WEBVTT

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This overview explains how expression-valued

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logical matrices become binary correspondence
matrices and combine through tensor construction.

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Explicit component order is essential when a larger
matrix is used to recover the intended Boolean result.

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Explicit component order is essential when a larger
matrix is used to recover the intended Boolean result.

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An expression-valued logical matrix contains formulas before values are substituted.

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Here its four cells identify the four possible states of A and B.

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Read each cell as a conjunction, including its negations.

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The entries are expressions, not four copies of one already computed bit.

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Substitute A one and B zero.

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The upper-left cell becomes one and zero, giving zero.

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The upper-right becomes one and one, giving one.

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Both lower cells contain not A, which is zero, so both are zero.

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Exactly one state-selecting cell is active.

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The active-cell behavior came from these particular formulas.

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A different logical matrix need not evaluate to the same pattern.

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Always identify the actual formulas and substitutions before making a claim about what a

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symbolic matrix represents.

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Take two numeric two-by-two factors.

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In the tensor product, each entry of M multiplies a complete copy of N.

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The first one creates an N block, the next zero creates a zero block.

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The lower row creates a zero block followed by another N block.

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The resulting four-by-four array is shown in full.

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A row address consists of the first-factor row followed by the second-factor row, and the

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column address is formed the same way.

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This is a tensor construction, not ordinary two-by-two matrix multiplication.

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Using zero-based numeric indices, select first-factor row one and second-factor row zero.

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That gives row two.

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Select column one then one, giving column three.

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The cell is M one one times N zero one, or one times one.

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This explicit address checks which factor contributes which index.

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Use the numeric tensor just constructed.

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Indices begin at zero.

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Row two is the third row and column three is the fourth column.

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The row address combines first-factor index one with second-factor index zero; the column address combines one with one.

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Select row two with row vector zero zero one zero.

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For each column, and each selector entry with the corresponding matrix entry, then combine the four products by exclusive-or.

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For column two the products are zero, zero, one, zero and reduce to one.

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Doing this for all columns gives the intermediate row zero zero one one.

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Now select column three with column vector zero zero zero one.

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Pairwise and gives zero, zero, zero, one; exclusive-or reduction gives one.

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This is a row, matrix, column contraction.

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The address also gives M at one one times N at zero one, or one.

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Keeping the first factor first in both addresses explains the tensor order.

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symbolic matrix represents.

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A logical matrix contains logical expressions in its cells.

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Ket X is X, not-X, and bra Y is Y, not-Y.

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Their outer product combines each row literal and column
literal using AND, producing four expression-valued cells.

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Set X and Y to one and their negations to zero.

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The base matrix evaluates to 1000, the AND operator CM.

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The equivalence logical matrix uses equivalence
on its diagonal and exclusive-or off the diagonal.

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Under the same valuation it gives 1001.

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For four variables, tensor the complete W-X
base matrix with the complete Y-Z base matrix.

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Each first-matrix entry combines with all four entries of the second.

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The W AND X entry fills the top-left 2×2 block.

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The other three entries fill the other blocks.

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Rows list W-Y, W-not-Y, not-W-Y, not-W-not-Y.

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Columns use X and Z in the same true-first order.

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The paper writes the left measurement factors as bra Y followed by bra W.

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The expanded components shown here define the W-Y row key.

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For the worked rule, W exclusive-or X implies not-Y AND Z.

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Call the inner operator matrices A and B.

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Implication accepts three cases, giving A tensor
B, not-A tensor B, and not-A tensor not-B.

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Their exclusive-or builds the result.

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At W zero, X one, Y one, Z one, the left
result is one and the right result is zero.

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Implication returns zero.

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The row key W-Y is 01,

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and column key X-Z is 11: the third row and first column in true-first order.

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The square construction extends to 2n logical subexpressions
with 2ⁿ rows and the same number of columns.

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Return to the opening expression-valued logical matrix.

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Here A and B again mean Boolean inputs, not the operator matrices named in the symbolic construction.

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Its cells are A and B, A and not B, not A and B, and not A and not B.

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Set A to zero and B to one.

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The entire upper row is zero because A is zero.

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In the lower row, not A is one.

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Combining it with B one gives one in the left cell, while combining it with not B zero gives zero in the right cell.

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The active cell has moved to the lower-left position.

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This case establishes a specific contrast.

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Substitution selects a different state.

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The symbolic formulas themselves are unchanged.

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Keep the input, intermediate values, and conclusion together when comparing it with the previous case.

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The calculated answer belongs to this complete assignment under the stated rule.

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This explicit address checks which factor contributes which index.

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Build and combine the required expression-valued terms.

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Evaluate them under the stated assignment
while preserving the component order.