Entanglement robustness via spatial deformation of identical particle wave functions
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Entanglement robustness via spatial deformation of identical particle wave functions
Matteo Piccolini,1, 2, ∗ Farzam Nosrati,1, 2, † Giuseppe Compagno,3
Patrizia Livreri,1 Roberto Morandotti,2 and Rosario Lo Franco1, ‡
1
Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy
2
INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec J3X 1S2, Canada
3
Dipartimento di Fisica e Chimica - Emilio Segrè, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy
We address the problem of entanglement protection against surrounding noise by a procedure suitably ex-
ploiting spatial indistinguishability of identical subsystems. To this purpose, we take two initially separated
and entangled identical qubits interacting with two independent noisy environments. Three typical models of
environments are considered: amplitude damping channel, phase damping channel and depolarizing channel.
After the interaction, we deform the wave functions of the two qubits to make them spatially overlap before
performing spatially localized operations and classical communication (sLOCC) and eventually computing the
arXiv:2104.09714v1 [quant-ph] 20 Apr 2021
entanglement of the resulting state. This way, we show that spatial indistinguishability of identical qubits can be
utilized within the sLOCC operational framework to partially recover the quantum correlations spoiled by the
environment. A general behavior emerges: the higher the spatial indistinguishability achieved via deformation,
the larger the amount of recovered entanglement.
I. INTRODUCTION measurements. An entropic measure has been recently intro-
duced [55] to quantify the degree of indistinguishability of
It is well known that the environment of an open quantum identical particles arising from their spatial overlap. Further-
system produces a detrimental noise which has to be dealt more, an operational framework based on spatially localized
with during the implementation of many useful quantum in- operations and classical communication (sLOCC), where the
formation processing schemes [1, 2]. One of the main goals no-label approach finds its natural application, has been firstly
in the development of fault-tolerant enhanced quantum tech- theorized [53] and later experimentally implemented [56, 57]
nologies is to provide a strategy to protect the entanglement as a way of activating physical entanglement. Such frame-
from such degradation. This challenge has been addressed, work has also been applied to fields such as the exploitation of
e.g., by the seminal works on quantum error corrections [3–6], the Hanbury Brown-Twiss effect with identical particles [58],
structured environments with memory effects [7–17], distilla- quantum entanglement in one-dimensional systems of anyons
tion protocols [18–20], decoherence-free subspaces [21, 22], [59], entanglement transfer in a quantum network [60], and
dynamical decoupling and control techniques [23–32]. quantum metrology [61, 62]. Moreover, in a recent paper [55]
It is not unusual to find identical particles (i.e., subsys- it has been shown that spatial indistinguishability, even partial,
tems such as photons, atoms, nuclei, electrons or any artifi- can be exploited to recover the entanglement spoiled from the
cial qubits of the same species) as building blocks of quan- preparation noise of a depolarizing channel.
tum information processing devices and quantum technolo-
gies [33, 34]. Nonetheless, the standard approach to identi- In this work, we aim to extend the results of Ref. [55]
cal particles based on unphysical labels is known to give rise to the wider scenario of different paradigmatic noise chan-
to formal problems when trying to asses the correlations be- nels, namely amplitude damping, phase damping and depo-
tween constituents with (partially or completely) overlapping larizing channels, under both Markovian and non-Markovian
spatial wave functions [35, 36]. For this reason, many alter- regimes. To do so, we introduce spatial deformations, i.e.,
native approaches have been developed to deal with the for- transformations turning initially spatially separated (and thus
mal aspects of the entanglement of identical particles [36– distinguishable) particles into indistinguishable ones by mak-
54]. Among these, the no-label approach [51–53] provides ing their wave functions spatially overlap. We then analyze
many advantages: for example, it allows to address the cor- the entanglement dynamics of two identical qubits interacting
relations between identical particles exploiting the same tools separately with their own environment, with the goal of show-
used for nonidentical ones (e.g., the von Neumann entropy ing that the application of the mentioned spatial deformation
of the reduced density matrix). Furthermore, it provides the at a given time of the evolution, immediately followed by the
known results for distinguishable particles in the limit of non- sLOCC measurement, constitutes a procedure capable of re-
overlapping (spatially separated) wave functions. Treating the covering quantum correlations.
global multiparticle state as a whole, indivisible object, in the
no-label approach entanglement strictly depends on both the This paper is organized as follows: in Section II we in-
spatial overlap of the wave functions and on spatially localized troduce the general framework of the analyzed dynamics and
the main tools used, namely the deformation operation and
the sLOCC protocol. The main results follow in Section III,
where we describe the considered model and study the scenar-
∗ matteo.piccolini@unipa.it ios of an amplitude damping channel, a phase damping chan-
† farzam.nosrati@unipa.it nel and a depolarizing channel. Finally, Section IV summa-
‡ rosario.lofranco@unipa.it rizes and discusses the main results.
2
ρAB(0) (a) (b) ρAB(t)
Noisy evolution
Deformation + sLOCC
A B A B
(c)
ρD(t) ρLR(t)
ψ1 ψ2 ψ1 ψ2
+ L local R
measurements
FIG. 1. State evolution in the considered scenario. (a) The two qubits are initially prepared in the pure entangled state ρAB (0). (b) They are left
to interact with a noisy environment, whose detrimental action produces the mixed state ρAB (t). (c) At time t a deformation of the two particles
wave functions is performed, immediately followed by a sLOCC measurement.
