Experimental quantum phase discrimination enhanced by controllable indistinguishability-based coherence
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Experimental quantum phase discrimination enhanced by controllable
indistinguishability-based coherence
Kai Sun,1, 2 Zheng-Hao Liu,1, 2 Yan Wang,1, 2 Ze-Yan Hao,1, 2 Xiao-Ye Xu,1, 2 Jin-Shi Xu,1, 2, ∗
Chuan-Feng Li,1, 2, † Guang-Can Guo,1, 2 Alessia Castellini,3 Ludovico Lami,4
Andreas Winter,5 Gerardo Adesso,6 Giuseppe Compagno,3 and Rosario Lo Franco7, ‡
1
CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei 230026, People’s Republic of China
2
CAS Centre For Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, People’s Republic of China
3
Dipartimento di Fisica e Chimica - Emilio Segrè,
Università di Palermo, via Archirafi 36, 90123 Palermo, Italy
4
Institut für Theoretische Physik und IQST, Universität Ulm,
arXiv:2103.14802v1 [quant-ph] 27 Mar 2021
Albert-Einstein-Allee 11, D-89069 Ulm, Germany
5
ICREA & Física Teórica: Informació i Fenómens Quàntics, Departament de Física,
Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
6
School of Mathematical Sciences and Centre for the Mathematics
and Theoretical Physics of Quantum Non-Equilibrium Systems,
University of Nottingham, University Park, Nottingham NG/ 2RD, United Kingdom
7
Dipartimento di Ingegneria, Università di Palermo,
Viale delle Scienze, Edificio 6, 90128 Palermo, Italy
(Dated: March 30, 2021)
Quantum coherence, a basic feature of quantum mechanics residing in superpositions of quantum
states, is a resource for quantum information processing. Coherence emerges in a fundamentally
different way for nonidentical and identical particles, in that for the latter a unique contribution
exists linked to indistinguishability which cannot occur for nonidentical particles. We experimentally
demonstrate by an optical setup this additional contribution to quantum coherence, showing that
its amount directly depends on the degree of indistinguishability and exploiting it to run a quantum
phase discrimination protocol. Furthermore, the designed setup allows for simulating Fermionic
particles with photons, thus assessing the role of particle statistics (Bosons or Fermions) in coherence
generation and utilization. Our experiment proves that independent indistinguishable particles can
supply a controllable resource of coherence for quantum metrology.
Introduction.—A quantum system can reside in coher- been found that the aptitude of spatial indistinguisha-
ent superpositions of states, giving rise to nonclassical- bility of identical particles can be exploited for entan-
ity [1, 2] which implies the intrinsic probabilistic nature glement generation [20], applicable even for spacelike-
of predictions in the quantum realm [3–7]. Besides this separated quanta [21] and against preparation and dy-
fundamental role, quantum coherence is also at the ba- namical noises [22–24]. The presence of entanglement
sis of quantum algorithms [8–13] and, from the modern is a signature that the bipartite system as a whole car-
information-theoretic perspective, constitutes a paradig- ries coherence even when the individual particles do
matic basis-dependent quantum resource [14–16], provid- not, the amount of this coherence being dependent on
ing a quantifiable advantage in certain quantum informa- the degree of indistinguishability. We name this spe-
tion protocols. cific contribution to quantumness of compound systems
as “indistinguishability-based coherence”, as a difference
For a single quantum particle, coherence emerges when with the more familiar “single-particle superposition-
the particle is found in a superposition of states in a based coherence”. Indistinguishability-based coherence
given basis of the Hilbert space. For multiparticle com- qualifies in principle as an exploitable resource for quan-
pound systems, the physics underlying the emergence tum metrology [17]. However, it requires sophisticated
of coherence is richer and strictly connected to the na- control techniques to be harnessed, especially in view of
ture of the particles, with fundamental differences for its nonlocal nature. Moreover, a crucial property of iden-
nonidentical and identical particles. In fact, states of tical particles is the exchange statistics, while operating
identical particle systems can manifest coherence even both Bosons and Fermions in the same setup is generally
when no particle resides in superposition states, pro- challenging.
