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.

Helpful background. Bra–ket notation explains vectors and inner products; tensor products introduces combined state spaces.

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.

Bounded operators acting on different ordinary tensor factors commute:

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

For unbounded operators, products need a common domain; the spectral projections of self-adjoint operators on separate factors commute. Fermionic odd mode operators use a graded algebra, so do not apply the displayed ordinary-tensor rule to them without representing the fermionic signs. This factorization statement does not itself assume spacetime separation.

Composite Hamiltonians

A noninteracting bipartite Hamiltonian is written

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

After local terms have been separated, an interaction is the part that cannot be written as a sum $A\otimes I+I\otimes B$:

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

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 bounded local observables (including every finite-dimensional observable),

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

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. Here each coordinate label includes any spin or other internal variables; exchanging only spatial arguments need not give the full exchange sign. For two identical fermions, the total wavefunction obeys

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

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.

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

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

In an orthonormal product basis, explain why

$$\operatorname{Tr}_B(|i\rangle\langle k|_A\otimes|j\rangle\langle\ell|_B) =\delta_{j\ell}|i\rangle\langle k|_A.$$

Solution

Tracing the $B$ factor gives $\operatorname{Tr}(|j\rangle\langle\ell|)=\langle\ell|j\rangle=\delta_{j\ell}$, while the $A$ factor remains. Thus a matrix element $\rho_{ij,k\ell}$ contributes to $(\rho_A)_{ik}$ only when $j=\ell$, and $(\rho_A)_{ik}=\sum_j\rho_{ij,kj}$. This is a sum over the discarded subsystem, not selection of one outcome.

References

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