Table Of ContentExperimental replication of single-qubit quantum phase gates
M. Miˇcuda, R. St´arek, I. Straka, M. Mikov´a, M. Sedl´ak, M. Jeˇzek, and J. Fiur´aˇsek
Department of Optics, Palack´y University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
Weexperimentallydemonstratetheunderlyingphysicalmechanismoftherecentlyproposedpro-
tocol for superreplication of quantum phase gates [W. Du¨r, P. Sekatski, and M. Skotiniotis, Phys.
Rev. Lett. 114, 120503 (2015)], which allows to produce up to N2 high-fidelity replicas from N
input copies in the limit of large N. Our implementation of 1 → 2 replication of the single-qubit
phase gates is based on linear optics and qubits encoded into states of single photons. We employ
the quantum Toffoli gate to imprint information about the structure of an input two-qubit state
ontoanauxiliaryqubit,applythereplicatedoperationtotheauxiliaryqubit,andthendisentangle
the auxiliary qubit from the other qubits by a suitable quantum measurement. We characterize
the replication protocol by full quantum process tomography and observe good agreement of the
experimental results with theory.
7
1
PACSnumbers: 03.67.-a,42.50.Ex,03.65.Ud,03.67.Hk
0
2
n I. INTRODUCTION the protocol experimentally feasible. Remarkably, this
a gate-replication scheme also simultaneously acts as a de-
J
vicewhichaddsacontroltoanarbitraryunknownsingle-
Laws of quantum mechanics impose fundamental lim-
5
qubitphasegateandconvertsittoatwo-qubitcontrolled
itsonourabilitytoreplicatequantuminformation[1,2].
1
phase gate [10–12].
The quantum no-cloning theorem for instance ensures
] thatquantumcorrelationscannotbeexploitedforsuper- Weexperimentallyimplementthegatereplicationpro-
h luminal signaling [2], and it guarantees the security of tocol using a linear optical setup, where qubits are en-
p
quantum key distribution [3]. These fundamental and coded into states of single photons. We perform full
-
t practicalimpactsoftheno-cloningtheoremstimulateda quantum process tomography of the replicated quantum
n
wide range of studies on approximate quantum cloning gatesandcharacterizetheperformanceofthereplication
a
u andnumerousoptimalquantumcloningmachinesforvar- protocol by quantum gate fidelity. The linear optical
q ious sets of input states were designed and experimen- quantum Toffoli gate which we utilize [13] is probabilis-
[ tally demonstrated [4]. Recently, the concept of quan- tic and operates in the coincidence basis, hence requires
tum cloning was extended from states to quantum op- postselection, similarly as other linear optical quantum
1
v erations [5, 6], which brings in new interesting aspects gates[14]. Althoughthedeterministicnatureofthegate
2 and features. In particular, it was shown very recently superreplicationprotocolisoneofitsimportantfeatures,
6 that starting from N copies of a single-qubit unitary op- our proof-of-principle probabilistic experiment still al-
0 eration U one can deterministically generate up to N2 lows usto successfully demonstratethe underlyingphys-
4 copies of this operation with an exponentially small er- ical mechanism of quantum gate replication.
0
ror [7, 8]. As pointed out in Ref. [9], the observed N2
.
1 limit of the number of achievable almost perfect clones
0 (a) (b)
is deeply connected to the Heisenberg limit on quantum
7
1 phase estimation. ψ ψ
v: derInlyitnhgispphaypseicrawl emeexchpaenriimsmentoaflltyhedesmupoenrsrterpaltiecatthioenuno-f | i V V† |0i U
Xi quantumgates,seeFig.1. Specifically,weconsider1 2 0 U⊗N | i
→ | i
replication of single-qubit unitary phase gates [7]
r
a
U(φ)= 0 0 +eiφ 1 1, φ [0,2π). (1) (c) (d)
| (cid:105)(cid:104) | | (cid:105)(cid:104) | ∈ ψ W
| i
As shown in Fig. 1(b), we impose suitable quantum cor-
relations between the two signal qubits and a third aux- 0 U U
| i
iliary qubit with the help of a three-qubit quantum Tof-
foli gate, we apply the operation U(φ) to the auxiliary
FIG. 1: (Color online) (a) Quantum circuit for superreplica-
qubit, and we finally erase the correlations between the
tionofsingle-qubitquantumphasegatesU(φ)[7]. (b)1→2
auxiliary qubit and the signal qubits. This latter part quantum-gate replication circuit involving two quantum Tof-
of the protocol can be in principle accomplished unitar- foli gates and one ancilla qubit. (c) The same circuit as in
ily by a second application of the Toffoli gate, see Fig. panel (b), but the second Toffoli gate is replaced with mea-
1(b), but we instead choose to erase the correlations by surement on ancilla qubit and feed-forward. (d) The circuits
means of a suitable measurement on the auxiliary qubit, in panels (b) and (c) convert a single-qubit phase gate U(φ)
see Fig. 1(c), which is much less demanding and makes into a two-qubit controlled-phase gate. For details, see text.