II. MATERIALS AND METHODS A. Deformations of identical particle states
Given a multipartite quantum system, a quantum transfor-
In this section we introduce the goal of this paper and the
mation acting differently on each subpart changing the rela-
main tools used to achieve it.
tions among them is called a deformation. In this section we
Let us consider the following process, illustrated in Fig. 1: focus on the specific set of continuous deformations which
at the beginning, two identical qubits in the entangled state modify the single spatial wave functions of identical particles.
ρAB (0) occupy two different regions of space A and B, thus In what follows, the no-label formalism [51] is used.
being distinguishable and individually addressable. Here, they Let us take a non-entangled state of two identical particles
locally interact with two spatially separated and independent |Φi = |φ1 ; φ2 i, where φi (i = 1, 2) is identified by the val-
noisy environments which spoil the initial correlations. At ues of a complete set of commuting observables describing a
time t, the two particles get decoupled from the environments spatial wave function ψi and an internal degree of freedom τi .
and undergo a deformation which makes their wave functions We suppose that the two particles are initially spatially sepa-
spatially overlap into the state ρD (t). Immediately after that, a rated, e.g., localized in two distinct regions A and B such that
sLOCC measurement is performed to generate the entangled 1 i = |Ai, |ψ2 i = |Bi and hA|Bi = 0. We want to modify
|ψ(0) (0)
state ρLR (t). In this work, we show that this procedure al- the spatial wave functions of the two particles in order to make
lows for the recovery of the entanglement spoiled by the pre- them overlap. Thus, we introduce a deformation D such that
viously introduced noise in an amount which depends on the
D
degree of spatial indistinguishability achieved with the defor- |φ1 ; φ2 i = |A, τ1 i ⊗ |B, τ2 i −→ |ψ1 , τ1 ; ψ2 , τ2 i , (1)
mation. Three different models of environmental noise shall
be considered: an amplitude damping channel, a phase damp- with hψ1 |ψ2 i , 0. Since the two spatially overlapping particles
ing channel and a depolarizing channel. are also identical, they are now indistinguishable: their final
global state cannot be written as the tensor product of single
Notice that here the system-environment interaction oc- particle states anymore and must be considered as a whole,
curs when the two particles are still distinguishable and no i.e. |ψ1 , τ1 ; ψ2 , τ2 i , |ψ1 , τ1 i ⊗ |ψ2 , τ2 i.
finite time interval separates the deformation from the imme- A deformation operator acting on identical particles is not,
diately subsequent sLOCC operation. It will thus be interest- in general, unitary, and its normalized action on a state ρ is
ing to compare the results of this work with those discussed in thus
Ref. [63], where the interaction with the noisy channels hap-
pens instead during a finite time interval between the defor- DρD† X
D[ρ] = = p̄i D[ρi ], (2)
mation and the sLOCC operation, that is when the qubits are Tr[DD ρ] †
i
indistinguishable in the frame of the localized environments.
where
The deformation process bringing two particles to spatially
overlap shall be now briefly introduced, followed by a recall Tr[DD† ρi ] Dρi D†
p̄i = , D[ρi ] = . (3)
of the sLOCC operational framework. Tr[DD† ρ] Tr[DD† ρi ]
3
B. sLOCC, Spatial Indistinguishability and Concurrence where Z = PLψ1 PRψ2 + PLψ2 PRψ1 . Notice that (9) ranges from
0 for spatially separated (thus distinguishable) particles (e.g.
The natural extension of the standard local operation when PLψ1 = PRψ2 = 1) to 1 for maximally indistinguishable
and classical communication framework (LOCC) for distin- particles (PLψ1 = PLψ2 , PRψ1 = PRψ2 ). Hereafter, we assume
guishable particles to the scenario of indistinguishable (and for convenience that the spatial wave functions of the single
thus individually unaddressable) particles is provided by the indistinguishable particles after the deformation have the form
spatially localized operations and classical communication
(sLOCC) environment [53]. Given a set of indistinguishable |ψ1 i = l |Li + r |Ri , |ψ2 i = l0 |Li + r0 |Ri , (10)
particles, sLOCC consist in a projective measurement of the
global state over distinct spatially separated regions, followed where
by a post-selection of the outcomes where only one particle
l = hL|ψ1 i , r = hR|ψ1 i , l0 = hL|ψ2 i , r0 = hR|ψ2 i (11)
is found in each location. The result of this operation is an
entangled state whose physical accessibility has been demon- are complex coefficients such that |l|2 + |r|2 = |l0 |2 + |r0 |2 = 1.
strated in a quantum teleportation experiment [56]. In the following analysis, we shall conveniently set l = r0 to
Suppose we are given a state ρ of two identical and indis- assure that the sLOCC probability PLR is different from zero.
tinguishable particles, e.g., obtained by the application (2) of As previously stated, the state ρLR obtained by the sLOCC
the deformation (1), and assume they have pseudo-spin 1/2. measurement is entangled. We recall that the entanglement of
The whole sLOCC operation (projection and post-selection) the bipartite quantum state ρLR of two distinguishable qubits
amounts to projecting the two qubits state on the subspace can be quantified by the Wootters concurrence [55, 64]
spanned by the basis
p p p
C(ρLR ) = max{0, λ4 − λ3 − λ2 − λ1 },
p
BLR = {|L ↑, R ↑i , |L ↑, R ↓i , |L ↓, R ↑i , |L ↓, R ↓i}, (4) (12)
via the projection operator where λi are the eigenvalues in decreasing order of the matrix
ξ = ρLR eρLR , with e
ρLR = (σLy ⊗ σRy ) ρ∗AB (σLy ⊗ σRy ) and σLy , σRy
X
Π̂LR = |Lσ, Rτi hLσ, Rτ| . (5)
σ,τ=↑,↓
being the usual Pauli matrix σy localized, respectively, on the
particle in L and in R.
