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ARTICLE OPEN
Robust entanglement preparation against noise by controlling
spatial indistinguishability
1,2 3,4 ✉
Farzam Nosrati , Alessia Castellini3, Giuseppe Compagno3 and Rosario Lo Franco
Initialization of composite quantum systems into highly entangled states is usually a must to enable their use for quantum
technologies. However, unavoidable noise in the preparation stage makes the system state mixed, hindering this goal. Here, we
address this problem in the context of identical particle systems within the operational framework of spatially localized operations
and classical communication (sLOCC). We define the entanglement of formation for an arbitrary state of two identical qubits. We
then introduce an entropic measure of spatial indistinguishability as an information resource. Thanks to these tools we find that
spatial indistinguishability, even partial, can be a property shielding nonlocal entanglement from preparation noise, independently
of the exact shape of spatial wave functions. These results prove quantum indistinguishability is an inherent control for noise-free
entanglement generation.
npj Quantum Information (2020)6:39 ; https://doi.org/10.1038/s41534-020-0271-7
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INTRODUCTION Here, we adopt the sLOCC framework to unveil further
The discovery and utilization of purely quantum resources is an fundamental traits of composite quantum systems. We first define
ongoing issue for basic research in quantum mechanics and the entanglement of formation for an arbitrary state of two
quantum information processing1,2. Processes of quantum metrol- indistinguishable qubits (bosons or fermions). We then introduce
ogy3, quantum key distribution4, teleportation5, or quantum the degree of indistinguishability as an entropic measure of
sensing6 essentially rely on the entanglement feature7,8. Unfortu- information, tunable by the shapes of spatial wave functions. We
nately, entanglement is fragile due to the inevitable interaction finally apply these tools to a situation of experimental interest,
between system and surrounding environment already in the that is noisy entangled state preparation. We find spatial
initial stage of pure state preparation, making the state mixed9,10. indistinguishability can act as a tailored property protecting
As a result, protecting entanglement from unavoidable noises entanglement generation against noise.
remains a main objective for quantum-enhanced technology11.
Many-body quantum networks usually employ identical quan-
tum subsystems (e.g., qubits) as building blocks12–19. Characteriz- RESULTS
ing peculiar features linked to particle indistinguishability in sLOCC-based entanglement of formation of an arbitrary state of
composite systems assumes importance from both the funda- two identical qubits
mental and technological points of view. Discriminating between We first focus on the quantification of entanglement for an
indistinguishable and distinguishable particles has always been a arbitrary state (pure or mixed) of identical particles. For identical
big challenge for which different theoretical20–23 and experimen- particles we in general mean identical constituents of a composite
tal24–28 techniques have been suggested. Recently, particle system. In quantum mechanics identical particles are not
identity and statistics have been shown to be a resource29–34 individually addressable, as are instead non-identical (distinguish-
and experiments which witness its presence have been per- able) particles, so that specific approaches are needed to treat
formed35. One aspect that remains unexplored is how the their collective properties39–46. Our goal is accomplished by
continuous control of the spatial configurations of one-particle straightforwardly redefining the entanglement of formation
wave functions, ruling the degree of indistinguishability of the known for distinguishable particles47 to the case of indistinguish-
particles, influences noisy entangled state preparation. Moreover, able particles, thanks to the sLOCC framework34.
a measure of the degree of indistinguishability lacks. The separability criterion in the standard theory of entangle-
Pursuing this study requires an entanglement quantifier for an ment for distinguishable particles8,47 maintains its validity also for
arbitrary (mixed) state of the system with tunable spatial a state of indistinguishable particles once it has been projected by
indistinguishability. It is desirable that this quantifier is defined sLOCC onto a subspace of two separated locations L and R. In
within a suitable operational framework reproducible in the fact, after the measurement, the particles are individually
laboratory. The natural approach to this aim is the recent addressable into these regions and the criteria known for
experimentally friendly framework based on spatially localized distinguishable particles can be adopted34,38.
