupings agree on the measurement result.
Your turn: change the right input · 2:43
Keep bra zero and the same exclusive-or C M, but replace ket one with ket zero. Ket zero is zero above one. Use the intermediate row from the left-first calculation. What single bit do you get now?
Prediction pause: 10 seconds.
Answer: equal inputs give zero · 3:08
The intermediate row is still one, zero. Pair it with zero above one. Both AND products are zero, so their exclusive-or is zero. That agrees with the meaning of exclusive-or: the two input states are now both zero.
From a numeric answer to a symbolic rule · 3:24
You can now calculate a logical measurement instead of only pointing at a cell. Next we will use expression-valued state vectors with a numeric implication C M. The same arithmetic will recover the whole implication expression from the paper.