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This research establishes Lie algebraic invariants as necessary conditions for state preparation in passive linear optical systems, effectively ruling out impossible transformations between input and output states. By providing a lower bound on the distance between target and actual states, these invariants offer a rigorous framework to narrow the search space for preparing entangled resources like NOON states without post-selection. Ultimately, this work sets fundamental limits on what can be achieved with linear optics, guiding future algorithmic approaches to approximate or heralded state generation.
Lie algebraic invariants in quantum linear optics
arXiv:2409.12223v3 Announce Type: replace 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, a deeper understanding of linear optical state preparation is needed. 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 spectrum of a density matrix mapped 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.
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, a deeper understanding of linear optical state preparation is needed. 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 spectrum of a density matrix mapped 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.
20 pages, 2 figures. Comments welcome!
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)
arXiv:2409.12223 [quant-ph]
(or arXiv:2409.12223v3 [quant-ph] for this version)
From: Pablo V. Parellada [view email] [v1] Wed, 18 Sep 2024 18:00:00 UTC (91 KB) [v2] Fri, 29 Aug 2025 15:04:13 UTC (127 KB) [v3] Fri, 5 Jun 2026 19:55:40 UTC (133 KB)
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