operations and classical communication (sLOCC), which encom- Consider two identical qubits, with spatial wave functions ψ1 and
passes entanglement under generic spatial overlap configura- ψ2, for which one desires to characterize the entanglement between
tions34. This approach has been shown to also enable remote the pseudospins between the separated operational regions.
entanglement36,37 and quantum coherence38. States of the system can be expressed by the elementary-state
1
Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy. 2INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec J3X 1S2,
Canada. 3Dipartimento di Fisica e Chimica—Emilio Segrè, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy. 4Dipartimento di Ingegneria, Università di Palermo, Viale
delle Scienze, Edificio 6, 90128 Palermo, Italy. ✉email: rosario.lofranco@unipa.it
Published in partnership with The University of New South Wales
F. Nosrati et al.
2
basis fjψ1 σ1 ; ψ2 σ 2 i; σ1 ; σ 2 ¼"; #g, expressed in the no-label containing a set of commuting observables such as spatial wave
particle-based approach44,45 where fermions and bosons are function jψi i and an internal degree of freedom jσi i (e.g.,
treated on the same footing. The density matrix of an arbitrary pseudospin with basis {↑, ↓}). The two-particle state Ψð2Þ i is thus
state of the two identical qubits can be written as ð2Þ
Ψ i ¼ jχ ; χ i ¼ jψ1 σ 1 ; ψ2 σ 2 i: (6)
X σ0 ;σ0 1 2
ρ¼ pσ11 σ22 jψ1 σ 1 ; ψ2 σ 2 i ψ1 σ01 ; ψ2 σ 02 =N ; (1) In general, the degree of indistinguishability depends on both the
σ1 ;σ2 ;σ01 ;σ02 ¼#;"
quantum state and the measurement performed on the system.
where N is a normalization constant. Projecting ρ onto the This means a given set of operations allows one to distinguish the
(operational) subspace spanned by the computational basis BLR ¼ particles while another set of operations does not. Let us narrow
fjL "; R "i; jL "; R #i; jL #; R "i; jL #; R #ig by the projector the analysis down to spatial indistinguishability within the sLOCC
ð2Þ
X framework, linked to the incapability of distinguish which one of
ΠLR ¼ jLτ 1 ; Rτ 2 ihLτ 1 ; Rτ 2 j; (2) the two particles is found in each of the separated operational
τ 1 ;τ 2 ¼";# region. This framework thus leads to the concept of remote spatial
one gets the distributed resource state indistinguishability of identical particles. The suitable class of
measurements to this aim is represented by local counting of
ð2Þ ð2Þ ð2Þ
ρLR ¼ ΠLR ρΠLR =TrðΠLR ρÞ; (3) particles, leaving the pseudospins untouched. Inside this class, the
ð2Þ
ð2Þ
joint projective measurement ΠLR defined in Eq. (2) represents the
with probability PLR ¼ TrðΠLR ρÞ. We call PLR sLOCC probability detection of one particle in L and of one particle in R. We indicate
since it is related to the post-selection procedure to find one with PXψi ¼ jhXjψi ij2 (X = L, R and i = 1, 2) the probability of finding
particle in L and one particle in R. The state ρLR is then one particle in the region X (X ¼ L; R) coming from jψi i. We then
exploitable for quantum information tasks by addressing the define the joint probabilities of the two possible events when one
individual pseudospins in the separated regions L and R, which particle is detected in each region: (i) P 12 ¼ PLψ1 PRψ2 related to the
represent the nodes of a quantum network. The state ρLR can be in event of finding a particle in L emerging from jψ1 i and a particle in
fact remotely entangled in the pseudospins and constitute the R emerging from jψ2 i, (ii) P 21 ¼ PLψ2 PRψ1 related to the vice versa.
distributed resource state. The trace operation is clearly performed The amount of the no-which way information emerging from the
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ð2Þ
in the LR-subspace (see Supplementary Notes 1 and 3 for details). outcomes of the joint sLOCC measurement ΠLR is a measure ð2Þof
The state ρLR, containing one particle Ψ i.
