Tensor Product Ordering

The default tensor-product convention is left-to-right subsystem ordering. If the composite Hilbert space is written

$$\mathcal H_A\otimes\mathcal H_B,$$

then the first ket label refers to subsystem $A$ and the second ket label refers to subsystem $B$.

Thus

$$\lvert ij\rangle \equiv \lvert i\rangle_A\otimes\lvert j\rangle_B$$

unless a page explicitly declares another convention.

Two-Qubit Basis Order

For two qubits in $\mathcal H_A\otimes\mathcal H_B$, the default computational basis order is

$$\lvert00\rangle,\quad \lvert01\rangle,\quad \lvert10\rangle,\quad \lvert11\rangle.$$

A state is represented as

$$\lvert\Psi\rangle =c_{00}\lvert00\rangle +c_{01}\lvert01\rangle +c_{10}\lvert10\rangle +c_{11}\lvert11\rangle,$$

with coefficient column

$$\begin{pmatrix} c_{00}\\ c_{01}\\ c_{10}\\ c_{11} \end{pmatrix}.$$

Equivalently, the left subsystem label changes more slowly than the right subsystem label in the displayed basis order.

Local Operators

An operator acting only on subsystem $A$ is written

$$O_A\otimes I_B.$$

An operator acting only on subsystem $B$ is written

$$I_A\otimes O_B.$$

In compact prose, pages may say ”$O_A$ acts on subsystem $A$” after declaring the composite space. In formulas where ambiguity is possible, the identity factors should be shown.

Quantum Circuits and Software Conventions

Quantum-circuit libraries sometimes use little-endian or register-specific conventions that differ from the pedagogical order above. Pages about algorithms, circuits, or numerical libraries must declare their ordering before writing state vectors, matrices, or measurement bit strings. Translation between conventions is a permutation of basis labels, not a physical change.

Common Mistakes

• Treating $\lvert01\rangle$ as meaningful without declaring which subsystem is first.
• Applying $O\otimes I$ to the wrong subsystem.
• Comparing two coefficient vectors from different ordering conventions without permuting entries.
• Dropping identity factors before the reader can tell which space an operator acts on.
• Assuming a software bit-string convention matches the mathematical convention on the page.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. In the default two-qubit order, write the diagonal of $\sigma_z\otimes I$.

Solution

The operator $\sigma_z\otimes I$ measures the first subsystem. It gives $+1$ on states whose first label is $0$ and $-1$ on states whose first label is $1$. In the basis

$$\lvert00\rangle,\quad \lvert01\rangle,\quad \lvert10\rangle,\quad \lvert11\rangle,$$

the diagonal is

$$(1,1,-1,-1).$$

2. In the same order, write the diagonal of $I\otimes\sigma_z$.

Solution

The operator $I\otimes\sigma_z$ measures the second subsystem. The diagonal is

$$(1,-1,1,-1).$$