Since the constituents are indistinguishable before the detec-
tion, it is impossible to know exactly which particle will be
found in which region. The sLOCC operation generates the III. INDISTINGUISHABILITY AS A FEATURE FOR
(normalized) two-particle entangled state RECOVERING ENTANGLEMENT
Π̂LR ρ(t) Π̂LR
ρLR (t) = h i , (6) In this section we report our main results. Each of the two
Tr Π̂LR ρ
independent environments is modeled as a bath of harmonic
with probability oscillators in the vacuum state except for one mode which is
h i coupled to the qubit interacting with it. Considering a qubit-
PLR = Tr Π̂LR ρ . (7) cavity model with just one excitation overall allows us to treat
After the sLOCC measurement, the two qubits occupy two the reservoir as characterized by a Lorentzian spectral density
distinct regions of space and are thus now distinguishable [65, 66]
and individually addressable. Furthermore, since in the no-
γ λ2
label formalism the inner product between two-particle states J(ω) = , (13)
is given by the rule [51] 2π (ω − ω0 )2 + λ2
hφ01 ; φ02 |φ1 ; φ2 i = hφ01 |φ1 i hφ02 |φ2 i + η hφ01 |φ2 i hφ02 |φ1 i , (8) where ω0 is the qubit transition frequency, γ is the micro-
scopic system-environment coupling constant related to the
with η = 1 for bosons and η = −1 for fermions, particle statis- decay of the excited state of the qubit in the Markovian limit of
tics naturally emerges within the sLOCC framework and is flat spectrum, and λ is the spectral width of the coupling quan-
thus expected to play a role in the dynamics. tifying the leakage of photons through the cavity walls. The
The sLOCC scenario also allows for the introduction of relaxation time τR on which the state of the system changes is
an entropic measure of the particles’ indistinguishability af- related to the coupling constant by the relation τR ≈ γ−1 , while
ter the deformation (1), which depends on the achieved spa- the reservoir correlation time τB is connected to the spectral
tial distribution of their wave functions ψ1 , ψ2 over the two width of the coupling by τB ≈ λ−1 . These coefficients regu-
regions L and R where sLOCC measurement occurs. Given late the behavior of the system: when γ < λ/2 (τR > 2τB ) the
the probability PXψi of finding the qubit having wave function system is weakly coupled to the environment, the reservoir
ψi (i = 1, 2) in the region X (X = L, R), the spatial indistin- correlation time is shorter than the relaxation time and we are
guishability measure is given by [55] in a Markovian regime; when γ > λ/2 (τR < 2τB ) instead,
PLψ1 PRψ2 PLψ1 PRψ2 PLψ2 PRψ1 PLψ2 PRψ1 we are in the strong coupling scenario, where the relaxation
I=− log2 − log2 , time is shorter than the bath correlation time and the regime
Z Z Z Z
(9) is non-Markovian. The way each qubit interacts with its own4
reservoir depends on the type of noise channel taken into ac- with A and B being two distinct spatial regions (hA|Bi = 0).
count. Thanks to the fact that the the two environmental interactions
The action of the three noisy channels considered in this are independent, the state after the noisy interaction is given
paper shall be computed within the usual Kraus operators by
formalism, or operator-sum representation [67]. The gen-
eral expression of the single-qubit evolved density matrix is
ρAB (t) = E0A ⊗ E0B ρAB (0) E0A ⊗ E0B
then given by ρ(t) = i Ei ρ(0)Ei† , where the Ei ’s are the
P
time-dependent Kraus operators corresponding to the spe- + E1A ⊗ E1B ρAB (0) E1A † ⊗ E1B †
cific channel and depend on the disturbance probability (de- (19)
coherence function) p(t). Each channel in fact introduces a + E0A ⊗ E1B ρAB (0) E0A ⊗ E1B †
time-dependent disturbance on the system with a probability
+ E1A ⊗ E0B ρAB (0) E1A † ⊗ E0B ,
p(t) = 1 − q(t) obtained by solving the differential equation where EiX (i = 1, 2, X = A, B) denotes the i-th single particle
[65, 68] Kraus operator of Eq. (17) acting on the qubit localized in
Z t region X, while ρAB (0) = |1− iAB h1− |AB is the initial density
q̇(t) = − dt1 f (t − t1 ) q(t1 ), (14) matrix. Using Eq. (17) in the above equation, one then finds
0
where the correlation function f (t − t1 ) is given by the Fourier
ρAB (t) = 1 − p(t) |1− iAB h1− |AB + p(t) |A ↑, B ↑i hA ↑, B ↑| .
transform of the spectral density J(ω) of the reservoir, namely
(20)
Z We now want to apply the deformation defined in Eq. (1) to
f (t − t1 ) = dω J(ω) e−i(ω−ω0 )(t−t1 ) . (15) the state (20) at time t. State |1− iAB gets mapped to
Solving Eq. (14) for the spectral density (13), one obtains the 1
disturbance (or error) probability [65] |1̄− iD = √ |ψ1 ↑, ψ2 ↓i − |ψ1 ↓, ψ2 ↑i , (21)
2
!#2
λ
" !