P in L and one particle in the spatial indistinguishability of the particles
in the state
R, can be diagonalized as ρLR ¼ i pi ΨLR LR
We thus use Z ð2Þ :¼ TrðΠLR Ψð2Þ i Ψð2Þ Þ ¼ P 12 þ P 21 , that
ð2Þ
i i Ψi , where pi is the
weight of each pure state ΨLR i i which is in general non-separable.
encloses the essence of this lack of information, to introduce the
Entanglement P of formation of ρLR is as usual48 entropic measure of the degree of remote spatial indistinguish-
E f ðρLR Þ ¼ min i pi EðΨLR
i iÞ, where minimization occurs over all
ability
the decompositions of ρ LR and EðΨi Þ is the entanglement of the
LR P 12 P 12 P 21 P 21
LR I LR :¼ log 2 log 2 : (7)
pure state Ψi i. We thus define the operational entanglement Z Z Z Z
ELR(ρ) of the original state ρ obtained by sLOCC as the The entropic expression above naturally arises from the require-
entanglement of formation of ρLR, that is ment of quantifying the no which-way information associated to
the uncertainty about the origin (spatial wave function) of the
E LR ðρÞ :¼ E f ðρLR Þ: (4) particle found in each of the operational regions. If particles do not
We can conveniently quantify the entanglement of spatially overlap in both remote regions, we have maximum
formation Ef(ρLR) by the concurrencepC(ρ LR), ffi according to the
information (P 12 ¼ 1, P 21 ¼ 0 or vice versa) and I LR ¼ 0 (the
ffiffiffiffiffiffiffiffiffiffiffiffiffi particles can be distinguished by their spatial location). On the
well-known relation E f ¼ h½ð1 þ 1 C 2 Þ=247,49, where
hðxÞ ¼ x log 2 x ð1 xÞ log 2 ð1 xÞ. The concurrence CLR(ρ) other hand, I LR ¼ 1 when there is no information at all about each
in the sLOCC framework can be easily introduced by particle origin (P 12 ¼ P 21 ) and the particles are maximally over-
n pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffio lapping in both regions. Notice that a given value of I LR
C LR ðρÞ :¼ CðρLR Þ ¼ max 0; λ4 λ3 λ2 λ1 ; (5) corresponds to a class of different shapes of the single-particle
spatial wave functions jψi i. Moreover, in an experiment which
where λi are the eigenvalues, in decreasing order, of the non- reconstructs the identical particle state by standard quantum
Hermitian matrix R ¼ ρLR e ρLR ¼ σ Ly σ Ry ρLR σLy σ Ry with
ρLR , being e tomography, the corresponding value of I LR can be indirectly
localized Pauli matrices σ y ¼ jLihLj σy , σ Ry ¼ jRihRj σ y . The
L obtained50.
The above definition of the degree of spatial indistinguishability
entanglement quantifier of ρ so obtained contains all the
for two identical particles allows us to defining a more general
information about spatial indistinguishability and statistics
degree of indistinguishability for N identical particles. In general,
(bosons or fermions) of the particles.
N different operational regions Ri (i = 1, …, N) are needed to
quantify the indistinguishability of N identical particles (see Fig. 1).
sLOCC-based entropic measure of indistinguishability LetðNÞus consider a N-identical particle elementary pure state
In quantum mechanics, identical particles can be given the Ψ i ¼ jχ ; χ ; :::; χ i, where jχ i is the ith single-particle state.
