Biorthogonal resource theory of genuine quantum superposition

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Onur Pusuluk

1, ∗

Faculty of Engineering and Natural Sciences, Kadir Has University, 34083, Fatih, Istanbul, Türkiye

The phenomenon of quantum superposition manifests in two distinct ways: it either spreads out across

non-orthogonal basis states or remains concealed within their overlaps. Despite its profound implications, the

resource theory of superposition often neglects the quantum superposition residing within these overlaps. However,

this component is intricately linked to a form of state indistinguishability and can give rise to quantum correlations.

In this paper, we introduce a pseudo-Hermitian representation of the density operator, wherein its diagonal

elements correspond to biorthogonal extensions of Kirkwood-Dirac quasi-probabilities. This representation

provides a uniﬁed framework for the inter-basis quantum superposition and basis state indistinguishability, giving

rise to what we term as genuine quantum superposition. Moreover, we propose appropriate generalizations of

current superposition measures to quantify genuine quantum superposition that serves as the fundamental notion

of nonclassicality from which both quantum coherence and correlations emerge. Finally, we explore potential

applications of our theoretical framework, particularly in the quantiﬁcation of electron delocalization in chemical

bonding and aromaticity.

I. INTRODUCTION

Quantum coherence is regarded as a basis-dependent notion

of nonclassicality in view of its deﬁnition as the quantum

superposition with respect to a ﬁxed orthonormal basis. On

the other hand, quantum correlations, encompassing quantum

entanglement [

] and more broadly quantum discord [

–

remain invariant under local unitary transformations. How-

ever, their essence is solely the quantum coherence shared in

composite systems. In a sense, these quantities appears as

basis-independent manifestations of quantum coherence [

–

Bell states, which exhibit maximum bipartite entanglement,

serve as an illustrative example. Within these states, quantum

coherence is entirely distributed over the joint system with-

out being localized in subsystems, irrespective of local basis

choices.

Interesting questions arise in any attempt to extend this

uniﬁed framework to include quantum superposition as shown

in Fig. 1. Does quantum superposition conceptually contain

quantum coherence and quantum correlations as particular

subsets? If so, can we argue that an entangled or discordant

state must possess not only non-zero coherence but also non-

zero superposition in any basis achievable through local unitary

transformations? Otherwise, should we seek other notions of

nonclassicality that can give rise to quantum coherence and

correlations apart from quantum superposition? Here, we will

approach these questions using quantum resource theories [

–

12].

Each resource theory categorizes all conceivable quantum

states into two groups: resource and free, based on their

behavior under a deﬁned set of restricted quantum operations

permitted for state manipulation. For example, in the resource

theory of entanglement [

], free states are separable, and it

remains impossible to generate an entangled state from these

states using solely local operations and classical communication.

Similarly, in the resource theory of quantum superposition,

∗onur.pusuluk@gmail.com

incoherent mixtures such as those depicted below:

ˆ𝜌𝑓=

𝑖

𝑝𝑖|𝑐𝑖⟩⟨𝑐𝑖|,(1)

are regarded as superposition-free states [

–

], where

{|𝑐𝑖⟩}

represents a set of normalized and linearly independent states

that are not necessarily orthogonal, and

{𝑝𝑖}

form a probability

distribution. Consequently, a system existing in state

ˆ𝜌𝑓

cannot

serve as a resource under quantum operations incapable of

creating or amplifying quantum superposition in the basis of

{|𝑐𝑖⟩}.

For the sake of simplicity, let us consider a four-level system

existing in a mixture that contains only two states

|𝑐1⟩

and

|𝑐2⟩

with

⟨𝑐1|𝑐2⟩=𝑠∈R

. We will focus on a particular bipartition

of this system into two two-level subsystems

𝐴

and

𝐵

such that

|𝑐𝑖⟩ ↦→ |𝑎𝑖⟩𝐴⊗ |𝑏𝑖⟩𝐵

with

⟨𝑎1|𝑎2⟩=⟨𝑏1|𝑏2⟩=√𝑠

. Then,

ˆ𝜌𝑓

reads

𝑝|𝑎1⟩⟨𝑎1|⊗|𝑏1⟩⟨𝑏1| + (1−𝑝) |𝑎2⟩⟨𝑎2|⊗|𝑏2⟩⟨𝑏2|,(2)

which is an example of quantum-quantum states. Despite ap-

FIG. 1. Hierarchy of nonclassicality between quantum entanglement

𝐸

, discord

𝐷

, coherence

𝐶

, superposition

𝑆

, and state indistinguisha-

bility

𝐼

. The balls depict physical systems, while the diagonal and

vertical arrows represent pure physical states. The angle between the

arrows quantiﬁes the overlap between the represented states such that

perpendicular arrows correspond to orthogonal states.

⇌

and

↔

stand

for a probabilistic mixture (or incoherent superposition) and coherent

superposition of the left and right states, respectively.

arXiv:2210.02398v2 [quant-ph] 14 Apr 2024

pearing as a superposition-free state in the basis of

{|𝑐1⟩,|𝑐2⟩}

this state possesses nonclassical correlations in the form of

quantum discord (please see Fig. 2).

A complete resource theory of quantum discord has yet to

emerge in the existing literature. Nonetheless, it is established

that incoherent operations cannot create discordant states such

as (2) without consuming local quantum coherences initially

present in the subsystems [

]. That is to say, the so-called

superposition-free state is not coherence-free in any orthogonal

basis accessible through local unitary transformations. There-

fore, the current resource-theoretical deﬁnition of quantum

superposition appears inadequate, in a notional sense, to cover

quantum coherence and quantum correlations as a subset.

