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The Forbidden Quantum Adder.

Alvarez-Rodriguez U, Sanz M, Lamata L, Solano E - Sci Rep (2015)

Bottom Line: We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces.This allows us to propose an approximate quantum adder that could be implemented in the lab.Finally, we discuss the distinct character of the forbidden quantum adder for quantum states and the allowed quantum adder for density matrices.

View Article: PubMed Central - PubMed

Affiliation: Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain.

ABSTRACT
Quantum information provides fundamentally different computational resources than classical information. We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces. This is an inherent restriction of quantum physics that is related to the impossibility of copying an arbitrary quantum state, i.e., the no-cloning theorem. Moreover, we demonstrate that a quantum adder, in absence of an ancillary system, is also forbidden for a known orthonormal basis. This allows us to propose an approximate quantum adder that could be implemented in the lab. Finally, we discuss the distinct character of the forbidden quantum adder for quantum states and the allowed quantum adder for density matrices.

No MeSH data available.


Related in: MedlinePlus

Scheme of the conjectured quantum adder.The inputs are two unknown quantum states,  and , while the outputs are proportional to the sum,  with an ancillary state .
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f1: Scheme of the conjectured quantum adder.The inputs are two unknown quantum states, and , while the outputs are proportional to the sum, with an ancillary state .

Mentions: Let be two quantum states of a finite-dimensional Hilbert space. The conjectured quantum adder, sketched in Fig. 1, would be a mathematical operation defined as the unitary , for every pair of unknown and ancillary vector ,where the ancillary state may depend on the input states. There are several ways of proving the unphysicality of Eq. (1). The simplest one is to note that the unobservable global phase on its l.h.s. could be distributed in infinite forms on its r.h.s., , with ϕ = ϕ1 + ϕ2, yielding an observable relative phase. When the ancillary state does not depend on the input quantum states, the (forbidden) quantum cloner becomes a particular case of this restricted quantum adder. This follows from applying U to two equal state vectors , since the inverse would generate a quantum cloning operation. Therefore, although the general case of the quantum adder is not equivalent to a quantum cloner, it is still forbidden.


The Forbidden Quantum Adder.

Alvarez-Rodriguez U, Sanz M, Lamata L, Solano E - Sci Rep (2015)

Scheme of the conjectured quantum adder.The inputs are two unknown quantum states,  and , while the outputs are proportional to the sum,  with an ancillary state .
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4495381&req=5

f1: Scheme of the conjectured quantum adder.The inputs are two unknown quantum states, and , while the outputs are proportional to the sum, with an ancillary state .
Mentions: Let be two quantum states of a finite-dimensional Hilbert space. The conjectured quantum adder, sketched in Fig. 1, would be a mathematical operation defined as the unitary , for every pair of unknown and ancillary vector ,where the ancillary state may depend on the input states. There are several ways of proving the unphysicality of Eq. (1). The simplest one is to note that the unobservable global phase on its l.h.s. could be distributed in infinite forms on its r.h.s., , with ϕ = ϕ1 + ϕ2, yielding an observable relative phase. When the ancillary state does not depend on the input quantum states, the (forbidden) quantum cloner becomes a particular case of this restricted quantum adder. This follows from applying U to two equal state vectors , since the inverse would generate a quantum cloning operation. Therefore, although the general case of the quantum adder is not equivalent to a quantum cloner, it is still forbidden.

Bottom Line: We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces.This allows us to propose an approximate quantum adder that could be implemented in the lab.Finally, we discuss the distinct character of the forbidden quantum adder for quantum states and the allowed quantum adder for density matrices.

View Article: PubMed Central - PubMed

Affiliation: Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain.

ABSTRACT
Quantum information provides fundamentally different computational resources than classical information. We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces. This is an inherent restriction of quantum physics that is related to the impossibility of copying an arbitrary quantum state, i.e., the no-cloning theorem. Moreover, we demonstrate that a quantum adder, in absence of an ancillary system, is also forbidden for a known orthonormal basis. This allows us to propose an approximate quantum adder that could be implemented in the lab. Finally, we discuss the distinct character of the forbidden quantum adder for quantum states and the allowed quantum adder for density matrices.

No MeSH data available.


Related in: MedlinePlus