Notation and Subsystem Labels

Subsystem notation should tell the reader which Hilbert-space factor a state or operator belongs to. In this volume, labels such as $A$, $B$, and $C$ identify tensor factors, not hidden classical identities carried by particles.

The default bipartite convention is

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B,$$

with $A$ written before $B$ unless stated otherwise.

Subsystem Labels

Subsystems are labeled by capital roman letters:

$$A,\ B,\ C,\ldots .$$

Their Hilbert spaces are written

$$\mathcal H_A,\qquad \mathcal H_B,\qquad \mathcal H_C.$$

The combined Hilbert space of $A$ and $B$ is

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B.$$

For three subsystems, the default ordering is

$$\mathcal H_{ABC} = \mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C.$$

Parentheses may be inserted for clarity, but finite tensor products are naturally associated up to canonical isomorphism. Ordering still matters for notation, matrices, and computational basis conventions.

Product States

A product state of two subsystems is written

$$\lvert\psi\rangle_A\otimes\lvert\phi\rangle_B.$$

When the tensor factors are unambiguous, the tensor-product symbol may be suppressed:

$$\lvert\psi\rangle_A\lvert\phi\rangle_B.$$

For basis states, compact notation may be used after the ordering is declared:

$$\lvert i\rangle_A\otimes\lvert j\rangle_B \equiv \lvert ij\rangle_{AB}.$$

The subscript on the compact ket is a reminder that the order is $A$ then $B$. If the order is changed, the same symbols can represent a different coordinate vector.

Product Bases

If one basis of $\mathcal H_A$ is labeled by $i$ and one basis of $\mathcal H_B$ is labeled by $j$, the product basis is

$$\lvert i\rangle_A\otimes\lvert j\rangle_B.$$

For two qubits, the default computational basis order is

$$\lvert 00\rangle,\quad \lvert 01\rangle,\quad \lvert 10\rangle,\quad \lvert 11\rangle.$$

In this ordering, a general two-qubit state is written

$$\lvert\Psi\rangle = c_{00}\lvert 00\rangle +c_{01}\lvert 01\rangle +c_{10}\lvert 10\rangle +c_{11}\lvert 11\rangle.$$

The corresponding coordinate column is

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

Some communities use different endian conventions for multi-qubit registers. A page or calculation that depends on bit ordering must state the convention before using compact labels.

Local Operators

An operator acting only on subsystem $A$ is written on the full Hilbert space as

$$O_A\otimes I_B.$$

An operator acting only on subsystem $B$ is

$$I_A\otimes O_B.$$

When no confusion can arise, authors often write $O_A$ for the embedded operator $O_A\otimes I_B$. This volume keeps identity factors explicit whenever the expression is new, when commutators are being computed, or when several tensor factors are present.

Operators on different subsystems commute:

$$[O_A\otimes I_B,\ I_A\otimes O_B] =0.$$

This commutation is a statement about tensor-factor support, not necessarily about spacetime separation. Relativistic locality is a later field-theoretic refinement.

Composite Hamiltonians

A noninteracting bipartite Hamiltonian is written

$$H_0 = H_A\otimes I_B +I_A\otimes H_B.$$

An interaction term is an operator that cannot be assigned to just one factor:

$$H = H_0+V_{AB}.$$

For two qubits, an example is

$$V_{AB} = J\,\sigma_x\otimes\sigma_x.$$

The placement of each Pauli matrix is part of the definition. With more qubits, writing identities explicitly avoids mistakes:

$$\sigma_x^{(2)} = I\otimes\sigma_x\otimes I$$

for a three-qubit register ordered as $1,2,3$.

Density Operators and Partial Trace

A joint density operator on $AB$ is written $\rho_{AB}$. Its reduced states are

$$\rho_A = \operatorname{Tr}_B\rho_{AB}, \qquad \rho_B = \operatorname{Tr}_A\rho_{AB}.$$

The subscript on the trace names the subsystem being discarded. Thus $\operatorname{Tr}_B$ leaves an operator on $\mathcal H_A$, while $\operatorname{Tr}_A$ leaves an operator on $\mathcal H_B$.

