Parts of quantum states

It is shown that generic $N$-party pure quantum states (with equidimensional subsystems) are uniquely determined by their reduced states of just over half the parties; in other words, all the information in almost all $N$-party pure states is in the set of reduced states of just over half the parties. For $N$ even, the reduced states in fewer than $N∕2$ parties are shown to be an insufficient description of almost all states (similar results hold when $N$ is odd). It is noted that real algebraic geometry is a natural framework for any analysis of parts of quantum states: two simple polynomials, a quadratic and a cubic, contain all of their structure. Algorithmic techniques are described which can provide conditions for sets of reduced states to belong to pure or mixed states.

Figure 1

The four overlapping, five-party reduced states ${\rho }_{AEFGH}$, ${\rho }_{BEFGH}$, ${\rho }_{CEFGH}$, ${\rho }_{DEFGH}$ are almost always a sufficient description of the eight-party pure state ${\mid \varphi ⟩}_{ABCDEFGH}$.