The power of noisy quantum states and the advantage of resource dilution

2025-05-06 0 0 309.5KB 13 页 10玖币

侵权投诉

Marek Miller, Manfredi Scalici, Marco Fellous Asiani, and Alexander Streltsov∗

Centre for Quantum Optical Technologies, Centre of New Technologies,

University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland

Entanglement distillation allows to convert noisy quantum states into singlets, which can in turn be used for

various quantum technological tasks, such as quantum teleportation and quantum key distribution. Entanglement

dilution is the inverse process: singlets are converted into quantum states with less entanglement. While the

usefulness of distillation is apparent, practical applications of entanglement dilution are less obvious. Here,

we show that entanglement dilution can increase the resilience of shared quantum states to local noise. The

increased resilience is observed even if diluting singlets into states with arbitrarily little entanglement. We

extend our analysis to other quantum resource theories, such as quantum coherence, quantum thermodynamics,

and purity. For these resource theories, we demonstrate that diluting pure quantum states into noisy ones can be

advantageous for protecting the system from noise. Our results demonstrate the usefulness of quantum resource

dilution, and provide a rare example for an advantage of noisy quantum states over pure states in quantum

information processing.

As has been realized in the early days of quantum infor-

mation theory, two remote parties sharing a pair of entangled

particles can perform information processing tasks which are

not possible in classical physics [1]. An important example

of that is quantum key distribution [2], allowing the parties to

establish a provably secure key. Typically, these tasks employ

singlets, highly entangled states of two quantum bits. If the

quantum states shared by the remote parties are noisy, it is still

possible to perform tasks based on singlets by applying entan-

glement distillation [3,4]. This procedure allows us to extract

singlets from a large number of copies of a noisy state, addi-

tionally making use of local operations and classical commu-

nication (LOCC) between the remote parties. Quantum states

which can be converted into singlets in this way are called dis-

tillable. Since most quantum information processing tasks are

based on singlets, this makes all distillable states also useful

for these tasks. However, not all entangled states are distill-

able, a phenomenon known as bound entanglement [5].

Conversely, it is possible to dilute singlets into quantum

states with less entanglement [4]. For pure entangled states,

optimal distillation and dilution procedures are known in the

limit where a large number of copies of the state is avail-

able [4]. Two remote parties, Alice and Bob, sharing a large

number of copies of a pure entangled state |ψ⟩AB can distill

them into singlets with the maximal rate S(ψA), where ψA=

TrB[ψAB] is the reduced state of Alice, S(ρ)=−Tr[ρlog2ρ] is

the von Neumann entropy, and ψAB =|ψ⟩⟨ψ|AB. The maximal

rate for diluting singlets into |ψ⟩AB is given by 1/S(ψA). For

pure entangled states the distillation and dilution procedures

are reversible, which means that in the asymptotic limit it is

possible to distill |ψ⟩AB into singlets and dilute them back into

|ψ⟩AB in a lossless way [4].

While the dilution procedure is possible in principle, it is

reasonable to believe that in practice it is never advantageous

to degrade singlets into weakly entangled states. As we will

see in this article, this intuition is not correct: there exist quan-

tum information processing tasks where entanglement dilu-

tion is essential, even if the diluted states contain arbitrarily

little entanglement. Distillation and dilution is not limited to

entanglement, and has also been considered in general quan-

tum resource theories [6]. The basis of any quantum resource

theory is the deﬁnition of free states and free operations, cor-

responding to states and transformations which can be created

or performed at no cost within reasonable physical constraints.

Important examples are the resource theories of quantum co-

herence [7], thermodynamics [8], and purity [9]. As we will

see, resource dilution provides an advantage in these quantum

resource theories as well.

Reducing entanglement loss under local noise. Con-

sider two remote parties, Alice and Bob, who share nsinglets

|ψ−⟩=(|01⟩−|10⟩)/√2. We assume that Bob’s quantum mem-

ory is not perfect, each qubit undergoing local noise Λ. After

the action of the noise, Alice and Bob end up with ncopies

of the noisy state ρ=11⊗Λ[ψ−]. For large n, they can dis-

till the states ρinto nEd(ρ) singlets, where Edis the distillable

entanglement [1,10]. Since Alice and Bob started with nsin-

glets, n[1 −Ed(ρ)] is the number of singlets lost due to the

imperfections of Bob’s quantum memory.

As we will now show, Alice and Bob can reduce the loss of

entanglement by diluting their singlets into states with less en-

tanglement, see also Fig. 1. By using LOCC, Alice and Bob

can dilute their nsinglets into n/S(ψA) copies of a weakly

entangled state |ψ⟩. We assume that this dilution procedure

can be achieved before the action of the noise. Note that the

number of diluted states |ψ⟩is larger than the number of sin-

glets n, and each of the additional qubits of Bob is also sub-

ject to the same noise Λ, see Fig. 1. After the action of the

noise, Alice and Bob end up sharing n/S(ψA) copies of the

state σ=11⊗Λ[ψ], which they can distill into singlets at

rate Ed(σ). Overall, in the limit of large n, Alice and Bob

can obtain nEd(11⊗Λ[ψ])/S(ψA) singlets using the dilution

procedure.

