Composite Systems and Entanglement Basics

To describe two distinguishable quantum subsystems together, combine their Hilbert spaces with a tensor product. Some joint pure states factor into one state for each part; entangled states do not. This chapter introduces that distinction and the reduced state used to predict measurements on only one part.

Two qubits and one joint state

For qubits $A$ and $B$, use the ordered basis $\lvert00\rangle,\lvert01\rangle,\lvert10\rangle,\lvert11\rangle$. The four-dimensional space is $\mathbb C^2\otimes\mathbb C^2$, not a space with only two alternatives. A state such as $\lvert0\rangle_A\lvert+\rangle_B$ is a product. The Bell state

$$\lvert\Phi^+\rangle= \frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2}$$

cannot be factored into separate kets for $A$ and $B$. Measuring both in the $0,1$ basis gives $00$ or $11$, each with probability $1/2$.

If you can measure only $A$, all its predictions come from

$$\rho_A=\operatorname{Tr}_B\!\left(\lvert\Phi^+\rangle\langle\Phi^+\rvert\right) =\frac{I_2}{2}.$$

The joint state is pure although its parts are mixed. The partial trace sums over the unobserved subsystem; it does not describe physically destroying it.

A mixed marginal detects entanglement when the joint state is pure. Without that assumption it is insufficient: the product state $(I_2/2)\otimes(I_2/2)$ has the same marginals and no entanglement.

Reading sequence

Question	Article	What you will learn
How are quantum systems composed?	Composite Systems	composition of distinguishable subsystems
What does the tensor product construct?	Tensor Products	tensor-product bases and dimensions
Which pure states factor?	Product States	factorization of a pure state
Which pure states do not factor?	Entangled States	nonfactorization and correlations
What notation organizes two-part systems?	Bipartite Systems	two-part notation and first calculations
What state describes one subsystem?	Reduced States	states for local predictions
How is a subsystem traced out?	Partial Trace: First Encounter	calculation by summing an unobserved index
What normal form diagnoses bipartite pure entanglement?	Schmidt Decomposition Overview	Schmidt coefficients and their meaning
How do local operators act?	Subsystems and Local Observables	operators on one or both subsystems

Build from products to correlations

Learn tensor-product notation and product states before entangled states. Then calculate local observables and partial traces, and use the Schmidt decomposition to organize a general bipartite pure state.

The full composite-systems volume develops mixed-state separability, multipartite systems, and entanglement measures. Identical particles require additional exchange-symmetry considerations beyond the distinguishable subsystems used here.

Try it. Can applying a unitary $U_B$ to qubit $B$ change the outcome probabilities of measurements on $A$ alone?

Solution

No. Tracing over $B$ is unchanged by conjugation with $I_A\otimes U_B$. Thus $\rho_A$ and every local probability stay fixed. Conditioning on a recorded outcome on $B$ is a different procedure and requires that record to identify the selected subensemble.

References

• I. Bengtsson and K. Życzkowski, Geometry of Quantum States, 2nd ed., Cambridge University Press, 2017.
• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Reviews of Modern Physics 81, 865–942, 2009, doi:10.1103/RevModPhys.81.865.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.