As a matter of fact, the correlations shared in state (2) arise

from the local indistinguishability in subsystems. With a com-

plete measurement on

𝐴

(

𝐵

), the states

|𝑎1⟩

and

|𝑎2⟩

(

|𝑏1⟩

and

|𝑏2⟩

) cannot be perfectly distinguished from each other. This

makes the global system sensitive to the local dynamics. Should

we take into account the basis state indistinguishability as an

independent notion of nonclassicality from which the quantum

coherence and correlations can arise as in Fig. 1? Or, should

we extend the conventional deﬁnition of quantum superposition

to include this kind of quantum indistinguishability as a special

case? This is what we would like to start discussing in this

paper.

To set the ground for this discussion, we propose utilizing

a pseudo-Hermitian matrix representation of a quantum state

to investigate its properties concerning a nonorthogonal basis.

This approach entails expressing the state within the biorthogo-

nal extension of the given basis, a well-established mathematical

technique in the literature (see, for example, Refs. [

–

]). Al-

though independently rediscovered to represent free operations

in the resource theory of superposition [

–

], this method

has not been extended to encompass quantum states. Here,

we will demonstrate that the diagonal elements of the pseudo-

Hermitian representation of quantum states are biorthogonal

Kirkwood-Dirac quasi-probabilities [

–

]. We will intro-

duce the term genuine quantum superposition by associating

the remaining matrix elements with both quantum superpo-

sition and basis state indistinguishability. Subsequently, we

will generalize existing superposition measures for genuine

quantum superposition. The measures introduced herein hold

promise for quantifying non-classicality in chemical bonding

phenomena within the realm of quantum chemistry, as evi-

denced by recent research where we successfully calculated

electron delocalization in aromatic molecules [35].

II. DEFINITION AND MOTIVATION

A generic density operator that lives in a

𝑑

-dimensional

Hilbert space can be expressed in the form of

ˆ𝜌=

𝑑



𝑗,𝑘=1

𝜌𝑗 𝑘 |𝑐𝑗⟩⟨𝑐𝑘|,(3)

where

{|𝑐𝑗⟩}

with

⟨𝑐𝑗|𝑐𝑘⟩=𝐺𝑗 𝑘

constitute a nonorthogonal

but complete and minimal basis.

FIG. 2. Geometric discord [

] of state (2). Right and left discords

are equal to each other, while the entanglement is always zero.

Theorem 1. In the case of nonorthogonal basis states, the

trace of an operator is calculated by

tr[ˆ𝜌]=

𝑑



𝑖=1⟨𝑐⊥

𝑖|ˆ𝜌|𝑐𝑖⟩,(4)

where

{|𝑐⊥

𝑖⟩}

with

⟨𝑐⊥

𝑖|𝑐𝑗⟩=𝛿𝑖, 𝑗

is called the dual of the basis

{|𝑐𝑖⟩}:

|𝑐⊥

𝑗⟩=

𝑖(𝐺−1)𝑖 𝑗 |𝑐𝑖⟩.(5)

The two dual bases are jointly called biorthogonal. When

all the overlaps go to zero, the dual basis becomes identical

|𝑐𝑖⟩

and consequently, Eq. (4) reduces to the conventional

deﬁnition of trace operation in orthogonal basis.

Proof. This trace deﬁnition can be simply veriﬁed consider-

ing the pure superposition states

|𝜓⟩=

𝑗

𝜓𝑗|𝑐𝑗⟩,(6)

for which

tr[|𝜓⟩⟨𝜓|] =⟨𝜓|𝜓⟩

becomes

Í𝑗 𝑘 𝜓𝑗𝜓∗

𝑘⟨𝑐𝑘|𝑐𝑗⟩=

Í𝑖⟨𝑐⊥

𝑖|𝜓⟩⟨𝜓|𝑐𝑖⟩

. Extending the same veriﬁcation to mixed

states is equally straightforward

tr[ˆ𝜌]=

𝑗 𝑘

𝜌𝑗 𝑘 tr[1|𝑐𝑗⟩⟨𝑐𝑘|] =

𝑖 𝑗 𝑘

𝜌𝑗 𝑘 ⟨𝑐⊥

𝑖|𝑐𝑗⟩⟨𝑐𝑘|𝑐𝑖⟩

=

𝑖⟨𝑐⊥

𝑖|ˆ𝜌|𝑐𝑖⟩,

(7)

as the identity operator can be expressed as

1=

𝑖|𝑐𝑖⟩⟨𝑐⊥

𝑖|.(8)

Corollary 2. Consider a density operator written in a

nonorthonormal basis

{|𝑐𝑖⟩}

as in Eq. (3). The overall non-

classicality of this state can be quantiﬁed by the

𝑙1

norm of the

operator, deﬁned by

𝑙1[ˆ𝜌]=

𝑗≠𝑗|⟨𝑐⊥

𝑗|ˆ𝜌|𝑐𝑘⟩|.(9)

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BiorthogonalresourcetheoryofgenuinequantumsuperpositionOnurPusuluk1,∗1FacultyofEngineeringandNaturalSciences,KadirHasUniversity,34083,Fatih,Istanbul,TürkiyeThephenomenonofquantumsuperpositionmanifestsintwodistinctways:iteitherspreadsoutacrossnon-orthogonalbasisstatesorremainsconcealedwithintheirover...

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