dt dt
p(t) = 1 − e−λt cos + sin , (16) which is not a normalized state since hψ1 |ψ2 i , 0. In order to
2 d 2
write it in terms of a normalized state |1̄− iN , we compute
with d = 2γλ − λ2 . Notice that this solution encompasses
p
q
both Markovian and non-Markovian regimes, depending on
h1̄− |1̄− iD = C12 , C1 := 1 − η|hψ1 |ψ2 i|2 , (22)
the ratio λ/γ. In particular, in the Markovian limit of flat spec-
trum which occurs for γ/λ 1, it is straightforward to see
that p(t) = 1 − e−γt/2 , as expected [67]. In general, the error and write it as
probability (16) is such that p(0) = 0 and lim p(t) = 1.
t→∞
|1̄− iD = C1 |1̄− iN . (23)
A. Amplitude Damping Channel
The same is done for the deformation of |A ↑, B ↑i, which gets
The action of the amplitude damping channel on a single mapped to
qubit in the operator-sum representation is given by the Kraus
operators [67]
q
|ψ1 ↑, ψ2 ↑iD = C2 |ψ1 ↑, ψ2 ↑iN , C2 := 1 + η|hψ1 |ψ2 i|2 ,
E0 = |↑i h↑| + 1 − p(t) |↓i h↓| = E0† , (24)
p
(17) where
E1 = p(t) |↑i h↓| , E1† = p(t) |↓i h↑| .
p p
Consider two identical qubits initially prepared in the Bell sin- hψ1 ↑, ψ2 ↑ |ψ1 ↑, ψ2 ↑iN = 1. (25)
glet state
1 The normalized state resulting from the spatial deforma-
|1− iAB = √ |A ↑, B ↓i − |A ↓, B ↑i , (18) tion (2) of the state (20) is thus
2
1 − p(t) C12 |1̄− iN h1̄− |N + p(t) C22 |ψ1 ↑, ψ2 ↑iN hψ1 ↑, ψ2 ↑|N
ρD (t) = . (26)
1 − p(t) C12 + p(t) C22
Following the scheme shown in Figure 1, we perform the sLOCC measurement immediately after the deformation, ap-5
plying the projection operator (5) onto the state (26), which finally gives
1 − p(t) |lr0 − η l0 r|2 |1− iLR h1− |LR + p(t) |lr0 + η l0 r|2 |L ↑, R ↑i hL ↑, R ↑|
ρLR (t) = , (27)
1 − p(t) |lr0 − η l0 r|2 + p(t) |lr0 + η l0 r|2
where l, r, l0 , r0 are the wave function coefficients defined from the ones for fermions (and vice versa) by simply chang-
in (11). ing sign to one of the coefficients l, r, l0 , r0 (that is, by shift-
In order to study the entanglement evolution of the state ing the phase of one of them by π). Therefore, in order to
ρLR (t) of Eq. (27), we calculate the concurrence defined in fix a framework to analyze the concurrence, we assume we
Eq. (12), which is are dealing with fermions whose spatial wave functions are
distributed over the regions L and R with positive real coeffi-
|lr0 − η l0 r|2 1 − p(t) cients. This reasoning shall hold for the other noisy channels,
C ρLR (t) = , (28)
so that the presented results are also valid for bosons. With
|lr0 − η l0 r|2 1 − p(t) + |lr0 + η l0 r|2 p(t)
where the statistics parameter η explicitly appears, as ex-
pected. As a first consideration, we notice that the results
about entanglement dynamics for bosons can be obtained
Markovian
Markovian ΔC(t)
C(ρLR (t)) 1.0
1.0
0.8
0.8 I=1
I=1
0.6 I=0.90
I=0.90
0.6 I=0.75
I=0.75
0.4 I=0.50
I=0.50
0.4 I=0
0.2
0.2
0.0 γt
0.0 γt 0 2 4 6 8 10
0 2 4 6 8 10
Non-Markovian
Non-Markovian ΔC(t)
C(ρLR (t)) 1.0
1.0
0.8
0.8 I=1
I=1
0.6 I=0.90
I=0.90
0.6 I=0.75
I=0.75
0.4 I=0.50
I=0.50
0.4 I=0
0.2
0.2
0.0 γt
0.0 γt 0 100 200 300 400 500
0 100 200 300 400 500
FIG. 3. Net gain in the entanglement recovery of two identical qubits
FIG. 2. Concurrence of two identical qubits (fermions with (fermions with l, l0 , r, r0 > 0, bosons with one of these four coeffi-
l, l0 , r, r0 > 0, bosons with one of these four coefficients negative) cients negative) in the initial state |1− iAB under localized amplitude
in the initial state |1− iAB subjected to localized amplitude damping damping channels, thanks to the deformation+sLOCC operation per-
channels, undergoing an instantaneous deformation+sLOCC opera- formed at time t. Results are reported for different degrees of spatial
tion at time t for different degrees of spatial indistinguishability I indistinguishability I (with |l| = |r0 |). Both the Markovian (λ = 5γ)
(with |l| = |r0 |). Both the Markovian (λ = 5γ) (upper panel) and (upper panel) and non-Markovian (λ = 0.01γ) (lower panel) regimes
non-Markovian (λ = 0.01γ) (lower panel) regimes are reported. are shown.6
this assumption, we get the concurrence as Markovian
PLR (t)
1.0
h i
(lr0 )2 + (l0 r)2 + 2 ll0 rr0 1 − p(t)
C ρLR (t) = .
