Lie algebraic invariants in quantum linear optics

TL;DR

This paper introduces Lie algebraic invariants in quantum linear optics to identify necessary conditions for state preparation, aiding the design of feasible entangled state generation without post-selection.

Contribution

It provides a method to derive conserved quantities in passive linear optical evolutions, establishing necessary conditions for exact and approximate state preparation.

Findings

Invariants are conserved quantities that constrain state transformations.

Different invariants indicate impossible state evolutions.

Lower bounds on state distance based on invariants aid in state approximation.

Abstract

Quantum linear optics without post-selection is not powerful enough to produce any quantum state from a given input state. This limits its utility since some applications require entangled resources that are difficult to prepare. Thus, we need a deeper understanding of linear optical state preparation. In this work, we give a recipe to derive conserved quantities in the evolution of arbitrary states along any possible passive linear interferometer. One example of such an invariant is the projection of a density operator onto the Lie algebra of passive linear optical Hamiltonians. These invariants give necessary conditions for exact state preparation: if the input and output states have different invariants, it is impossible to design a passive linear interferometer that evolves one into the other. Moreover, we provide a lower bound to the distance between an output and target state based on the distance between their invariants. This gives a necessary condition for approximate or heralded state preparations. Therefore, the invariants allow us to narrow the search when trying to prepare useful entangled states, like NOON states, from easy-to-prepare states, like Fock states. We conclude that future exact and approximate state preparation methods will need to consider the necessary conditions given by our invariants to weed out impossible linear optical evolutions.