Complexity of Quantum Information from Chromatic Homotopy theory

In Strings of Qubits and Factorization Algebras, we saw that the algebra $A:=\mathbb{C}⟨\downarrow ,\uparrow ⟩$ encodes quantum superpositions of finite length bit strings. We can study invariants of this $E_1$ algebra.

• $E_1$ cotangent complex ${\mathbb{L}}_A^{(1)}$: this encodes global states which do not admit a bipartite decomposition. In the language of materials, we imagine that an example of such a state comes from a state that’s a tensor product of two states whose support on the real line don’t overlap. That is, there exists an entanglement surface across which relative entropy vanishes.

  • In our example, $A$ is a free algebra, hence ${\mathbb{L}}_A\simeq {\mathbb{C}}^{\oplus 2}$. (this might need a shift, we’ll come back to reexamine this).
  • There’s a universal derivation $d:A\to {\mathbb{L}}_A^{(1)}$. This induces a map of abelian groups
    • $\delta :{\pi }_0(A)\to {\pi }_0({\mathbb{L}}_A)$
  • See Relative Entanglement.

• Topological Hochschild Homology $THH(A)$: this should act like a generalized entropy measure. The $S^1$ action encodes cyclic invariance of trace. We get associated invariants like $TP$ and $TC$.

  • There’s a comparison map $A\to THH(A)$, this induces a map of abelian groups $\mathcal{T}:{\pi }_0(A)\to TH{H}_0(A)$
  • To do computations, we can literally plug in an explicit superposition of bit strings, like $|\varphi ⟩:=a|\uparrow ⟩+b|\uparrow \downarrow ⟩$ and compute its entropy $\mathcal{T}(\varphi )$ as an element of $TH{H}_0(A)$.
  • Even for basis states like $|\uparrow \downarrow \downarrow ⟩$, $\mathcal{T}(100)$ is a new invariant of the bit-string: it measures how the classical bit string can be built by quantum-mechanically superposing shorter strings.

• Euler characteristics of both of the spectra above (alternating sum of homotopy groups) provide a family of invariants parametrized by (suitably finite) anima.

• Chromatic localizations of these invariants.