vided that the wavefunctions of the particles overlap [17–
19]. In general, a special contribution to quantum co- In this work, we experimentally investigate the op-
herence arises thanks to the spatial indistinguishability erational contribution to quantum coherence stemming
of identical particles which cannot exist for nonidenti- from spatial indistinguishability of identical particles. By
cal (or distinguishable) particles [17]. Recently, it has virtue of our recently developed photonic architecture
2
nonlocally-encoded qubit state in the compound basis B
under spatially local operations and classical communi-
cation (sLOCC). Also, considering that this coherence
vanishes when the two particles are nonidentical thus in-
dividually addressable [17], the emergence of coherence in
|ΨLR i essentially hinges on the spatial indistinguishabil-
ity of the identical particles, in strict analogy to the emer-
gence of entanglement between pseudospins [20, 25, 29].
Fig. 1. Illustration of the indistinguishability-activated phase
The coherence of the state of Eq. (1) is independent
discrimination task. A resource state ρin that contains coher-
ence on a computational basis is distilled from spatial indis- of the Bosonic or Fermionic nature of the particles be-
tinguishability. The state then enters a black box which im- cause of the specific choice of the initial single-particle
plements a phase unitary Ûk = eiĜφk , k ∈ {1, . . . , n} on ρin . states. However, in general, particle statistics plays a
The goal is to determine the φk actually applied through the role in determining the allowed spatial overlap proper-
output state ρout : indistinguishability-based coherence pro- ties of identical particles and is thus crucial for the co-
vides operational advantage to the task. herence of the overall state of the system. Hence, we
shall extend our experimental investigation to a state
where these fundamental aspects can be observed. Tak-
capable of tuning the indistinguishability of two uncor- ing again a scenario with two indistinguishable particles,
related photons [25], we observe the direct connection one of the particles is now initialized with innate coher-
between degree of indistinguishability and amount of co- ence in the pseudospin basis, i.e., the initial two-particle
herence, and show that indistinguishability-based coher- state reads |Ψi = |ψ ↓, ψ 0 s0 i, where |s0 i = a |↑i + b |↓i
ence can be concurrent with single-particle superposition- with |a|2 + |b|2 = 1. Projecting onto B generates the
based coherence. In particular, we demonstrate that it three-level distributed state [17]
has operational implications, providing a quantifiable ad-
vantage in a phase discrimination task [26, 27], as de- 1
picted in Fig. 1. Furthermore, we design a setup capable |ΦLR i = Φ
(alr0 |L ↓, R ↑i + b(lr0 + ηl0 r) |L ↓, R ↓i
NLR
to test the impact of particle statistics in coherence pro-
+aηl0 r |L ↑, R ↓i), (2)
duction and phase discrimination for both Bosons and
Fermions. This is accomplished by compensating for Φ
p
where NLR = a2 (|lr0 |2 + |l0 r|2 ) + b2 |lr0 + ηl0 r|2 . In
the exchange phase during state preparation, simulating
this state, indistinguishability-based coherence coexists
Fermionic states with photons, which leads to statistics-
with single-particle superposition-based coherence, giv-
dependent efficiency of the quantum task.
ing rise to an overall multilevel coherence in the opera-
Indistinguishability-based coherence—To formally re-
tional basis B.
call the idea of coherence activated by spatial indis-
A photonic coherence synthesizer.—We demon-
tinguishability [17], we first consider the basic scenario
strate the preparation of two-level and three-level
where the wavefunctions of two identical particles with
indistinguishability-based coherence by means of the
orthogonal pseudospins, ↓ and ↑ overlap at two spatially-
photonic configuration shown in Fig. 2(a). The corre-
separated sites, L and R. Omitting the unphysical label-
spondence between photon’s polarization and pseudospin
ing of identical particles [28], the state is described as
reads |Hi ↔ |↑i, |V i ↔ |↓i, with |Hi and |V i identi-
|Ψi = |ψ ↓, ψ 0 ↑i, with |ψi = l |Li + r |Ri and |ψ 0 i =
fying horizontal and vertical polarization, respectively.