From the above discussion, it is clear that the dilution pro-

vides an advantage whenever the inequality

Ed(11⊗Λ[ψ])

S(ψA)>Ed(11⊗Λ[ψ−]) (1)

holds for some state |ψ⟩. As we will now see, the dilution pro-

arXiv:2210.14192v2 [quant-ph] 3 Jul 2023

Figure 1. Applying entanglement dilution to reduce the loss of entanglement under local noise. Figure a) shows the setup without dilution:

a singlet |ψ−⟩is subject to local noise on Bob’s side, resulting in the state ρ=11⊗Λ[ψ−]. In our example, Alice and Bob can distill ρinto

singlets at rate 1/3. In ﬁgure b), Alice and Bob ﬁrst dilute their singlets into weakly entangled states |ψ⟩. Each of these states undergoes the

same local noise as in ﬁgure a), resulting in the states σ=11⊗Λ[ψ], which can then again be distilled into singlets. The overall singlet rate is

2/3, showing an improvement over the setup in ﬁgure a).

cedure can indeed provide an advantage, even if the diluted

states |ψ⟩exhibit arbitrarily little entanglement. Suppose for

example that Bob’s qubits are subject to phase damping de-

scribed by Λ[ρ]=K0ρK†

0+K1ρK†

1, with Kraus operators

K0= 1 0

0√1−λ!,K1= 0 0

0√λ!,(2)

and 0 < λ < 1. We consider a situation when Alice and Bob

dilute their singlets into pure states |ψ⟩=cos α|00⟩+sin α|11⟩.

In Fig. 2, we show both sides of the inequality (1) for λ=

1/2 as a function of α, see Supplemental Material for a more

0.0 0.2 0.4 0.6 0.8

0.40

0.42

0.44

0.46

0.48

0.50

Figure 2. Reducing entanglement loss under local phase damping

by diluting into pure states |ψ⟩=cos α|00⟩+sin α|11⟩. Solid curve

shows R=Ed(11⊗Λ[ψ])/S(ψA) as a function of αfor noise parameter

λ=1/2. This corresponds to the singlet rate achievable via dilution

into |ψ⟩. Dashed line shows the corresponding singlet rate Ed(11⊗

Λ[ψ−]) if no dilution is applied. Maximal performance is achieved in

the limit α→0.

detailed analysis. We see that diluting the singlets provides an

advantage for all αin the range 0 ≤α < π/4. Moreover, the

performance of dilution increases with decreasing α, reaching

its maximal value for α→0. This behavior is surprising, as

for α=0 the state is not entangled.

Reducing the loss of coherence. We will now investigate

the usefulness of resource dilution for preserving quantum co-

herence. Here, we assume that an initial collection of nqubits

in a maximally coherent state |+⟩=(|0⟩+|1⟩)√2 undergoes

a local decoherence process Λ(which we will specify later),

leading to ncopies of a ﬁnal state ρ= Λ[|+⟩⟨+|]. Similar

to entanglement, we will now see that the loss of coherence

can be reduced by diluting the maximally coherent states into

weakly coherent ones.

In the following, we will focus on the resource theory of

coherence based on maximally incoherent operations [7,11].

We take the free states to be diagonal in the reference basis

{|i⟩}, and the free operations to be all operations which do not

create coherence in the reference basis. This is the largest

set of operations which is compatible with any reasonable re-

source theory of quantum coherence, we refer to the Supple-

mental Material for more details.

After each of the qubits undergoes the decoherence pro-

cess, in the limit of large nit is possible to distill the result-

ing states ρinto maximally coherent states at rate C(ρ)=

S(∆[ρ]) −S(ρ) [12] with ∆[ρ]=Pi⟨i|ρ|i⟩|i⟩⟨i|. Similar to

entanglement dilution, it is possible to perform coherence di-

lution, converting the maximally coherent states into weakly

coherent states µat rate 1/C(µ). Letting these states undergo

the decoherence process Λ, the overall rate of maximally co-

herent states obtainable after the action of the noise is given

by C(Λ[µ])/C(µ). As we will see in the following, dilution

is useful for protecting a system from a decoherence process.

Moreover, we will see that for some types of noise it is ad-

0.5 1.0 1.5

0.10

0.12

0.14

0.16

Figure 3. Reducing the loss of coherence by dilution for single-qubit

amplitude damping noise. We show the rate R=C(Λ[µ])/C(µ) by

dilution into pure states cos α|0⟩+sin α|1⟩(solid curve) and mixed

states sin2α|+⟩⟨+|+cos2α11/2 (dashed curve) for γ=0.9 as a func-

tion of α. Dotted line shows the coherence rate R=C(Λ[|+⟩⟨+|]) ≈

0.13 achievable without dilution.

vantageous to dilute the states |+⟩into mixed states with little

coherence.

