Construction of ecient Schmidt number witnesses for high-dimensional quantum states Nikolai Wyderka1Giovanni Chesi2Hermann Kampermann1Chiara Macchiavello2and Dagmar Bru1

2025-04-27 0 0 982.02KB 7 页 10玖币

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Construction of eﬃcient Schmidt number witnesses for

high-dimensional quantum states

Nikolai Wyderka,1Giovanni Chesi,2Hermann Kampermann,1Chiara Macchiavello,2and Dagmar Bruß1

1Institut f¨ur Theoretische Physik III, Heinrich-Heine-Universit¨at D¨usseldorf,

Universit¨atsstr. 1, D-40225 D¨usseldorf, Germany

2Istituto Nazionale di Fisica Nucleare Sezione di Pavia, Via Agostino Bassi 6, I-27100 Pavia, Italy

(Dated: March 1, 2023)

Recent progress in quantum optics has led to setups that are able to prepare high-dimensional

quantum states for quantum information processing tasks. As such, it is of importance to benchmark

the states generated by these setups in terms of their quantum mechanical properties, such as their

Schmidt numbers, i.e., the number of entangled degrees of freedom.

In this paper, we develop an iterative algorithm that ﬁnds Schmidt number witnesses tailored to

the measurements available in speciﬁc experimental setups. We then apply the algorithm to ﬁnd a

witness that requires the measurement of a number of density matrix elements that scales linearly

with the local dimension of the system. As a concrete example, we apply our construction method

to an implementation with photonic temporal modes.

I. INTRODUCTION

Over the course of the last decades, diﬀerent physical

platforms have been developed to perform quantum in-

formational tasks. The requirements for such platforms

have been formalized at various occasions. One of the

most often used list of requirements was formulated by

DiVincenzo, and includes among other things the capa-

bility of robust state preparation, state manipulation and

measurements [1]. While most platforms available today

are limited to the generation and manipulation of qubits,

it has become more and more evident over the last years

that higher-dimensional systems can provide advantages

in speciﬁc tasks including quantum communication [2–

7] and quantum computing (see Ref. [8] and references

therein). However, embedding these additional degrees

of freedom in trapped ions or superconducting qubits

has been challenging [9, 10]. In contrast, time-energy

degrees of freedom in photonic states can be used to

encode very high-dimensional quantum states. Further-

more, using quantum pulse gates allows for robust and

precise manipulation and measurement of these signals,

and therefore provides a viable candidate for a platform

for high-dimensional quantum tasks [11].

In this paper, we develop theoretical tools that help

to characterize the state preparation capabilities of these

and similar setups. Recent work focused on the con-

struction of high-dimensional entanglement witnesses for

these setups that allow for eﬃcient entanglement detec-

tion of the prepared quantum states [12, 13]. In this

work, we construct easily measurable Schmidt number

witnesses for such setups. Apart from detecting whether

a state is entangled, the Schmidt number of a state car-

ries additional information about the dimensionality of

the entanglement. We develop an algorithm to generate

witness candidates that use only few of the experimen-

tally available measurement settings. Consequently, we

then apply the algorithm to the measurements available

in the setup described in Ref. [11] and show that the

obtained observable indeed is a proper Schmidt number

witness, i.e., it certiﬁes the Schmidt number of the gen-

erated quantum state. While proposals exist to certify

Schmidt numbers with measurements in only two local

bases [14], these ideas do not apply to the setup at hand,

as here only certain linear combinations of the entries of

the density matrix can be measured.

The paper is organized as follows. In section II we re-

view the notion of Schmidt numbers and Schmidt num-

ber witnesses, in section III we develop and explain the

algorithm that generates Schmidt number witness candi-

dates. In Section IV we apply the algorithm to obtain a

Schmidt number witness using only O(d) measurement

settings to certify the Schmidt number of the generated

states, and compare its noise robustness to that of other

Schmidt number witnesses. In the Appendix, we provide

an explicit construction for an experiment using photonic

temporal modes.

II. SCHMIDT NUMBERS AND SCHMIDT

NUMBER WITNESSES

Throughout this paper, we consider bipartite quantum

systems in Cd⊗Cd. We denote the set of linear maps

from Cdto itself by Md.

The Schmidt number of a bipartite quantum state is

an entanglement measure that is related to the hardness

of generating a quantum state using local operations and

classical communication [15]. For pure states, it is de-

ﬁned as the number of non-vanishing Schmidt coeﬃcients

in the Schmidt decomposition of the state, i.e., for ev-

ery bipartite quantum state |ψi, written in the compu-

tational basis as

|ψi=

d−1

i,j=0

cij |iiA|jiB,(1)

one can ﬁnd local orthonormnal bases, {|iiA},{|jiB}for

arXiv:2210.05272v2 [quant-ph] 28 Feb 2023

subsystems Aand B, respectively, such that

|ψi=

k−1

i=0

λi|iiA|iiB,(2)

where λi∈R, λi>0 and Pk−1

i=0 λ2

i= 1. The k≤dnon-

vanishing numbers λiare called Schmidt coeﬃcients of

|ψi, and kis called the Schmidt rank of |ψi, or SR(|ψi) in

short [16]. In order to generalize this measure to mixed

states ρ, one uses the convex roof construction of the

Schmidt rank, called Schmidt number, or SN(ρ) [17]:

SN(ρ) := min

ρ=Pipi|ψiihψi|max

iSR(|ψii),(3)

i.e., it is given by the maximal Schmidt rank within a

given decomposition of ρ, minimized over all decompo-

sitions. Due to the minimization, it is usually hard to

calculate the Schmidt number of a given quantum state.