l0 |Li + r0 |Ri denoting the spatial wavefunctions corre-
Frequency-degenerate photon pairs are generated by
sponding to the two pseudospins. Let us use spatially lo-
pumping a beamlike type-II β-barium borate (BBO)
calized operations and classical communication, i.e., the
crystal via spontaneous parametric down-conversion [30],
sLOCC-framework [20], to activate and exploit the oper-
and sent to the main setup via two single-mode fibers,
ational coherence. Projecting onto the operational sub-
respectively. The two-photon initial state |Hi ⊗ |V i
space B = {|Lσ, Rτ i ; σ, τ =↓, ↑} yields the normalized
is uncorrelated, and two half-wave plates (HWPs, #1
conditional state [17]
and #2) with their orientation set at 22.5◦ and θ/2,
1 respectively, are utilized to adjust their polarization.
|ΨLR i = (lr0 |L ↓, R ↑i + ηl0 r |L ↑, R ↓i), (1)
Ψ
NLR The two-level state |ΨLR i is effectively prepared by the
p setup already employed to demonstrate polarization-
Ψ
with NLR = |lr0 |2 + |l0 r|2 , and the exchange phase fac- entanglement activation by spatial indistinguishability
tor η = 1(−1) originates from the Bosonic (Fermionic) [25]. Each of the two initially uncorrelated photons
nature of the indistinguishable particles. We see that, passes through a polarizing beam splitter (PBS), which
although each particle starts from an incoherent state distributes their spatial wavefunction between two re-
(namely, |ψ ↓i, |ψ 0 ↑i) in the pseudospin computational mote sites, L and R, according to the polarization state.
basis, the final state |ΨLR i overall resembles a coherent, Next, additional HWPs are added in different paths to
3
(a) (b)
L
PAD
PAD
R
PAD
PAD
PAD
BBO HWP QWP BS PBS BD
Fig. 2. Experimental configuration. (a) Preparation of coherent resource states by implementing sLOCC on indistinguishable
particles. Photon pairs with opposite polarizations are prepared by pumping a β-barium borate (BBO) crystal. The two photon
wavefunctions are distributed in two spatial regions, with the indistinguishability tuned by the half-wave plates (HWPs) #1
and #2. The elements in the dashed box are inserted only to prepare the three-level state. (b) Discrimination of different
phases. The Franson interferometer creates two phase channels with different configurations, which is adjusted by the HWP
sandwiched between two quarter-wave plates (QWPs). Inset: the polarization analysis device (PAD) comprising a QWP, a
HWP, a polarizing beam splitter (PBS) and a single-photon detector. BS beam splitter, BD beam displacer.