(29)
(lr0 )2 + (l0 r)2 + 2 ll0 rr0 1 − 2p(t) 0.8
I=1
I=0.90
We point out that when no deformation is performed and 0.6
I=0.75
the particles remain distinguishable in two distinct regions
I=0.50
(I = 0), the sLOCC projector (5) is equivalent to the iden- 0.4 I=0
tity operator. This implies that when the particles are not
brought to spatially overlap, our procedure gives the same en- 0.2
tanglement we would have without performing the sLOCC
operation. For this reason, we take the results for I = 0 as 0.0 γt
0 2 4 6 8 10
the term of comparison to quantify the entanglement gained
due to the deformation + sLOCC procedure, i.e., ∆C(t) :=
C ρLR (t) − C ρAB (t) . Figure 2 shows the concurrence (29) Non-Markovian
for both the Markovian and the non-Markovian regimes, while PLR (t)
1.0
Figure 3 displays ∆C(t).
As can be seen in Figure 2, spatial indistinguishability (9)
0.8
has a direct influence on the general behavior: when the par- I=1
ticles are not perfectly indistinguishable (I , 1), the entan- I=0.90
0.6
glement vanishes with a monotonic decay in the Markovian I=0.75
regime and with a periodic one in the non-Markovian regime. I=0.50
0.4 I=0
From Figure 3, we can see that when I , 1 the deformation
and sLOCC procedure becomes inefficient in recovering the
0.2
correlations as time grows. Nonetheless, it is interesting to
notice that it provides an initial effective advantage as a con-
0.0 γt
sequence of the fact that the decay rate shown in Figure 2 gets 0 100 200 300 400 500
lower as the indistinguishability increases. However, when
the particles wave functions maximally overlap (I = 1), the FIG. 4. Success probability of obtaining a nonzero outcome from
entanglement remains stable at its initial maximum value, thus the sLOCC projection for fermions (l, l0 , r, r0 > 0 and l = r0 ) in-
becoming unaffected by the noise. These results show that, in teracting with localized amplitude damping channels. Different de-
the scenario of the amplitude damping channel, we have pro- grees of spatial indistinguishability are reported in both the Marko-
vided an operational framework where spatial indistinguisha- vian (λ = 5γ) (upper panel) and non-Markovian (λ = 0.01γ) (lower
bility, even imperfect, of two identical qubits can be exploited panel) regimes.
as a scheme to recover quantum correlations spoiled by a
short-time interaction with the noisy environment.
Finally, to check whether such procedure would be of any the concurrence plotted in Fig. 2 is PLR (t) = 1 − p(t) (notice,
practical interest we have to analyze its theoretical probabil- however, that this success probability can be improved by dif-
ity of success. This strictly depends on the probability for the ferently setting the coefficients of the spatial wave functions).
sLOCC projection (6) to produce a non-null result, physically
representing a state which does not get discarded during the
postselection. Such probability is defined in Eq. (7) and, for B. Phase Damping channel
identical qubits undergoing a local interaction with an ampli-
tude damping channel, it is equal to
A phase damping channel acting on a single qubit is de-
scribed by the Kraus operators
(lr0 )2 + (l0 r)2 − 2η ll0 rr0 1 − 2 p(t)
PLR (t) = . (30)
C12 1 − p(t) + C22 p(t) E0 = |↑i h↑i + 1 − p(t) |↓i h↓| = E0† ,
p
(31)
E1 = p(t) |↓i h↓| = E1† .
p
Figure 4 shows the success probability (30) for different de-
grees of spatial indistinguishability in both the Markovian and
non-Markovian regime in the case of fermions. As can be Once again, we consider the Bell state |1− iAB of two identical
seen, when the indistinguishability is not maximum, the prob- qubits defined in (18) as our initial state. The evolved state
ability of success tends to 1 as time passes in both regimes, ρAB (t) after the interaction with the two independent environ-
thus giving rise to a trade-off with the concurrence. The trade- ments is computed as in Eq. (19), which for the phase damping
off is confirmed by the probability being constant and equal to channel described by the above Kraus operators gives
1/2 when the concurrence is maximum, i.e. for I = 1. For !
bosons, the time-dependent success probability correspond- p(t) p(t)
ρAB (t) = 1 − |1− iAB h1− |AB + |1+ iAB h1+ |AB , (32)
ing to I = 1 (with the constraint l = r0 = l0 = −r) and to 2 27
Markovian where |1+ iAB is the Bell state defined as
C(ρLR (t))
1.0
1
0.8
I=1
|1+ iAB = √ |A ↑, B ↓i + |A ↓, B ↑i . (33)
I=0.90
2
0.6
I=0.75
I=0.50
0.4 I=0
At time t, deformation (1) is applied to the state (32) to make
the two particles spatially overlap. Deformation of |1− iAB
0.2 gives the state (21), while |1+ iAB gets mapped to
0.0 γt
0 2 4 6 8 10
1
|1̄+ iD = √ |ψ1 ↑, ψ2 ↓i + |ψ1 ↓, ψ2 ↑i . (34)
2
Non-Markovian
C(ρLR (t))
1.0
Once again, state |1̄+ iD is not normalized: it is indeed easy to
0.8
I=1
show that
I=0.90
0.6
I=0.75
I=0.50 |1̄+ iD = C2 |1̄+ iN , (35)
0.4 I=0
0.2
where h1̄+ |1̄+ iN = 1 and C2 is defined in (24). Thus, the global
normalized state after the deformation is
0.0 γt
0 100 200 300 400 500
FIG. 5. Concurrence of two identical qubits (fermions with 1− 1
2 p(t) C12 |1̄− iN h1̄− |N + 1
2 p(t) C22 |1̄+ iN h1̄+ |N
l, l0 , r, r0 > 0, bosons with one of these four coefficients negative) ρD (t) = .