For a bipartite system of dimension d×d, the maximal

Schmidt number is given by d, and one can deﬁne the sets

of Schmidt number kas

Sk={ρ: SN(ρ)≤k}.(4)

Clearly, Sk⊂Sk+1, and S1is the usual set of sep-

arable states. The maximally entangled state |φ+i=

√dPd−1

i=0 |iiiis member of Sd, but not Sd−1, and it can

be shown that [17, 18]

hφ+|ρk|φ+i ≤ k

d(5)

for all states ρk∈Sk.

In order to certify a speciﬁc Schmidt number experi-

mentally, it is useful to deﬁne an analogon to entangle-

ment witnesses for Schmidt numbers [19]. We say that

an observable Wkis a Schmidt-number-(k+ 1) witness,

•Tr(Wkρk)≥0 for all ρk∈Sk,

•Tr(Wkρ)<0 for at least one ρ.

Thus, whenever one ﬁnds in an experiment that

Tr(Wkρ)<0, then ρmust have at least Schmidt number

k+ 1. Note that for k= 1, we recover the usual notion

of an entanglement witness [16]. An important exper-

imental advantage of using theses witnesses lies in the

fact that a negative expectation value certiﬁes a certain

Schmidt number for any experimental input state, pure

or mixed. In order to certify that a given observable is a

Schmidt-number-(k+ 1) witness, it is suﬃcient to min-

imize its overlap w.r.t. pure states in Skand show that

the minimum is non-negative. This can be seen from the

fact that the sets Skare compact, thus, one can ﬁnd a

(potentially mixed) optimal quantum state ρ?

ksuch that

ck:= minρk∈SkTr(Wkρk) = Tr(Wkρ?

k). As SN(ρ?

k)≤k,

it exhibits a decomposition ρ?

k=Pipi|ψiihψi|with

SR(|ψii)≤k. Thus, ck=Pipihψi|Wk|ψii ≥ Pipick=

Figure 1. The sets Skof states of Schmidt number kfor d= 3,

as well as their outer approximations SR

kdeﬁned in Eq. (8).

ck, implying that the inequality is an equality and each

of the pure |ψiiachieves the same minimal value of ck.

The observation of Eq. (5) can be directly transformed

into a Schmidt-number-(k+1) witness via the observable

[19]

Wk=1d2−d

k|φ+ihφ+|.(6)

We refer to this witness as the standard Schmidt-number

witness, and we will compare our constructions to this

one in the end.

The problem remains that in general, the minimization

over pure states with ﬁxed Schmidt rank remains chal-

lenging, and in many cases no analytical solution can be

found. This can be remedied by relaxing the optimiza-

tion slightly.

To that end, note that there exists a characterization

of the set Skin terms of positive maps [17]: It holds that

ρ∈Skif and only if (1d⊗Λk)(ρ)≥0 for all k-positive

maps Λk:Md→ Md, where 1ddenotes the identity

map in dimension d. A map Λkis called k-positive, if

1k⊗Λkis a positive map.

This characterization is useful, as it allows to deﬁne

slightly larger sets than Sk, which can be characterized

with less eﬀort. To that end, we deﬁne the generalized

reduction map [16, 17]

Rp(ρA) = Tr(ρA)1d−pρA,(7)

where ρA∈ Mdis a single qudit mixed state. It was

shown that Rpis k-positive, but not k+ 1-positive, iﬀ

p∈(1

k+1 ,1

k] [17, 20]. For p= 1 one recovers the usual

reduction map. We use it to deﬁne

k={ρ: (1d⊗R1

k)(ρ)≥0}.(8)

Using the relation between k-positive maps and states

of Schmidt number k, it is clear that Sk⊂SR

k. Thus,

we can use these sets as an outer approximation of Sk,

as they can be characterized by semi-deﬁnite constraints.

The general embedding situation is displayed in Fig. 1.

We use this fact to ﬁnd a lower bound on the optimal

value of ck= minρk∈SkTr(Wkρk)≥minρk∈SR

kTr(Wkρk).

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ConstructionofecientSchmidtnumberwitnessesforhigh-dimensionalquantumstatesNikolaiWyderka,1GiovanniChesi,2HermannKampermann,1ChiaraMacchiavello,2andDagmarBru11InstitutfurTheoretischePhysikIII,Heinrich-Heine-UniversitatDusseldorf,Universitatsstr.1,D-40225Dusseldorf,Germany2IstitutoNazionalediFi...

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Construction of ecient Schmidt number witnesses for high-dimensional quantum states Nikolai Wyderka1Giovanni Chesi2Hermann Kampermann1Chiara Macchiavello2and Dagmar Bru1

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