revert the photons’ polarization, and a beam displacer nience, we omit the site delimiters of the distributed
(BD) is inserted on each site to combine the propagating state and simply denote it using polarization. The system
directions of the two photons. At this point, the spatial is prepared in |ΨLR (θ)i = cos θ |HV i + sin θ |V Hi, and
wavefunctions of the two photons become overlapped, its measure of coherence in the basis B is Cl1 (ΨLR ) =
allowing for preparation of the state |ΨLR i via sLOCC. | sin 2θ |. The coherence completely stems from the in-
Explicitly, a pair of polarization analysis devices (PADs) distinguishability of the photons, as it vanishes at the
are inserted to cast polarization measurement, and the limit θ = kπ/2 (k integer number), i.e., when the
coincidence photon counting process realizes the desired two photons are distinguishable. To quantify the spa-
projection onto the distributed basis B (for more details, tial indistinguishability of the two photons we use the
P2 (i) (i)
see Ref. [25]). entropic measure [22] I = − i=1 pLR log pLR , where
(1) (2)
To prepare the three-level state |ΦLR i, an additional pLR = |lr0 /NLR | (pLR = |l0 r/NLR
Φ 2 Φ 2
| ) refers to the prob-
part of setup consisting of an HWP set at 22.5◦ and a BD ability of finding the photon from ψ and ψ 0 (ψ 0 and ψ)
is appended on each site, L and R, the orientations of the ending at L and R, respectively. For our setup, one has
HWPs #3 and #4 being also adjusted to prepare one of I = − cos2 θ log(cos2 θ) − sin2 θ log(sin2 θ). The exper-
the photons in the polarization-superposition state (see imental result for the measurement of coherence versus
dashed box in Fig. 2(a)). The coherence underpinning indistinguishability is plotted in Fig. 3(a), clearly reveal-
the system is finally activated and detected via sLOCC. ing the monotonic dependence in accord with theoretical
As a first observation, we want to prove the direct predictions. Here and after, the error bars represent the
quantitative connection between produced coherence and 1σ standard deviation of data points, which is deduced
spatial indistinguishability of photons, in analogy to what by assuming a Poisson distribution for counting statistics,
has been done for the entanglement [25]. In fact, in and resampling over recorded data. The inset shows the
the present experimental study, the resource of inter- result of quantum state tomography at θ = π/4, which
est is quantum coherence and such a preliminary anal- has a fidelity of 0.988 to the maximally coherent state.
ysis is essential in view of its controllable exploitation Phase discrimination.—Having generated tunable co-
for the specific quantum metrology protocol. This anal- herence using sLOCC, we apply it in the phase dis-
ysis is performed for the two-level state |ΨLR i resulting crimination task to demonstrate the operational advan-
from the original elementary state |Ψi. Various meth- tage due to indistinguishability and the role of parti-
ods have been proposed to quantify coherence [26, 31– cle statistics. The formal definition of phase discrim-
34]. Here, we adopt the P l1 norm of the density ma- ination task is as follows: a phase unitary among n
trix ρ, that is Cl1 (ρ) = i6=j |ρij | [31]. For conve- possible choices Uk = eiĜφk , k ∈ {1, . . . , n} is ran-
4
(a) (b) is placed in one of the arms to adjust the parameters
p1 and p2 . After the MZI, the photons are projected
on the desired state. Since |ΨLR i is a two-level coher-
ent state, the measurement projectors Π̂1 and Π̂2 de-
fined in the basis {|LV, RHi , |LH, RV i} are realized
by drawing the corresponding subspace from the prod-
uct (single-particle) state measurement. This procedure
is as follows. On the site L (R), the polarization projector
0
Fig. 3. Experimental result for the two-level state |ΨLR i. (a) is ÔL = |χi hχ| with |χi = α |Hi + β |V i (ÔR = |χ0 i hχ0 |
0 0 0
Quantification of coherence Cl1 versus the two-photon indis- with |χ i = α |Hi+β |V i); the product projector is thus
0
tinguishability I. The inset shows the real part of the density ÔL ⊗ÔR , leading to the two-photon projector |Ψαβ i hΨαβ |
matrix for the input state |ΨLR (π/4)i deduced by quantum with |Ψαβ i = αβ 0 |LH, RV i + βα0 |LV, RHi in the sub-
state tomography. (b) The error probability Perr of phase space of interest {|LV, RHi , |LH, RV i}. Thanks to the
discrimination versus the phase parameter φ, with θ = π/4
final PAD unit of the setup of Fig. 2(b), the parameters
to give maximal coherence and p1 = 0.44. The dashed line
shows the Helstrom-Holevo bound without coherence. {α, β, α0 , β 0 } can be adjusted to perform the desired
projective measurements Π̂1 , Π̂2 and eventually obtain
the error probability of discrimination Perr .