1 2 N i
property of indistinguishability associated to a specific set of Each jχ i i is characterized by the set of values χ i ¼ χ ai ; χ bi ; ¼ ; χ ni
quantum measurements, being different from identity that is an corresponding to a complete set of commuting observables
intrinsic property of the system. With respect to the set of a ^ ¼;n
^; b; ^. For example, if a ^ describes the spatial distribution of
measurements, it seems natural to define a continuous degree of the single-particle states, χ ai is a spatial wavefunction ψi. To define
indistinguishability, which quantifies how much the measurement a suitable class of measurements, we take the N-particle state
process can distinguish the particles. In this section, we deal with
jαβiN :¼ jα1 β1 ; α2 β2 ; ¼ ; αN βN i; (8)
this aspect within sLOCC. For simplicity, the treatment is first
presented for a two-particle pure state and then generalized to N- where the ith single-particle state jαi βi i is characterized by a
particle pure states. It is worth to mention that the framework is subset a ^ ¼ ; ^j of the a
^; b; ^ ¼;n
^; b; ^ commuting observables with
universal and also valid for mixed states. eigenvalues αi ¼ αai ; αbi ; ¼ ; αji , and by the remaining observables
Let us consider an elementary pure state of two identical ^k; ¼ ; n
^ with eigenvalues βi ¼ βki ; ¼ ; βni . In the first member of
particles Ψð2Þ i ¼ jχ 1 ; χ 2 i, where jχ i i is a generic one-particle state Eq. (8) we have set α ≔ {α1, …, αN} and β ≔ {β1, …, βN}. The
npj Quantum Information (2020) 39 Published in partnership with The University of New South Wales
F. Nosrati et al.
3
Fig. 1 Projective measurements based on sLOCC. Illustration of different single-particle spatial wave functions ψi (i = 1, …, N) associated to N
identical particles in a generic spatial configuration. The amount of spatial indistinguishability of the particles can be defined by using
spatially localized single-particle measurements in N separated regions Ri .
N-particle projector on outcomes (α, β) of the complete set of In the usual formulation, it is defined as a mixture of a pure
ðNÞ
observables is Παβ ¼ jαβiN hαβj, while the projector on outcomes maximally entangled (Bell) state and of the maximally mixed state
α of the partial set of observables is (white noise). Its explicit expression, AB assuming to be interested pffiffiffi in
X ðNÞ generating the Bell state Ψ i ¼ ðj"A ; #B i ± j#A ; "B iÞ= 2, is
ΠαðNÞ ¼
±
Παβ : ¼ ð1 pÞΨAB AB
± i Ψ ± þ pI4 =4, where I4 is the 4 × 4 identity
±
(9) W AB
β matrix and p is the noise probability which accounts for the amount
Within the sLOCC framework, we can quantify to which extent of white noise in the system during the pure state preparation
particles in the state ΨðNÞ i can be distinguished by knowing the
±
stage. The Werner state W AB is also the product of a single-particle
ðNÞ depolarizing channel induced by the environment applied to an
results α of the local measurements described by Πα of Eq. (9),
considering that single-particle spatial wave functions {ψi} can initial Bell state7,8. It is known that the concurrence for such state is
overlap (see Fig. 1). The (sLOCC) measurements have to satisfy the CðW AB±
Þ ¼ 1 3p=2 when 0 ≤ p < 2/3, being zero otherwise8 (see
following properties: (1) the N single-particle states fjαi βi ig are black dot-dashed line of Fig. 3a).
peaked in separated spatial regions fRi g (see Fig. 1); (2) In perfect analogy, the Werner state for two identical qubits
ðNÞ
hΨðNÞ jΠα jΨðNÞ i ≠ 0, i.e., the probability of obtaining the projected with spatial wave functions ψ1, ψ2 can be defined by
state must be different from zero (see Methods and Supplemen- W ± ¼ ð1 pÞj1± ih1± j þ pI 4 =4; (11)
tary Note 1 for the general formulas).
P
We indicate with Pαi χ j the single-particle
probability that the where I 4 ¼ i¼1;2;s¼ ± ji s ihis j, having used the orthogonal Bell-
result αi comes from the state χ j i. We then define the joint state basis Bf1 ± ;2 ± g ¼ fj1þ i; j1 i; j2þ i; j2 ig with
probability PαP :¼ Pα1 χ p1 Pα2 χ p2 PαN χ pN , where P ¼ fp1 ; p2 ; :::; pN g pffiffiffi
j1 ± i :¼ ðjψ1 "; ψ2 #i ± jψ1 #; ψ2 "iÞ= 2;
is one of the N! permutations of the N single-particle states fjχ i ig.