For expectation values of local observables,

$$\operatorname{Tr}_{AB} \bigl[ \rho_{AB}(O_A\otimes I_B) \bigr] = \operatorname{Tr}_A(\rho_A O_A).$$

This identity is one of the main reasons reduced states are useful: they encode all local statistics for their subsystem.

Tensor Products Versus Ordinary Products

The expression

$$\lvert\psi\rangle_A\otimes\lvert\phi\rangle_B$$

is a vector in a new Hilbert space. It is not a product of two numbers. The scalar product of two product vectors is

$$\bigl( {}_A\langle\psi_1\vert\otimes{}_B\langle\phi_1\vert \bigr) \bigl( \lvert\psi_2\rangle_A\otimes\lvert\phi_2\rangle_B \bigr) = \langle\psi_1\vert\psi_2\rangle_A \langle\phi_1\vert\phi_2\rangle_B.$$

The right-hand side is an ordinary product of complex numbers. The left-hand side uses a tensor product of vectors and covectors.

Identical-Particle Caution

For distinguishable subsystems, labels such as $A$ and $B$ may refer to two laboratories, two qubits, two atoms in separated traps, or two different species. For identical particles, labels in a wavefunction such as $x_1$ and $x_2$ are coordinate arguments in a formal description, not directly trackable particle names.

The symmetrization postulate fixes the allowed exchange sectors. For two identical fermions, the exchange rule is

$$\Psi(x_2,x_1) = -\Psi(x_1,x_2).$$

For two identical bosons, it is

$$\Psi(x_2,x_1) = \Psi(x_1,x_2).$$

Questions about entanglement for identical particles therefore require care: one must specify whether the relevant split is by modes, spatial regions, spin degrees of freedom, species, or an algebra of observables.

Common Mistakes

• Dropping subsystem labels before the tensor-product order has been declared.
• Writing a local operator without the identity factors in a calculation where the full Hilbert space matters.
• Reading a compact ket as basis-independent notation.
• Confusing the partial trace over $B$ with a projection onto a particular state of $B$.
• Treating identical-particle coordinate labels as hidden particle names.
• Mixing qubit endian conventions without warning.

Cross-Links

• Why Composite Systems Matter
• Concept Map
• Tensor Products of Hilbert Spaces
• Tensor Product Ordering
• Composite Hamiltonians
• Entanglement Depends on a Decomposition
• Indistinguishability
• Symmetrization Postulate
• Core Formalism: Tensor Products
• Mathematical Toolkit: Tensor Products
• Density Operators

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. In a three-qubit register ordered as $A,B,C$, write the operator that applies $\sigma_z$ to $B$ only.

Solution

The operator is

$$I_A\otimes\sigma_z^{(B)}\otimes I_C.$$

If the subsystem labels are clear, this may also be written as $I\otimes\sigma_z\otimes I$.

2. A two-qubit state is written in the default basis order as the column vector

$$\begin{pmatrix} 0\\ 1/\sqrt2\\ 1/\sqrt2\\ 0 \end{pmatrix}.$$

Write the corresponding ket.

Solution

Using the default order

$$\lvert 00\rangle,\quad \lvert 01\rangle,\quad \lvert 10\rangle,\quad \lvert 11\rangle,$$

the ket is

$$\frac{1}{\sqrt2} \bigl( \lvert 01\rangle+\lvert 10\rangle \bigr).$$

3. If $\rho_A=\operatorname{Tr}_B\rho_{AB}$, on which Hilbert space does $\rho_A$ act?

Solution

It acts on $\mathcal H_A$. The notation $\operatorname{Tr}_B$ means that subsystem $B$ has been traced out, leaving an operator on the remaining subsystem.