domly applied on an initial state ρin with a probabil- We directly measure the error probability of phase dis-
ity ofPpk , where the generator of the transformation crimination for various φ at p1 = 0.44 by employing
Ĝ = στ =↑,↓ ωστ |Lσ, Rτ i hL ↑, R ↓| is diagonal on the the maximally coherent state |ΨLR (π/4)i and optimiz-
computational
Pn basis (ωστ are arbitrary coefficients) and ing over the measurement settings of Π̂1 and Π̂2 . The
k=1 pk = 1. We shall identify the φk that is actually ap- experimental result, matching well with the theoretical
plied with maximal confidence from the output state ρout , prediction,
by casting on it positive operator-valued measurements
(POVMs). Here, we focus on the n = 2 scenario with 1 p
Perr = 1 − 1 − 2p1 (1 − p1 )(1 + cos φ) , (4)
φ1 = 0, φ2 = φ, and solving the task using the experi- 2
mentally feasible minimum-error discrimination [35, 36]. is shown in Fig. 3(b). Note that without coherence, the
We first investigate phase discrimination with the two- best strategy of phase discrimination is to constantly
level state and, without loss of generality, choose the gen- guess the phase with greater probability, yielding P̄err =
erator Ĝ = |L ↑, R ↓i hL ↑, R ↓| (obtained fixing ω↑↓ = 1 p1 (top dashed line). The reduced Perr thus unravels
and ω↑↑ = ω↓↑ = ω↓↓ = 0). Consequently, the output the almost ubiquitous advantage of indistinguishability-
states after being affected by Uk read based coherence.
1 Particle statistics matters.—The symmetric form of
Ψk = Ψ
(lr0 |L ↓, R ↑i + ηl0 rei(k−1)φ |L ↑, R ↓i), (3) Eq. (3) prevents the exchange phase factor η from af-
NLR
fecting the outcome of |ΨLR i-based phase discrimination
and they are discriminated by a POVM (a von Neumann task. However, when |ΦLR i is utilized in the same task,
projective measurement in this case) comprising two pro- the intrinsic statistics of the indistinguishable particles
jectors Π = {Π̂1 , Π̂2 }: when Π̂k clicks, the phase is iden- renders the situation more complicated. In our optical
tified as φk . By this definition, the chance of making setup, any state prepared necessarily has η = +1. For
an error is Perr = p1 hΨ1 |Π̂2 |Ψ1 i + p2 hΨ2 |Π̂1 |Ψ2 i, and is simplicity, we choose a = b and set l = l0 = r = r0
lower bounded by the Helstrom-Holevo bound[37, 38], (l0 = r = 0) to maximize (destroy) indistinguishability.
This is experimentally achieved by setting the orientation
q
2
namely, Perr > 12 1 − 1 − 4p1 p2 |hΨ1 |Ψ2 i| . For a
of HWPs #3 and #4 be 22.5◦ and θ = π/4(0). However,
two-level coherent state, it is straightforward to identify investigation of Fermionic systems with η = −1 is also
the measurement projectors Π̂1 and Π̂2 [17]. possible, which follows from the observation that η in
The phase discrimination game is experimentally real- Eq. 2 can be absorbed into l0 . By setting θ = −π/4,
ized using the setup of Fig. 2(b). The photons in the we invert the sign of l0 to simulate indistinguishability-
state |ΨLR i on the site R are sent into a unbalanced activated coherence of Fermionic particles.