in the initial state |1− iAB interacting with localized phase damping 1 − 21 p(t) C12 + 1
2 p(t) C22
channels, undergoing an instantaneous deformation+sLOCC opera- (36)
tion at time t for different degrees of spatial indistinguishability I Finally, the sLOCC operation is performed: the action of the
(with |l| = |r0 |). Both the Markovian (λ = 5γ) (upper panel) and projection operator (5) on the state (36), as defined in Eq. (6),
non-Markovian (λ = 0.01γ) (lower panel) regimes are reported. gives
1− 1
2 p(t) |lr0 − η l0 r|2 |1− iLR h1− |LR + 1
2 p(t)|lr0 + η l0 r|2 |1+ iLR h1+ |LR
ρLR (t) = . (37)
1 − 12 p(t) |lr0 − η l0 r|2 + 1
2 p(t)|lr0 + η l0 r|2
We now study the entanglement evolution of such a state by Focusing the analysis once again on fermions with real and
the concurrence C(ρLR (t)), which is readily found to be positive coefficients l, r, l0 , r0 to fix a framework, concur-
rence (38) is then equal to
C ρLR (t) = max {0, λ1 (t) − λ2 (t)} ,
h i
λ1 (t) := max {λA (t), λB (t)} , λ2 (t) := min {λA (t), λB (t)} , 1 − p(t) (lr0 )2 + (l0 r)2 + 2ll0 rr0
C ρLR (t) = .
(39)
(38)
(lr0 )2 + (l0 r)2 + 1 − p(t) 2ll0 rr0
with
The time behavior of the concurrence of Eq. (39) is plotted
1 − 21 p(t) |lr0 − η l0 r|2 in Figure 5 for both the Markovian and the non-Markovian
λA (t) := , regime, while the net gain due to the deformation and sLOCC
1 − 12 p(t) |lr0 − η l0 r|2 + 21 p(t)|lr0 + η l0 r|2
operation is depicted in Figure 6. Once again, the entangle-
ment recovered is found to decrease as the interaction time in-
creases where the generated spatial indistinguishability is not
1
p(t) |lr0 + η l0 r|2
λB (t) := 2 . maximum. As in the amplitude damping scenario, such de-
1− 1
2 p(t) |lr0 − η l0 r|2 + 21 p(t)|lr0 + η l0 r|2 phasing is monotonic in the Markovian regime and periodic8
Markovian Markovian
ΔC(t) PLR (t)
1.0 1.0
0.8 0.8
I=1
I=1
I=0.90
0.6 I=0.90 0.6
I=0.75
I=0.75
I=0.50
0.4 I=0.50 0.4 I=0
0.2 0.2
0.0 γt 0.0 γt
0 2 4 6 8 10 0 2 4 6 8 10
Non-Markovian Non-Markovian
ΔC(t) PLR (t)
1.0 1.0
0.8 0.8
I=1
I=1
I=0.90
0.6 I=0.90 0.6
I=0.75
I=0.75
I=0.50
0.4 I=0.50 0.4 I=0
0.2 0.2
0.0 γt 0.0 γt
0 100 200 300 400 500 0 100 200 300 400 500
FIG. 6. Net gain in the entanglement recovery of two identical qubits FIG. 7. Probability of obtaining a non-zero outcome from the sLOCC
(fermions with l, l0 , r, r0 > 0, bosons with one of these four co- projection for fermions (with l, l0 , r, r0 > 0 and l = r0 ) interacting with
efficients negative) in the initial state |1− iAB under localized phase localized phase damping channels. Different degrees of spatial indis-
damping channels, thanks to the deformation+sLOCC operation per- tinguishability are reported in both the Markovian (λ = 5γ) (upper
formed at time t. Results are reported for different degrees of spatial panel) and non-Markovian (λ = 0.01γ) (lower panel) regimes.
indistinguishability I (with |l| = |r0 |). Both the Markovian (λ = 5γ)
(upper panel) and non-Markovian (λ = 0.01γ) (lower panel) regimes
are shown. Finally, the success (sLOCC) probability of obtaining the
outcome ρLR (t) for two identical qubits undergoing local
phase damping channels is
in the non-Markovian one, with a decay rate which decreases
as particle indistinguishability increases. Nonetheless, differ-
(lr0 )2 + (l0 r)2 − 2η ll0 rr0 1 − p(t)
ently from that case, the entanglement now does not vanish. PLR (t) = . (41)
Indeed, for t → ∞ it reaches a constant value which, under 1 − 12 p(t) C12 + 12 p(t)C22
the above assumptions, is given by
2ll0 rr0 Figure 7 depicts the behavior of the sLOCC probability of
C∞ = . (40) success (7) for fermions (with real and positive coefficients
(lr0 )2 + (l0 r)2 of the spatial wave functions) for different values of I. Once
Furthermore, when the indistinguishability is maximum (I = again, there is a trade-off between the probability of success
1) quantum correlations after the sLOCC measurement result and the concurrence, with PLR (t) = 1 when the particles are
to be completely immune to the action of the noisy environ- distinguishable and PLR = 1/2 for perfectly indistinguishable
ment and maintain their initial value. Is is important to high- qubits. A similar general behavior is found for bosons (with
light that the existence of such a steady value for the entan- the constraint l = r0 = l0 = −r), having PLR (t) = 1 − p(t)/2 in
glement of identical particles is only due to the spatial in- the case of maximal indistinguishability I = 1 .