2
II. GATE SUPERREPLICATION PROTOCOL fidelity of the corresponding Choi-Jamiolkowski states,
F = Φ Φ 2, which can be expressed as
U1U2 |(cid:104) U1| U2(cid:105)|
repLleictautisonfirpstrobtoriceofllyforrevpiehwasethgeatgeesneUri(cφ)N[7→]. WMesiunptreor-- FlimU1iUt2o=f la|Trgre[UN2†U1th]|e2/fi2d2Meli.tyItofcatnrabnesfoshrmowatniotnha(t4)inwtihthe
duce the following notation for M-qubit computational theidealoperationU(φ)⊗M isexponentiallyclosetoone
basisstates, m = m m ...m ,wherem [0,2M−1] when M = N2−α, α > 0. Consequently, N copies of
1 2 M
and m 0,|1 (cid:105)rep|resent digits of(cid:105)binary rep∈resentation U(φ) suffice to faithfully implement up to N2 copies of
i
of m. It∈h{olds}that U(φ). An intuitive explanation for this result is that in
the asymptotic limit the function CM becomes Gaus-
U(φ)⊗M m =ei|m|φ m , (2) |m|
| (cid:105) | (cid:105) sian with a width that scales as √M. If M < N2,
where m =(cid:80)M m denotestheHammingweightofan then the Hamming weights of an exponentially large
| | i=1 i majority of the basis states are located in the interval
M-bit binary string m m ...m . All basis states m
with the same Hammi1ng2weightM m acquire the sa|me(cid:105) mmin ≤ |m| < mmax, and the superrepliction protocol
phase shift mφ and the numbe|r o|f such states CM induces correct relative phase shifts for all such states.
| | |m| Although experimental implementation of the super-
obeys binomial distribution, CM =(cid:0)M (cid:1). replication protocol for large M is beyond the reach of
|m| |m|
This degeneracy is exploited in the superreplication current technology, the main mechanism of the super-
scheme illustrated in Fig. 1(a) [7]. The whole protocol replication can be demonstrated already for N = 1 and
canbedividedintothreesteps. First,aunitaryoperation M = 2. In this case we have m = 1 and m = 2,
min max
V imprintsinformationabouttheHammingweightofthe anditfollowsfromEq.(3)thatV isthethree-qubitquan-
M-qubit signal state of system A onto an auxiliary N- tum Toffoli gate [11, 13, 17–21], which flips the state of
qubitsystemB, whichisinitiallypreparedinstate 0 . the auxiliary qubit if and only if both signal qubits are
B
| (cid:105)
In the joint computational basis m n we explicitly in state 1 , see Fig. 1(b). Assuming a pure input state
A B
| (cid:105) | (cid:105) | (cid:105)
have ψ =c 00 +c 01 +c 10 +c 11 ofthesignal
in A 00 01 10 11
| (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105)
qubits and recalling that the auxiliary qubit is initially
V m 0 = m 0 , m <m ,
| (cid:105)A| (cid:105)B | (cid:105)A| (cid:105)B | | min in state 0 , the three-qubit state after the application of
| (cid:105)
V m 0 = m k(m) , m m <m , the Toffoli gate reads
A B A B min max
| (cid:105) | (cid:105) | (cid:105) | (cid:105) ≤| |
V m 0 = m 2N −1 , m m .
