Entanglement entropy and area laws

For a bipartite pure state, the Schmidt decomposition pairs orthogonal states in the two subsystems with nonnegative probabilities summing to one. Tracing out one subsystem leaves those probabilities as eigenvalues of the other subsystem's density matrix. The von Neumann entropy is their Shannon entropy. A relative phase between two Schmidt terms does not change these eigenvalues, whereas changing their magnitudes does. This is why amplitudes must be squared before they enter an entropy.

In the two-term example, p equal to zero or one gives a product state and zero entropy. Equal weights maximize the entropy at the natural logarithm of two. These statements concern a pure global state with a specified tensor-product split. If the global state is mixed, a subsystem's entropy can include ordinary thermal or classical uncertainty, so a nonzero entropy alone no longer certifies entanglement. Distinguishing those contributions requires additional information or an appropriate entanglement measure.

In a local field-theory vacuum, degrees of freedom on opposite sides of a region's boundary correlate at arbitrarily short distances. A spatial cutoff epsilon makes the entropy finite. In more than one spatial dimension, a common leading behavior is proportional to boundary area divided by epsilon raised to the boundary dimension. Its coefficient generally depends on the regulator and the microscopic field content. Calling this an area law does not make its leading divergent coefficient a universal observable.

A useful special result occurs for a single interval of length ell in the vacuum of a unitary conformal field theory on an infinite one-dimensional line. The entropy has logarithmic growth with coefficient c/3, where c is the central charge, plus a nonuniversal constant. Finite temperature, finite system size, boundaries, excited states, or multiple intervals change the formula. The logarithm is not obtained by setting the boundary area to zero in a higher-dimensional power law; it is a distinct dimensional case.

For separated regions, mutual information combines entropies as $I(A:B)=S_A+S_B-S_{A\cup B}$. Local boundary divergences can cancel, giving a finite quantity under appropriate separation and state assumptions. Adjacent regions can reintroduce short-distance divergences. Relative entropy is another powerful finite comparison in suitable settings, but its mathematical domain and reference state matter. Subtracting two enormous regulated numbers numerically is not by itself a proof that a continuum limit exists.

Gauge constraints add a further subtlety: the physical Hilbert space need not factor into a simple product of independent spatial-region Hilbert spaces. One must specify the observable algebra or an extended-space prescription and account for boundary or center degrees of freedom. Entropy can depend on that choice. Universal terms, mutual information, and carefully defined relative entropy often provide more robust comparisons. Extending such tools through interacting continuum and gravitational systems remains an active research direction, rather than a reason to omit the subsystem definition.