Mach-Zehnder interferometer (UMZI), while the photons The state preparation in all above cases is character-
on the site L are directly detected. We put a HWP be- ized via quantum state tomography, and the results are
tween two QWPs fixed at 45◦ to build a phase gate, and presented in Fig. 4(a), the magnitude of the imaginary
situate one phase gate into each of the arms after a non- part of the density matrices are smaller than 0.07. For the
polarization beam splitter (BS). In fact, in the short arm Bosonic case, the outcome authenticates the presence of
of UMZI, the state |ΨLR i remains unchanged, while in coherence between all three vectors of the computational
the long arm, a relative phase φ between |LV, RHi and basis shown in Eq. (2). For the distinguishable case, the
|LH, RV i is imported. A movable shutter (not shown) coherence is in contrast solely inherited from one of the
5
(a) level system, to minimize the error probability of dis-
crimination Perr , three projectors Π̂1 , Π̂2 and Π̂3 are
required where Σ3i Π̂i = I and Tr[Π̂3 Φ1LR Φ1LR ] =
Tr[Π̂3 Φ2LR Φ2LR ] = 0. The projectors Π̂i (i =
1, 2, 3) consist of three linearly independent basis vectors
B 0 = {|L ↑, R ↓i , |L ↓, R ↑i , |L ↓, R ↓i}. Similarly to the
(b) method used above for the two-level state, these three
Bosonic projectors are also extracted from the subspace of the
product projectors on the two sites L and R and imple-
Distingui- mented by the PAD unit of the setup.
shable Fig. 4(b) reports the measured error probabilities for
phase discrimination with the three-level states, where
Fermionic we omit the experimental result for the Fermionic case,
because it is identical to the two-level case already given
in the earlier text (see Fig. 3(b)). A clear discrep-
ancy between the credibility of phase discrimination us-
Fig. 4. Experimental result for three-level state |ΦLR i with ing different kinds of particles can be observed. Partic-
p1 = 0.50. (a) The real part of the density matrix for the
ularly, both types of indistinguishable particles provide
input states |ΦLR i of Bosonic, distinguishable and Fermionic
particles (simulated), deduced by quantum state tomography, advantage over distinguishable ones within the range of
with θ = ±π/4 to give maximal coherence. (b) The error φ ∈ ( 2π 4π
3 , 3 ), but Fermions further outperform Bosons by
probability Perr of phase discrimination versus φ. a difference in Perr of 0.119 at φ = π. This can be intu-
itively interpreted by recalling that the exchange interac-
tion of Fermions prevent them from occupying the same
particles, and localized on the site R. For the Fermionic state, so the wavefunction amplitude disperses between
case, the destructive interference completely eliminates different states and produces large amount of coherence.
the amplitude on the symmetric basis |V V i ∼ |L ↓, R ↓i, In contrast, Bosons tend to bunch on a single state, so
and the resulted state interestingly resembles the two- the applicable coherence is reduced.
level state, |ΨLR (π/4)i, as is substantiated by the de- Discussion.—Coherence activated from spatial indis-
duced density matrices. In the experiment, we simulate tinguishability is a fundamental contribution to quan-
this case with the average fidelity of 95.9(2)% compared tumness of multiparticle composite systems intimately
with |ΨLR (±π/4)i. Notice that a minus sign appears in related to the presence of identical particles (subsys-
the coefficient of the |V Hi terms, which is attributed to tems). It cannot exist between different types of quanta,
the π-phase acquired by the photons upon reflected by that is, in systems made of nonidentical (or distinguish-
PBS. able) particles. Due to its intrinsic nonlocal trait, in order
We are now in the position to demonstrate the role of to exploit indistinguishability-based coherence for quan-
particle statistics in the phase discrimination task. The tum information tasks, transformations and measure-
corresponding operations Uk are again realized using the ments on the resource state must admit direct product
phase gates within the UMZI, yielding two output states decomposition into local operations, which are achieved
Φk [17] written as by sLOCC. We note that in the case of two identical
particles, Schmidt decomposition recovers our capabil-
Φk = (a(lr0 eiω↓↑ φk |L ↓, R ↑i + ηl0 reiω↑↓ φk |L ↑, R ↓i) ity to perform all possible measurements [41]. There-
+b(lr0 + ηl0 r)eiω↓↓ φk |L ↓, R ↓i)/NLR
Φ
. (5) fore, the application of indistinguishability-based coher-
ence between three or more quanta will be an open re-
Here, we set ω↓↑ = 1, ω↑↓ = 2 and ω↓↓ = 3 in the search route.