As an example demonstrating this eﬀect, consider the

single-qubit amplitude damping, which is represented by the

Kraus operators

K0= 1 0

0p1−γ!,K1= 0√γ

0 0 !.(3)

In Fig. 3, we show the ﬁnal rate of maximally coherent states

achievable without dilution, with dilution into pure qubit

states of the form |ψ⟩=cos α|0⟩+sin α|1⟩, and into mixed

states of the form µ=sin2α|+⟩⟨+|+cos2α11/2 for γ=0.9.

As we see from Fig. 3, by diluting into pure qubit states it

is possible to extract maximally coherent states at an overall

rate of 0.15, which is achieved for α≈0.34. This is also the

maximal possible rate achievable by dilution into pure qubit

states, see Supplemental Material for more details. In contrast

to this, diluting into the mixed state µachieves maximal per-

formance in the limit α→0, leading to an overall coherence

rate log(19)/18 ≈0.16. Noting that the state µis maximally

mixed in this limit, this result is highly counterintuitive, as

it means that the best performance is obtained by creating a

large number of states which are almost maximally mixed.

Reducing the loss of energy and coherence in quantum

thermodynamics. As we will now see, the ideas presented

above are also applicable to the resource theory of quan-

tum thermodynamics. Here, we consider a quantum system

Swith Hamiltonian HS, and the corresponding Gibbs state

γS=e−βHS/Tr[e−βHS] at the inverse temperature β=1/kT .

The Gibbs state is the free state of the resource theory of quan-

tum thermodynamics, and the free transformations are known

as thermal operations [13]. These are transformation of the

system which can be implemented by coupling the system

to a thermal bath with Hamiltonian HBand applying an en-

ergy preserving unitary: Λ[ρS]=TrB[U(ρS⊗γB)U†], where

[U,HS+HB]=0. Thermal operations preserve the Gibbs

state and do not increase the Helmholtz free energy of the sys-

tem [14,15].

If ncopies of a quantum state ρare available, then in the

limit n→ ∞ by using thermal operations it is possible to con-

vert ρinto a state σwhich is diagonal in the eigenbasis of HS

at rate [15]

R(ρ→σ)=S(ρ||γ)

S(σ||γ)(4)

with the quantum relative entropy S(ρ||γ)=Tr[ρlog2ρ]−

Tr[ρlog2γ]. If σis not diagonal in the energy eigenbasis, the

conversion is possible at the same rate by using thermal oper-

ations together with a sublinear number of qubits with coher-

ence [15]. It is possible to relax the set of free transformation

to be Gibbs-preserving, i.e., making the only requirement that

they leave the Gibbs state invariant [16]. In this case, asymp-

totic transformations are also characterized by Eq. (4), as fol-

lows from results in [17]. In contrast to thermal operations,

no additional coherence is required in this setup.

In the following, we will focus on qubit systems with

Hamiltonian H=E0|E0⟩⟨E0|+E1|E1⟩⟨E1|at temperature T,

where |Ei⟩are the eigenstates of the Hamiltonian with eigen-

values Ei, and E0is the ground state energy. Consider now

nqubits, initialized in the excited state |E1⟩. We assume that

each of the qubits is subject to a thermal noise Λ, i.e., noise

which does not create coherence in the eigenbasis of HSand

does not increase the Helmholtz free energy of the system.

As we will now see, diluting the excited state |E1⟩into noisy

states can provide an advantage, making the overall nqubit

system more robust against the action of thermal noise.

In analogy to our previous results for entanglement and co-

herence, by diluting ncopies of the excited states into µbefore

the action of the noise it is possible to obtain excited states

at the overall rate S(Λ[µ]||γ)/S(µ||γ). Dilution of the excited

state into µprovides an advantage whenever

S(Λ[µ]||γ)

S(µ||γ)>S(Λ[|E1⟩⟨E1|]||γ)

S(|E1⟩⟨E1|||γ).(5)

As an example, consider now noise of the form

Λ[ρ]=pγ+(1 −p)∆[ρ],(6)

with ∆[ρ]=Pi⟨Ei|ρ|Ei⟩|Ei⟩⟨Ei|and 0 ≤p≤1. It is clear that

the noise Λis thermal, as it destroys all coherence eventually

available in the state ρand does not increase the Helmholtz

free energy of the system. For this type of noise, optimal per-

formance can be achieved by diluting the excited state into a

diagonal state:

µ=(1 −q)|E0⟩⟨E0|+q|E1⟩⟨E1|,(7)

we refer to the Supplemental Material for more details.

So far we have focused on transformations between states

which are diagonal in the energy eigenbasis. We will now go

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ThepowerofnoisyquantumstatesandtheadvantageofresourcedilutionMarekMiller,ManfrediScalici,MarcoFellousAsiani,andAlexanderStreltsov∗CentreforQuantumOpticalTechnologies,CentreofNewTechnologies,UniversityofWarsaw,Banacha2c,02-097Warsaw,PolandEntanglementdistillationallowstoconvertnoisyquantumstatesintos...

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