distinguishability of the qubits and to the procedure used to
produce the entangled state, i.e. the sLOCC operation. This
result clearly shows that spatial indistinguishability of iden- C. Depolarizing Channel
tical qubits can be exploited to recover quantum correlations
spoiled by the detrimental noise of a phase damping-like en- In this section we reconsider and expand the results on en-
vironment interacting independently with the constituents, as tanglement protection at the preparation stage presented in
shown in Figure 6. Ref. [55]. A depolarizing channel acting on a system of two9
Markovian ity 1 − p(t) and of introducing a white noise which drives it
C(ρLR (t)) into the maximally mixed state with probability p(t). This is,
1.0
for instance, a typical noise occurring when quantum states
are initialized. Supposing once again that our system of two
0.8
I=1 identical particles is initially in the Bell state |1− iAB , it is well
I=0.90 known that this kind of noisy interaction produces the Werner
0.6
I=0.75 state [67]
I=0.50
0.4 I=0 1
ρAB (t) = WAB
−
(t) := 1 − p(t) |1− iLR h1− |LR + p(t) 11, (42)
4
0.2
where 11 is the 4 × 4 identity operator. Hereafter, we work for
0.0 γt convenience on the Bell states basis
0 2 4 6 8 10
BB = {|1+ iAB , |1− iAB , |2+ iAB , |2− iAB },
Non-Markovian
C(ρLR (t)) where |1+ iAB , |1− iAB have been previously defined respec-
1.0
tively in (18) and (33), while |2+ iAB and |2− iAB are given by
0.8 1
I=1
|2+ iAB = √ |A ↑, B ↑i + |A ↓, B ↓i ,
I=0.90 2
0.6
I=0.75
(43)
1
I=0.50 |2− iAB = √ |A ↑, B ↑i − |A ↓, B ↓i .
0.4 I=0
2
We recall that since such basis is orthonormal, the identity
0.2 operator can be written as
X
0.0 γt 11 = | j s iAB h j s |AB .
0 100 200 300 400 500 j=1,2
s=↑,↓
FIG. 8. Concurrence of two identical qubits (fermions with
l, l0 , r, r0 > 0, bosons with one of these four coefficients negative) At time t we deform the two qubits wave functions. The
in the initial state |1− iAB subjected to localized depolarizing chan- deformation of states |1+ iAB and |1− iAB has already been dis-
nels, undergoing an instantaneous deformation+sLOCC operation at cussed in (34) and (21), while states |2+ iAB and |2− iAB get
time t for different degrees of spatial indistinguishability I (with mapped respectively to
|l| = |r0 |). Both Markovian (λ = 5γ) (upper panel) and non-
Markovian (λ = 0.01γ) (lower panel) regimes are reported. |2̄+ iD = C2 |2̄+ iN , |2̄− iD = C2 |2̄− iN , (44)
where h2̄+ |2̄+ iN = h2̄− |2̄− iN = 1 and C2 is defined in (24). The
result of the deformation of state (42) is thus the deformed
Werner state of two indistinguishable qubits ρD (t) = W̄D− (t)
qubits has the effect of leaving it untouched with probabil- [55], where
3 " #
2 1 3
W̄D− (t) := 1 − p(t) C1 |1̄− iN h1̄− |N + C2 p(t) |1̄+ iN h1̄+ |N + |2̄+ iN h2̄+ |N + |2̄− iN h2̄− |N / 1 − η |hψ1 |ψ2 i| 1 − p(t) .
2 2
4 4 2
(45)
To perform the final sLOCC measurement we assume that the projection operator on the state (45) as defined in Eq. (6)
|ψ1 i , |ψ2 i have the usual structure given in Eq. (10). Applying we get
3 1
ρLR (t) = 1 − p(t) |lr0 − η l0 r|2 |1− iLR h1− |LR + p(t) |lr0 + η l0 r|2 |1+ iLR h1+ |LR + |2+ iLR h2+ |LR + |2− iLR h2− |LR
4 4
" # (46)
3 3
/ 1 − p(t) |lr − η l r| + p(t) |lr + η l0 r|2 .
0 0 2 0
4 4
Before computing the concurrence we notice that, as for the phase damping channel, the state of Eq. (46) is real and di-10
Markovian Markovian
ΔC(t) PLR (t)
1.0 1.0
0.8 0.8
I=1
I=1
I=0.90
0.6 I=0.90 0.6
I=0.75
I=0.75
I=0.50
0.4 I=0.50 0.4 I=0
0.2 0.2
0.0 γt 0.0 γt
0 2 4 6 8 10 0 2 4 6 8 10
Non-Markovian Non-Markovian
ΔC(t) PLR (t)
1.0 1.0
0.8 0.8
I=1
I=1
I=0.90
0.6 I=0.90 0.6
I=0.75
I=0.75
I=0.50
0.4 I=0.50 0.4 I=0
0.2 0.2
0.0 γt 0.0 γt
0 100 200 300 400 500 0 100 200 300 400 500
FIG. 9. Net gain in the entanglement recovery of two identical qubits FIG. 10. Probability of obtaining a nonzero outcome from the
(fermions with l, l0 , r, r0 > 0, bosons with one of these four coef- sLOCC projection for fermions with real and positive coefficients
ficients negative) in the initial state |1− iAB interacting with a depo- (l = r0 ) under a depolarizing channel. Different degrees of spatial in-
larizing channel, thanks to the deformation+sLOCC operation per- distinguishability I are reported in both Markovian (λ = 5γ) (upper
formed at time t. Results are reported for different degrees of spatial panel) and non-Markovian (λ = 0.01γ) (lower panel) regimes.
indistinguishability I (with |l| = |r0 |). Both the Markovian (λ = 5γ)
(upper panel) and non-Markovian (λ = 0.01γ) (lower panel) regimes
are shown.