Notice that PαP can be nonzero for each of the N! permutations, pffiffiffi
the outcome αi can P
since in general come from any of the single- j2 ± i :¼ ðjψ1 "; ψ2 "i ± jψ1 #; ψ2 #iÞ= 2: (12)
particle state χ j i. The quantity Z ¼ P PαP thus accounts for this
no which-way effect concerning the probabilities. The degree of The Werner state of Eq. (11) is justified as a model of noisy state. In
indistinguishability is finally given by fact, it is straightforward to see that W ± is produced by a localized
depolarizing channel acting on one of two initially separated
X PαP PαP identical qubits, followed by a quick single-particle spatial
I α :¼ ðNÞ
log 2 ðNÞ : (10)
P Z Z deformation procedure which makes the two identical qubits
spatially overlap (see Supplementary Note 2). Hence, in Eq. (11),
This quantity depends on measurements performed on the state. j1 ± i is the target pure state to be prepared and I 4 =4 is the noise
If the particles are initially all spatially separated, each in a as a mixture of the four Bell states.
different measurement region, only one permutation remains and Given the configuration of the spatial wave functions and using
α ¼ 0: we have complete knowledge on the single-particle state
I the sLOCC framework, the amount of operational entanglement
χ j i which gives the outcome αi, meaning that the particles are contained in W ± can be obtained by the concurrence C LR ðW ± Þ ¼
distinguishable with respect to the measurement Πα . On the
ðNÞ CðW LR ±
Þ of Eq. (5). Notice that the state of Eq. (11) is in general not
other hand, if for any possible permutation P 0 ≠ P one has normalized, depending on the specific spatial degrees of free-
P0αP ¼ PαP , indistinguishability is maximum and reaches the value dom45. This is irrelevant at this stage, since the entanglement of
W ± is calculated on the final distributed state W LR ±
, which is
I α ¼ log 2 N!. As a specific example, when χ ai ¼ ψi (spatial wave
obtained from W after sLOCC and is normalized (see Eq. (3).
±
functions) and χ bi ¼ σ i (pseudospins), I α of Eq. (10) is the direct Focusing on the observation of entanglement, a well-suited
generalization of I LR of Eq. (7) and provides the degree of spatial configuration for the spatial wave functions is jψ1 i ¼ l jLi þ r jRi
indistinguishability under sLOCC for N identical particles. and jψ2 i ¼ l0 jLi þ r 0 eiθ jRi, where l, r, l0 , r 0 are non-negative real
numbers (l2 þ r 2 ¼ l0 2 þ r 0 2 ¼ 1) and θ is a phase. The wave
Application: noisy preparation of pure entangled state functions are thus peaked in the two localized measurement
We now apply the tools above to a situation of experimental regions L and R, as depicted in Fig. 2. The degree of spatial
interest, namely noisy entanglement generation with identical indistinguishability I LR of Eq. (7) is tailored by adjusting the
particles. shapes of jψ1 i, jψ2 i, with PLψ1 ¼ l 2 , PLψ2 ¼ l0 2 (implying PRψ1 ¼ r 2 ,
Werner state51 W AB
±
for two nonidentical qubits A and B is PRψ2 ¼ r 0 2 ). The interplay between C LR ðW ± Þ and I LR vs. noise
considered as the paradigmatic example of realistic noisy prepara- probability p can be then investigated (see Supplementary Note 3
tion of a pure entangled state subject to the action of white noise. for some explicit expressions of C LR ðW ± Þ).
Published in partnership with The University of New South Wales npj Quantum Information (2020) 39F. Nosrati et al.
