13.21.5 Tensor Contraction Sequence Notation

Tensor Contraction Sequence Notation simplifies tensor operations by sequentially contracting indices through repeated summation.

Tensor Contraction Sequence Notation is the written convention for denoting a multi-step contraction sequence procedure as an ordered chain of labeled arrows or numbered steps connecting successive intermediate tensors, making the order and identity of each contraction step explicit within a single compact expression rather than describing the steps only in prose.

Definition

A contraction sequence is denoted by a chain of arrows, each labeled with the specific contraction applied at that step:

T_{0} \to_{}^{C_{1}} T_{1} \to_{}^{C_{2}} T_{2} \dots \to_{}^{C_{k}} T_{k}

where each $C_{i}$ labels the index pair contracted to pass from $T_{i - 1}$ to $T_{i}$ .

Components of the Notation

Intermediate Labels

Each intermediate tensor along the chain receives its own symbol, typically subscripted by its position in the sequence, allowing any single stage of the sequence to be referenced individually without needing to restate the entire chain up to that point.

Step Labels

The label attached to each arrow specifies precisely which index pair was contracted, providing a compact record of the contraction sequence pair order without requiring separate prose description of each step.

Compact Alternative for Long Chains

For sequences with many steps, an ellipsis notation compresses the middle of the chain while still explicitly displaying the first and last few labeled steps, preserving readability without listing every intermediate stage in full.

Reading a Sequence Notation

Left-to-Right Order

The chain is read left to right, with each arrow indicating that the tensor to its left is transformed into the tensor to its right by the labeled contraction, matching the temporal order in which the steps of the sequence procedure are actually to be carried out.

Verifying Type Progression

Because each arrow corresponds to exactly one contraction, the type pair recorded at each successive intermediate label must decrease by one in both entries relative to the previous label, providing a built-in consistency check readable directly from the notation.

Diagram

Position Within Notation

Sequence notation extends the basic tensor contraction notation from describing a single contraction to describing an entire ordered chain of contractions, providing the written form in which a contraction sequence procedure, together with its specific sequence pair order and its resulting intermediate tensors, is most naturally expressed and communicated.