generator Ĝ. Differently from the two-level situation, In this paper, we have experimentally investigated
in this three-level coherent case, we need to place each indistinguishability-based coherence, demonstrating its
UMZI on each site L and R. The UMZI has a path dif- operational usefulness in a quantum metrology proto-
ference equivalent to 2.7ns between the long and short col. Our photonic architecture is capable of tuning
paths, and the coincidence interval is set at 0.8ns. The the degree of spatial indistinguishability of two uncor-
quantum states affected by the two phase operations in related photons, and adjusting the interplay between
the UMZIs are registered separately [39, 40]. We ad- indistinguishability-based coherence and single-particle
just the electronic delay of the coincidence module to superposition-based coherence. This has allowed us to
pick out the events that the two photons had taken prepare via sLOCC various types of resource states and
the long/short and short/long paths, which correspond exploit them in the phase discrimination task to charac-
to the state after being affected by U1 and U2 , respec- terize the operational coherence. Interestingly, our setup
tively. Moreover, for the measurement of the three- has been designed in such a way that both Bosonic and
6
Fermionic statistics can occur in the resource states, thus Buell, et al., Nature 574, 505 (2019).
enabling the possibility to directly observe how the na- [14] A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404
ture of the employed particles affects the efficiency of the (2016).
quantum task. Our results highlight, in a comprehensive [15] A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod.
fashion, the fundamental and practical aspects of con- Phys. 89, 041003 (2017).
[16] E. Chitambar and G. Gour, Rev. Mod. Phys. 91, 025001
trollable indistinguishability of identical building blocks (2019).
for quantum-enhanced technologies. [17] A. Castellini, R. Lo Franco, L. Lami, A. Winter,
This work was supported by National Key Re- G. Adesso, and G. Compagno, Phys. Rev. A 100, 012308
search and Development Program of China (Grants (2019).
Nos. 2016YFA0302700, 2017YFA0304100), the Na- [18] S. Chin and J. Huh, Phys. Rev. A 99, 052345 (2019).
tional Natural Science Foundation of China (Grants [19] J. Sperling, A. Perez-Leija, K. Busch, and I. A. Walmsley,
Nos. 61725504, U19A2075, 61805227, 61975195, Phys. Rev. A 96, 032334 (2017).
[20] R. Lo Franco and G. Compagno, Phys. Rev. Lett. 120,
11774335, and 11821404), Key Research Program 240403 (2018).
of Frontier Sciences, CAS (Grant No. QYZDY- [21] A. Castellini, B. Bellomo, G. Compagno, and R. L.
SSW-SLH003), Science Foundation of the CAS Franco, Physical Review A 99, 062322 (2019).
(Grant No. ZDRW-XH-2019-1), the Fundamental [22] F. Nosrati, A. Castellini, G. Compagno, and
Research Funds for the Central Universities (Grant R. Lo Franco, npj Quantum Inf. 6, 1 (2020).
No. WK2470000026, No. WK2030380017), Anhui Ini- [23] F. Nosrati, A. Castellini, G. Compagno, and
tiative in Quantum Information Technologies (Grants R. Lo Franco, Phys. Rev. A 102, 062429 (2020).
[24] A. Perez-Leija, D. Guzmán-Silva, R. d. J. León-Montiel,
No. AHY020100, and No. AHY060300). L.L. acknowl- M. Gräfe, M. Heinrich, H. Moya-Cessa, K. Busch, and
edges support from the Alexander von Humboldt A. Szameit, npj Quant. Inf. 4, 45 (2018).
Foundation. [25] K. Sun, Y. Wang, Z.-H. Liu, X.-Y. Xu, J.-S. Xu, C.-F.