4
Generally, the entanglement amount is conditional since the bosons are, respectively
state is obtained by postselection. As a result, the entangled
ðfÞ ðbÞ 2l2 1 l 2 ð4 3pÞ
state ρLR is detectable if the sLOCC probability PLR is high enough PLR ¼ 2l2 ð1 l 2 Þ; PLR ¼ 2 : (14)
to be of experimental relevance. Let us see what happens for 2 1 2l2 ð2 3pÞ
I LR ¼ 1 (l ¼ l0 ). When the target pure state in Eq. (11) is j1 i, ðfÞ
using in Eq. (12) the explicit expressions of jψ1 i, jψ2 i with θ = 0 Notice that the sLOCC probability for fermions, PLR , is in this case
(θ = π) for fermions (bosons), from W we obtain by sLOCC the independent of the noise probability. Fixing l2 = 1/2 we maximize
distributed Bell state W ¼ 1LR i 1LR , with 1LR i ¼ ðjL "; R #i the sLOCC probability, which is 1/2 for fermions and 1/4 for
pffiffiffi
LR
bosons in the worst scenario of maximum noise probability p = 1.
jL #; R "iÞ= 2, therefore (see blue solid line of Fig. 3a) Differently, targeting the pure state j1þ i in Eq. (11), for fermions
(bosons) with θ = π (θ = 0), one gets a p-dependent W þ LR by sLOCC
C LR ðW Þ ¼ CðW
LR Þ ¼ 1; for any noise probability p (13)
with C LR ðW þ Þ ¼ CðW þ LR Þ ¼ ð4 5pÞ=ð4 pÞ when 0 ≤ p < 4/5,
for which the probabilities of detecting this state for fermions and being zero elsewhere. The entanglement now decreases with
increasing noise, remaining however larger than that for
nonidentical qubits (see red dashed line of Fig. 3a). The choice
of the state to generate makes a difference concerning noise
protection by indistinguishability. However, we remark that
identical qubits in the distributed resource state after sLOCC,
W LR
±
, are individually addressable. Local unitary operations
(rotations) in L and R can be applied to each qubit to8transform
the noise-free prepared 1LR i into any other Bell state . Another
relevant aspect is that the phase θ in jψ2 i acts as a switch between
fermionic and bosonic behavior of entanglement (see Supple-
mentary Note 3 for details on more general instances). The result
for nonidentical particles is retrieved when the qubits become
distinguishable (I LR ¼ 0, l ¼ r 0 ¼ 1 or l ¼ r 0 ¼ 0).
Since the preparation of j1 i, as represented by Eq. (11), results
to be noise-free for both fermions and bosons when I LR ¼ 1, it is
important to know what occurs for a realistic imperfect degree of
spatial indistinguishability. In Fig. 3b, we display entanglement as
a function of both p and I LR . The plot reveals that entanglement
preparation can be efficiently protected against noise also for
I LR < 1. A crucial information in this scenario is the minimum
degree of I LR that guarantees nonlocal entanglement in L and R,
Fig. 2 Noisy entanglement preparation with tailored spatial by violating a CHSH-Bell inequality8, whatever the noise prob-
indistinguishability. Illustration of two controllable spatially over- ability p. We remark that a Bell inequality violation based on
lapping wave functions ψ1, ψ2 peaked in the two localized regions of sLOCC provides a faithful test of local realism52. Using the
measurement L and R. The two identical qubits are prepared in an Horodecki criterion53, we find that the Bell inequality is violated
entangled state under noisy conditions, giving W ± . The degree of for any p whenever 0:76 < I LR 1, implying 0:56F. Nosrati et al.
5
from the case of distinguishable qubits where, as known8, W AB ±
angular momentum as spin-like degree of freedom68–70. Setups
violates Bell inequality only for small white noise probabilities using integrated quantum optics can also simulate fermionic
0 ≤ p < 0.292 (giving 0:68ðW AB ±
Þ 1). These results show robust statistics using photons71. Other suitable platforms for fermionic
quantum entanglement preparation against noise through spatial subsystems can be supplied either by superconducting quantum
indistinguishability, even partial. In fact, rather than addressing circuits with Ramsey interferometry72, or by quantum electronics
individual qubits, here one controls the shapes of their spatial with quantum point contacts as electronic beam splitters73–75. The
wave functions jψ1 i, jψ2 i. Significant changes in these shapes can results of this work are expected to stimulate further theoretical
occur that anyway maintain I LR of Eq. (7) beyond the threshold and experimental studies concerning the multiple facets of
(≈0.76) assuring noise-free generation of nonlocal entanglement. indistinguishability as a controllable fundamental quantum trait
Indistinguishability here emerges as a property of composite and its exploitation for quantum technologies.
quantum systems inherently robust to surrounding-induced
disorder, protecting exploitable quantum correlations.
