Semi-Algebraic Entanglement Consistency Relations: Fundamental and Dynamical
Encoding the state of physical systems, the many-body wavefunction lives in a Hilbert space that conveniently offers a tensor factor to each constituent of the system. Consequentially, the wavefunction inherits structural properties such as entanglement, a new type of quantum correlation across the tensor factors. Entanglement can be quantified by forming polynomials in the wavefunction that remain invariant under local operations, which in turn naturally imply the properties we intuit about entanglement. It follows from the Tarski-Seidenberg theorem that these polynomial entanglement measures must obey polynomial constraints. It is among these constraints that the core features separating quantum correlations from classical correlations are illuminated, such as their monogamous nature. The smallest non-trivial system which exhibits entanglement monogamy tradeoffs governed by polynomial constraints is three qubits, for which we find novel constraints that are stronger than the previously known constraints, and which may as well be a complete description of three-qubit entanglement since the number of degrees of freedom matches the number of entanglement measures. A remarkable redundancy known in three qubits is that any party's total entanglement decomposes exactly into bipartite and tripartite type entanglement; we find that this behavior is merely part of the GHZ family tradition, and that any party within the n-qubit GHZ SLOCC equivalence class also has total entanglement which decomposes exactly into all k-partite type entanglements, thus providing insight into strong versions of monogamy relations. Refocusing back to tripartite systems, we then suppose arbitrarily many degrees of freedom into our subsystems, which appear to refuse all temptations of polygamy. Up to this point we have only been considering idealized isolated systems as it is known that monogamy relations aren't sensitive to the total purity, however we then find entanglement constraints which are violated by globally mixed states, ultimately suggesting that environments can enable entanglement to grow beyond what is otherwise allowed. Finally, we address the dynamical behavior of entanglement under several key interactions, and characterize the couplings according to their ability to generate entanglement.