Logical matrices and tensor construction

This overview explains how expression-valued logical matrices become binary correspondence matrices and combine through tensor construction. Explicit component order is essential when a larger matrix is used to recover the intended Boolean result. An expression-valued logical matrix contains formulas before values are substituted. Here its four cells identify the four possible states of A and B. Read each cell as a conjunction, including its negations. The entries are expressions, not four copies of one already computed bit. Substitute A one and B zero. The upper-left cell becomes one and zero, giving zero. The upper-right becomes one and one, giving one. Both lower cells contain not A, which is zero, so both are zero. Exactly one state-selecting cell is active. The active-cell behavior came from these particular formulas. A different logical matrix need not evaluate to the same pattern. Always identify the actual formulas and substitutions before making a claim about what a symbolic matrix represents. Take two numeric two-by-two factors. In the tensor product, each entry of M multiplies a complete copy of N. The first one creates an N block, the next zero creates a zero block. The lower row creates a zero block followed by another N block. The resulting four-by-four array is shown in full. A row address consists of the first-factor row followed by the second-factor row, and the column address is formed the same way. This is a tensor construction, not ordinary two-by-two matrix multiplication. Using zero-based numeric indices, select first-factor row one and second-factor row zero. That gives row two. Select column one then one, giving column three. The cell is M one one times N zero one, or one times one. This explicit address checks which factor contributes which index. Use the numeric tensor just constructed. Indices begin at zero. Row two is the third row and column three is the fourth column. The row address combines first-factor index one with second-factor index zero; the column address combines one with one. Select row two with row vector zero zero one zero. For each column, and each selector entry with the corresponding matrix entry, then combine the four products by exclusive-or. For column two the products are zero, zero, one, zero and reduce to one. Doing this for all columns gives the intermediate row zero zero one one. Now select column three with column vector zero zero zero one. Pairwise and gives zero, zero, zero, one; exclusive-or reduction gives one. This is a row, matrix, column contraction. The address also gives M at one one times N at zero one, or one. Keeping the first factor first in both addresses explains the tensor order. symbolic matrix represents. A logical matrix contains logical expressions in its cells. Ket X is X, not-X, and bra Y is Y, not-Y. Their outer product combines each row literal and column literal using AND, producing four expression-valued cells. Set X and Y to one and their negations to zero. The base matrix evaluates to 1000, the AND operator CM. The equivalence logical matrix uses equivalence on its diagonal and exclusive-or off the diagonal. Under the same valuation it gives 1001. For four variables, tensor the complete W-X base matrix with the complete Y-Z base matrix. Each first-matrix entry combines with all four entries of the second. The W AND X entry fills the top-left 2×2 block. The other three entries fill the other blocks. Rows list W-Y, W-not-Y, not-W-Y, not-W-not-Y. Columns use X and Z in the same true-first order. The paper writes the left measurement factors as bra Y followed by bra W. The expanded components shown here define the W-Y row key. For the worked rule, W exclusive-or X implies not-Y AND Z. Call the inner operator matrices A and B. Implication accepts three cases, giving A tensor B, not-A tensor B, and not-A tensor not-B. Their exclusive-or builds the result. At W zero, X one, Y one, Z one, the left result is one and the right result is zero. Implication returns zero. The row key W-Y is 01, and column key X-Z is 11: the third row and first column in true-first order. The square construction extends to 2n logical subexpressions with 2ⁿ rows and the same number of columns. Return to the opening expression-valued logical matrix. Here A and B again mean Boolean inputs, not the operator matrices named in the symbolic construction. Its cells are A and B, A and not B, not A and B, and not A and not B. Set A to zero and B to one. The entire upper row is zero because A is zero. In the lower row, not A is one. Combining it with B one gives one in the left cell, while combining it with not B zero gives zero in the right cell. The active cell has moved to the lower-left position. This case establishes a specific contrast. Substitution selects a different state. The symbolic formulas themselves are unchanged. Keep the input, intermediate values, and conclusion together when comparing it with the previous case. The calculated answer belongs to this complete assignment under the stated rule. This explicit address checks which factor contributes which index. Build and combine the required expression-valued terms. Evaluate them under the stated assignment while preserving the component order.