all, we emphasize that, differently from the amplitude damp-
ing channel and the phase damping channel, a sudden death
agonal on the Bell states basis, thus being invariant under the phenomenon occurs when no deformation and sLOCC are
localized action of the Pauli matrices σyL ⊗ σRy . Therefore, the performed: indeed, when I = 0 the entanglement vanishes
concurrence is evaluated in terms of the four eigenvalues of at the finite time e t) = 2/3. However, when
t such that p(e
ρLR (t), namely 0 < I < 1, the state emerging from the sLOCC procedure re-
covers an amount of entanglement which decreases monoton-
1 − 43 p(t) |lr0 − η l0 r|2
λA (t) = , ically with t in the Markovian regime and periodically in the
1 − 43 p(t) |lr0 − η l0 r|2 + 43 p(t) |lr0 + η l0 r|2 non-Markovian regime. Nonetheless, as in the phase damping
case, such decrease approaches a constant value given by
1
p(t) |lr0 + η l0 r|2
λ j (t) = 4 ,
1− 3
4 p(t) |lr0 − η l0 r|2 + 43 p(t) |lr0 + η l0 r|2 (
(lr0 )2 + (l0 r)2 − 4 ll0 rr0
)
C∞ = max 0, − 0 2 . (48)
where the index j = B, C, D. Considering once again fermions 2 (lr ) + (l0 r)2 − ll0 rr0
with real and positive coefficients l, r, l0 , r0 , the concurrence
has the expression Furthermore, we notice once again that when the maximum
h i spatial indistinguishability (I = 1) is achieved, our procedure
1 − 32 p(t) (lr0 )2 + (l0 r)2 + 2ll0 rr0
allows for a complete entanglement recovery independently
C ρLR (t) = max .
0,
on t.
(lr ) + (l r) + 1 − 2 p(t) 2ll rr
3
0 2 0 2 0 0
(47) As a final quantity of interest we obtain the sLOCC prob-
Figure 8 shows the time behavior of entanglement quan- ability of success, defined in Eq. (7), for two identical qubits
tified by Eq. (47), while Figure 9 depicts ∆C(t). First of whose correlations have been spoiled by a local depolarizing11
channel, that is independently on how long the qubits have been interacting
with the detrimental environment.
(lr0 )2 + (l0 r)2 − 2η ll0 rr0 1 − 23 p(t) If the indistinguishability is not maximum, instead, our re-
PLR (t) = h i . (49) sults show that for an amplitude damping channel-like envi-
1 − η (ll0 )2 + (rr0 )2 + 2ll0 rr0 1 − 23 p(t) ronment the entanglement after the sLOCC drops to zero af-
ter a short interaction time; nonetheless, the interval of time
In Figure (10), PLR (t) is plotted in the case of two fermions where the amount of recovered entanglement is significant
(with real and positive coefficients and l = r0 ) for different increases with the indistinguishability in both the Markovian
degrees of spatial indistinguishability. Again, as expected, and the non-Markovian regimes. When the environment acts
a trade-off exists between the probability of success and the as a phase damping channel, instead, the recovered correla-
concurrence, with the higher probability achieved when the tions are always nonzero and our protocol provides an ex-
qubits are perfectly distinguishable. Nonetheless, as hap- ploitable resource independently on the interaction time (sta-
pens in the previous channels, such probability reaches a sta- tionary entanglement). This behavior also holds in the de-
tionary value which decreases as the indistinguishability in- polarizing channel scenario, where the deformation+sLOCC
creases, with PLR = 1/2 as the minimum value when I = 1. protocol achieves a special usefulness since it allows to re-
For bosons, a similar behavior is found (with the constraint cover quantum correlations destroyed at finite time by a sud-
l = r0 = l0 = −r), having PLR (t) = 1 − 3p(t)/4 when I = 1 den death phenomena.
[55]. We point out that the results reported in Figs. 2, 5, 8 show
a similar behavior to the ones discussed in Ref. [63] (for a
Markovian regime) where, in contrast to the present analysis,
IV. DISCUSSION the system-environment interaction occurs between the defor-
mation bringing the particles to spatially overlap and the final
In this paper we have shown that spatially localized opera- sLOCC measurements. Nonetheless, the decay rate is much
tions and classical communication (sLOCC) provide an opera- larger in the situation considered here: the sLOCC operational
tional framework to successfully recover the quantum correla- framework for entanglement recovery performs better when
tions between two identical qubits spoiled by the independent the environment is not able to distinguish the particle it is in-
interaction with two noisy environments. The performance of teracting with, as happens in Ref. [63]. Despite this, in a real
such procedure is found to be strictly dependent on the de- world application it is likely that the system-environment in-
gree of spatial indistinguishability reached by the spatial de- teraction will occur both before the (spatial) deformation and
formation of the particles wave functions. A general behav- between the deformation and the sLOCC. Therefore, an in-
ior has emerged: the higher is the degree of spatial indistin- teresting possible prospect of this work would be to investi-
guishability, the better is the efficacy of the protocol, quan- gate the general open quantum system framework provided
tified by the difference between the amount of entanglement in Ref. [63] when applied to noisy initial states such as those
present at time t with and without the application of our pro- given in Eqs. (20), (32), and (42).
cedure. In particular, when the two particles are brought to Our findings ultimately provide further insights about pro-
perfectly overlap and the maximum degree of indistinguisha- tection techniques of entangled states from the detrimental ef-
bility is achieved, the initial (maximum) amount of entangle- fects of surrounding environments by suitably manipulating
ment is completely recovered in all the considered scenarios, the inherent indistinguishability of identical particle systems.
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