K.S. and Z.-H.L. contributed equally to this work. Li, G.-C. Guo, A. Castellini, F. Nosrati, G. Compagno,
et al., Opt. Lett. 45, 6410 (2020).
[26] C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani,
N. Johnston, and G. Adesso, Phys. Rev. Lett. 116,
150502 (2016).
∗
jsxu@ustc.edu.cn [27] M. Ringbauer, T. R. Bromley, M. Cianciaruso, L. Lami,
†
cfli@ustc.edu.cn W. S. Lau, G. Adesso, A. G. White, A. Fedrizzi, and
‡
rosario.lofranco@unipa.it M. Piani, Phys. Rev. X 8, 041007 (2018).
[1] A. M. Gleason, J. Math. Mech. 6, 885 (1957). [28] R. Lo Franco and G. Compagno, Sci. Rep. 6, 20603
[2] S. Kochen and E. Specker, J. Math. Mech. 17, 59 (1967). (2016).
[3] E. Schrödinger, Naturwissenschaften 23, 823 (1935). [29] M. R. Barros, S. Chin, T. Pramanik, H.-T. Lim, Y.-W.
[4] E. P. Wigner, in Philosophical reflections and syntheses Cho, J. Huh, and Y.-S. Kim, Opt. Express 28, 38083
(Springer, 1995) pp. 247–260. (2020).
[5] D. Frauchiger and R. Renner, Nat. Commun. 9, 3711 [30] S. Takeuchi, Opt. Lett. 26, 843 (2001).
(2018). [31] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev.
[6] K.-W. Bong, A. Utreras-Alarcón, F. Ghafari, Y.-C. Lett. 113, 140401 (2014).
Liang, N. Tischler, E. G. Cavalcanti, G. J. Pryde, and [32] M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli,
H. M. Wiseman, Nat. Phys. 16, 1199 (2020). N. Johnston, and G. Adesso, Phys. Rev. A 93, 042107
[7] M. N. Bera, A. Acín, M. Kuś, M. W. Mitchell, and (2016).
M. Lewenstein, Rep. Prog. Phys. 80, 124001 (2017). [33] I. Marvian and R. W. Spekkens, Physical Review A 94,
[8] P. W. Shor, in Proceedings 35th annual symposium on 052324 (2016).
foundations of computer science (Ieee, 1994) pp. 124– [34] Y.-T. Wang, J.-S. Tang, Z.-Y. Wei, S. Yu, Z.-J. Ke, X.-
134. Y. Xu, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 118,
[9] E. Martin-Lopez, A. Laing, T. Lawson, R. Alvarez, X.-Q. 020403 (2017).
Zhou, and J. L. O’brien, Nat. Photon. 6, 773 (2012). [35] S. M. Barnett and S. Croke, Adv. Opt. Photonics 1, 238
[10] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Proc. (2009).
R. Soc. Lond. A 454, 339 (1998). [36] P. J. Mosley, S. Croke, I. A. Walmsley, and S. M. Barnett,
[11] L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, Phys. Rev. Lett. 97, 193601 (2006).
I. A. Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, [37] C. W. Helstrom, J. Stat. Phys. 1, 231 (1969).
Č. Brukner, and P. Walther, Nat. Commun. 6, 7913 [38] A. S. Holevo, Proc. Steklov Inst. Math. 124, 1 (1978).
(2015). [39] S. M. Barnett and E. Riis, J. Mod. Opt. 44, 1061 (1997).
[12] S. Aaronson and A. Arkhipov, in Proceedings of the forty- [40] M. Mohseni, A. M. Steinberg, and J. A. Bergou, Phys.
third annual ACM symposium on Theory of computing Rev. Lett. 93, 200403 (2004).
(2011) pp. 333–342. [41] S. Sciara, R. Lo Franco, and G. Compagno, Sci. Rep. 7,
[13] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, 44675 (2017).
R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A.
You can also read
