Product States and the Composition Postulate

A pure product state assigns a state vector to each subsystem separately. Its coefficients factor into local amplitudes, and all local measurement probabilities are independent. This first encounter gives a quick factorization test; product states develops the broader theory, mixed states, and dynamics.

Helpful background. Tensor products explains the product-basis ordering used below.

The Factorization Test

For a specified split $A|B$, a normalized pure state is product when

$$|\Psi\rangle_{AB}=|\psi\rangle_A\otimes|\phi\rangle_B.$$

If the local amplitudes are $a_i$ and $b_j$, its product-basis coefficients satisfy $C_{ij}=a_i b_j$. In finite dimensions this says that the nonzero coefficient matrix has rank one. For two qubits,

$$|\Psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$$

is product exactly when $ad-bc=0$. Use the complex amplitudes themselves in this test, including their phases.

For example,

$$\frac{|00\rangle-|01\rangle+|10\rangle-|11\rangle}{2} =|+\rangle_A| -\rangle_B.$$

Four nonzero basis coefficients do not imply entanglement: local superpositions can multiply into a product state.

What Factorization Predicts

For local effects $E$ and $F$,

$$p(E,F)=\langle\psi|E|\psi\rangle \langle\phi|F|\phi\rangle.$$

Thus all product measurements have independent outcome statistics. Independence in just one chosen measurement basis is insufficient to establish a product state.

A mixed product state has the form $\rho_A\otimes\rho_B$. A mixture of different product states need not itself factor, even though it is separable. Keep that mixed-state distinction separate from the pure-state rank-one test above.

Quick Check

Replace the last minus sign in the displayed four-term state by a plus sign. Does the new normalized state factor? Compare its computational-basis probabilities with the original state’s.

Solution

The new coefficient matrix is $C=\tfrac12\begin{pmatrix}1&-1\\1&1\end{pmatrix}$, whose determinant is $1/2$. The state is entangled. Both states nevertheless give probability $1/4$ to every computational-basis outcome. Their equal populations hide different relative phases, so this one basis cannot distinguish product structure from entanglement.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, ch. 2.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, ch. 6. Author’s manuscript.