Composite Systems and Entanglement

Composite quantum systems are described by tensor-product Hilbert spaces. If subsystem $A$ has Hilbert space $\mathcal H_A$ and subsystem $B$ has Hilbert space $\mathcal H_B$, the joint system is described by

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

That rule organizes product states, local observables, entangled states, and reduced density operators. Identical-particle statistics and Fock space add further structure, developed in Many-Body and Quantum Statistical Mechanics.

This volume is the canonical home for the structure of quantum composition. The short Core Formalism pages introduce the postulate; this volume develops the calculation language and the conceptual distinctions in depth.

Required background. State Vectors supplies pure-state representations and basis expansions. Density Operators supplies positive trace-one operators for mixed and reduced states. Familiarity with tensor-product linear algebra is assumed for forming composite spaces, bases, and operators.

What This Volume Covers

The main themes are:

• tensor-product Hilbert spaces, product bases, subsystem labels, and operators on composite systems;
• product states, separable mixed states, Bell states, singlet and triplet states, local-unitary equivalence, decomposition-dependent cautions, and entangled states;
• reduced density operators, partial traces, local measurement statistics, purification, and subsystem entropy;
• Schmidt decomposition, Schmidt rank, entanglement entropy, Renyi entropies, mutual information, GHZ and W states, and elementary entanglement diagnostics;
• the distinction between subsystem entanglement and exchange symmetry, with links to the many-particle foundations;
• bridges to quantum information, many-body physics, quantum chemistry, AMO physics, foundations, and QFT.

The organizing distinction is:

$$\text{composition} \quad\neq\quad \text{correlation} \quad\neq\quad \text{entanglement} \quad\neq\quad \text{exchange symmetry}.$$

Composition is the tensor-product structure. Correlation is a property of measurement statistics. Entanglement is nonseparability relative to a chosen subsystem decomposition. Exchange symmetry is the constraint imposed when the particles are identical.

Why Tensor Products Are Not Optional

Classically, the state of two systems is often represented by an ordered pair of states or by a Cartesian product of phase spaces. Quantum theory needs a vector space in which amplitudes can superpose. The tensor product supplies that vector space.

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

$$|i\rangle_A\otimes |j\rangle_B$$

form a basis of $\mathcal H_A\otimes\mathcal H_B$. A general vector is

$$|\Psi\rangle = \sum_{ij} c_{ij}\, |i\rangle_A\otimes |j\rangle_B.$$

Only special coefficient arrays factor as $c_{ij}=a_i b_j$. Those special states are product states. The generic state in a tensor-product Hilbert space is not a product state, and this is why entanglement is structural rather than exotic.

Local States and Lost Information

A subsystem of an entangled pure state is usually not described by a state vector. Its autonomous statistics are described by a reduced density operator:

$$\rho_A = \operatorname{Tr}_B \rho_{AB}.$$

For the Bell state

$$|\Phi^+\rangle = \frac{|00\rangle+|11\rangle}{\sqrt 2},$$

the reduced state of either qubit is maximally mixed:

$$\rho_A = \rho_B = \frac12 I.$$

This does not mean the pair is classically uncertain about one matching computational-basis outcome or the other. The pure joint state contains phase-coherent correlations that are invisible to either subsystem alone. This is the first serious warning that a subsystem state is not always a list of properties the subsystem carries independently.

Connection to identical particles and Fock space

For distinguishable subsystems, tensor factors can often be labeled by apparatus, location, spin register, atom, or mode. Identical particles require a different layer of structure. Physical states of identical bosons live in symmetric subspaces, while physical states of identical fermions live in antisymmetric subspaces.

For a one-particle Hilbert space $\mathcal h$, fixed-$N$ bosonic and fermionic spaces are

$$\operatorname{Sym}^N \mathcal h, \qquad \wedge^N \mathcal h.$$

Allowing variable particle number gives Fock spaces:

$$\mathcal F_B(\mathcal h) = \bigoplus_{N=0}^{\infty} \operatorname{Sym}^N\mathcal h, \qquad \mathcal F_F(\mathcal h) = \bigoplus_{N=0}^{\infty} \wedge^N\mathcal h.$$

Creation and annihilation operators then replace explicit symmetrized wavefunctions with mode-occupation algebra. The full construction now belongs to Identical Particles, Fock Space, and Second Quantization. This overview retains only the distinction needed to interpret subsystem entanglement.

Reading paths

Composition and local states. Read Density Operators, then use the state-classification gateway and Entangled States. Continue through the reduced-state gateway to Partial Trace.

Processes and information limits. After partial trace, study Quantum Operations and then the No-Broadcasting Theorem.

Foundations. After entangled states, the Bell-locality gateway gives the sequence Local Hidden Variables → CHSH Inequality → Bell’s Theorem.

What This Volume Does Not Own

The basic postulates belong to Core Formalism. This volume uses the Hilbert-space construction of tensor products developed above. Measurement theory, POVMs, decoherence, and open-system master equations continue in Measurement and Open Quantum Systems. Quantum circuits, algorithms, and error correction continue in Quantum Information; specialist many-body phase theory belongs to Many-Body and Quantum Statistical Mechanics. Material-facing treatments continue in Quantum Matter. Full relativistic field quantization lies beyond this quantum-mechanics bridge.

Common Mistakes

• Treating a tensor product as an ordered pair rather than a vector space.
• Calling every correlation entanglement.
• Treating a reduced density operator as ordinary ignorance in all contexts.
• Forgetting to state the subsystem decomposition before asking whether a state is entangled.
• Treating identical-particle labels as directly observable particle identities.
• Assuming second quantization means quantizing an already quantized theory a second time.

Suggested starting points

These pages connect the main ideas of composition, reduction, and entanglement:

• Product, Separable, and Entangled States
• Entangled States
• Reduced States and Partial Trace
• Partial Trace
• Quantum Operations
• Bell Locality and Quantum Correlations

The sidebar gives the wider chapter structure. Empty scaffolds identify unwritten topics and do not supply a treatment.

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• E. Schrodinger, “Discussion of Probability Relations between Separated Systems,” Mathematical Proceedings of the Cambridge Philosophical Society 31, 555-563, 1935.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.

Exercises

1. Suppose $\dim\mathcal H_A=2$ and $\dim\mathcal H_B=3$. What is the dimension of $\mathcal H_A\otimes\mathcal H_B$, and why is the answer not $2+3$?

Solution

The dimension is $2\cdot 3=6$. A basis is built from all pairs of subsystem basis vectors, so each basis vector of $A$ combines with each basis vector of $B$. The direct sum would have dimension $2+3$, but it describes an either/or alternative of vector spaces, not a composite system with both subsystems present.

2. Why does the plus Bell state have maximally mixed one-qubit reduced states even though the two-qubit state is pure?

Solution

The joint density operator is

$$\rho_{AB} = \frac12 \Big( |00\rangle\langle 00| +|00\rangle\langle 11| +|11\rangle\langle 00| +|11\rangle\langle 11| \Big).$$

Tracing over $B$ removes the off-diagonal terms because $\langle 0\vert 1\rangle=0$, leaving

$$\rho_A = \frac12 \bigl(|0\rangle\langle 0| +|1\rangle\langle 1|\bigr) = \frac12 I.$$

The purity is a property of the joint state; the subsystem alone is mixed because it is entangled with the other subsystem.