METHODS
Amplitudes and probabilities in the no-label approach
DISCUSSION For calculating all the necessary probabilities and traces to obtain the
In this work, we have studied the effect of spatial indistinguish- results of the work, under different spatial configurations of the wave
ability on entanglement preparation under noise, within the functions, we need to compute scalar products (amplitudes) between
sLOCC framework. Firstly, thanks to the analogy with known states of N identical particles.
methods for distinguishable particles, the entanglement of The N-particle probability amplitude has been defined in the literature
formation, and the related concurrence, has been defined for an by means of the no-label particle-based approach, here adopted, to deal
with systems of identical particles45. Indicating with χi, χ 0i (i = 1, …, N)
arbitrary pure or mixed state of two identical qubits. Secondly, we single-particle states containing all the degrees of freedom of the particle,
have introduced the degree of spatial indistinguishability of the general expression of the N-particle probability amplitude is
identical particles by an entropic measure of information. This X
achievement entails a continuous quantitative identification of hχ 01 ; χ 02 ; ¼ ; χ 0n jχ 1 ; χ 2 ; ¼ ; χ n i :¼ ηP hχ 01 jχ P1 ihχ 02 jχ P2 i ¼ hχ 0n jχ Pn i; (15)
P
indistinguishability as an informational resource. Hence, one can
evaluate the amount of entanglement exploitable by sLOCC into where P = {P1, P2, …, Pn} in the sum runs over all the one-particle state
two separated operational sites under general conditions of permutations, η = ±1 for bosons and fermions, respectively, and ηP is 1 for
spatial indistinguishability and state mixedness. bosons and 1 (−1) for even (odd) permutations for fermions. Notice that
the explicit dependence on the particle statistics appears, as expected.
The Werner state W ± has been then chosen as a typical Along our manuscript, we especially need two-particle probabilities and
instance of noisy mixed state of two identical qubits, with tunable trace operations. For N = 2, the general expression above reduces to the
spatial overlap of their wave functions on the two remote following two-particle probability amplitude
operational regions. The tunable spatial overlap rules the
hχ 01 ; χ 02 jχ 1 ; χ 2 i ¼ hχ 01 jχ 1 ihχ 02 jχ 2 i þ ηhχ 01 jχ 2 ihχ 02 jχ 1 i: (16)
indistinguishability degree. We have found that, under conditions
of complete spatial indistinguishability, maximally entangled pure
states between internal (spin-like) degrees of freedom can be
prepared unaffected by noise. Even in the more realistic scenario DATA AVAILABILITY
of experimental errors in controlling particle spatial overlap, we The main results of this manuscript are analytical. All data generated or analyzed
have supplied a lower bound for the degree of spatial during this study are included in this article (and in its Supplementary
indistinguishability beyond which the generated entangled state information file).
violates the CHSH-Bell inequality independently of the amount of
noise. These findings are independent of particle statistics, holding Received: 5 August 2019; Accepted: 7 April 2020;
for both bosons and fermions. One reasonably may expect that
also coherence can be protected by spatial indistinguishability,
based on a previous work showing that the latter enables
quantum coherence38. This supports the observed effects in an
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ACKNOWLEDGEMENTS
42. Killoran, N., Cramer, M. & Plenio, M. B. Extracting entanglement from identical
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ticles and their entanglement. Philos. Trans. R. Soc. A 376, 20170317 (2018).
R.L.F. proposed the ideas, discussed the results, and contributed to the preparation of
46. Lourenç, A. C., Debarba, T. & Duzzioni, E. I. Entanglement of indistinguishable
the paper.
particles: a comparative study. Phys. Rev. A 99, 012341 (2019).
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48. Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state COMPETING INTERESTS
entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996). The authors declare no competing interests.
npj Quantum Information (2020) 39 Published in partnership with The University of New South WalesF. Nosrati et al.
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