In a recent work, authors prove a yet another no-go theorem that forbids the existence of a universal probabilistic quantum protocol producing a superposition of two unknown quantum states. In this short note, we show that in the presence of closed time-like curves (CTCs), one can indeed create superposition of unknown quantum states and evade the no-go result.

arXiv:1605.06005Modern Physics Letters AAug 31, 201610 citations

Topologically-driven impossibility of superposing unknown states.

Zuzana Gavorová

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arXiv:2011.02275 · Physical Review A

Impossibility of creating a superposition of unknown quantum states

Somshubhro Bandyopadhyay

Published Nov 23, 2020• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.102.050202

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Topologically-driven impossibility of superposing unknown states.

Zuzana Gavorová

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\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.