Topologically-driven impossibility of superposing unknown states.
Year: 2021
Type: preprint
Abstract: Our understanding of quantum mechanics broadens by discovering tasks that are easy classically but their quantum versions are impossible, such as cloning. Another such task is the superposition task: Given two input states $\left|u\right>,\left|v\right>\in\mathcal{H}$ create a superposition $\alpha e^{i\phi(\left|u\right>,\left|v\right>)} \left|u\right>+\beta\left|v\right>$ for some real function $\phi$. This is the quantum version of the classical adder. Oszmaniec, Grudka, Horodecki and W\'ojcik [1] showed that a quantum circuit acting on a single pair of input states cannot output an exact superposition of them. Here we prove that a quantum circuit cannot approximate a superposition for every pair of states in $\mathcal{H}$ - even if it accesses any number of copies of the input states. Our proof follows the [2]: Any quantum circuit defines a continuous function on the input states. Using new topological arguments we prove that no continuous function can have the desired output. Compared to methods that view quantum circuits as linear [3-5] or polynomial [6] functions, the topological method excludes any number of input-state copies. We give some intuition of why the method is applicable to the superposition task. We consider the following protocol: given a large number of copies of the two input states, use state tomography to obtain a classical description of each, then prepare their superposition from scratch. Comparing our impossibility to this tomography protocol, we find a separation between two types of measurement.
Source: arXiv (Cornell University)
Author: Zuzana Gavorová
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3209569753