A B A B max Ψ =c 00 0 +c 01 0 +c 10 0 +c 11 1 .
| (cid:105) | (cid:105) | (cid:105) | (cid:105) | |≥ AB 00 01 10 11
(3) | (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) | (cid:105)|(6(cid:105))
Here each k(m) is an N-qubit computational basis IfwenowapplytheunitaryoperationU(φ)totheauxil-
| (cid:105)
state chosen such that its Hamming weight satisfies iary qubit, and then disentangle this qubit from the rest
k = m mmin,andthelowerandupperthresholdson by another application of the Toffoli gate, we obtain a
| | | |−
theHammingweightarechosensymmetricallyaboutthe pure output state of the two signal qubits,
valueM/2forwhichCM ismaximized,m = M−N ,
|m| min (cid:100) 2 (cid:101)
mmax =(cid:100)M+2N(cid:101). |ψout(cid:105)A =c00|00(cid:105)+c01|01(cid:105)+c10|10(cid:105)+eiφc11|11(cid:105). (7)
In the second step of the protocol, the N replicated
gates U(φ)⊗N are applied to the N auxiliary qubits. Fi- The resulting two-qubit unitary operation thus reads
nally,inthethirdstep,thesystemAisdisentangledfrom
CU(φ)= 00 00 + 01 01 + 10 10 +eiφ 11 11. (8)
the auxiliary qubits by applying an inverse unitary op- | (cid:105)(cid:104) | | (cid:105)(cid:104) | | (cid:105)(cid:104) | | (cid:105)(cid:104) |
eration V−1. This sequence of operations results in the
The theoretical scheme in Fig. 1(b) can be further sim-
following unitary transformation on the M-qubit system
plifiedbyreplacingthesecondToffoligatewithmeasure-
A,
ment of the output auxiliary qubit in the superposition
m m m <m , basis = √1 (0 1 ) followed by a feed-forward, see
min |±(cid:105) 2 | (cid:105)±| (cid:105)
| (cid:105)→| (cid:105) | | Fig. 1(c). In particular, a two-qubit controlled-Z gate
m ei(|m|−mmin)φ m mmin m <mmax,
| (cid:105)→ | (cid:105) ≤| | should be applied to the signal qubits if the auxiliary
m eiNφ m m m .
max qubit is projected onto state while no action is re-
| (cid:105)→ | (cid:105) | |≥
(4) |−(cid:105)
quired when it is projected onto state + .
According to the Choi-Jamiolkowski isomorphism [15, | (cid:105)
Fidelity of transformation (8) with the two replicas of
16], an M-qubit unitary operation U can be represented
the phase gate U(φ) U(φ) reads
by a pure maximally entangled state of 2M qubits, ⊗
1
1 2(cid:88)M−1 FUU(φ)= 8(5+3cosφ). (9)
Φ =I U Φ = m U m , (5)
| U(cid:105) ⊗ | (cid:105) 2M/2 | (cid:105)⊗ | (cid:105)
m=0 Wenotethatthefidelitycanbemadephaseindependent
andequaltoameanfidelityF¯ = 5 forallφbytwirling
where Φ = 2−M/2(cid:80)2M−1 m m . Similarity of two [22], which would make the reUpUlicati8on procedure phase-
| (cid:105) m=0 | (cid:105)| (cid:105)
unitary operations U and U can be quantified by the covariant. To apply the twirling, one has to generate a
1 2
3
followed by a suitable feed-forward on the signal qubits,
ψ
| i similarly as in Fig. 1(c). However, the two CH gates are
unavoidable,whichmakesthecircuitinFig.2muchmore
0 H H U(φ) difficult to implement than the circuit in Fig. 1(c).
| i
FIG. 2: (Color online) Quantum circuit for optimal 1 → 2
replication of single-qubit unitary phase gates U(φ) [6]. III. EXPERIMENTAL SETUP
We experimentally implement the circuit in Fig. 1(c)
uniformly distributed random phase shift θ and apply a which demonstrates the underlying physical mechanism
transformation U(θ) to the ancilla qubit, in addition to of superreplication of quantum phase gates. The experi-
the replicated operation U(φ). Subsequently, an inverse mentalsetupisshowninFig. 3anditscoreisformedby
unitary operation U†(θ) U†(θ) is applied to the two ourrecentlydemonstratedlinearopticalquantumToffoli
⊗
output signal qubits. Since U(φ)U(θ) = U(φ + θ), it gate [13]. We utilize time-correlated photon pairs gen-
holds that for a fixed θ the fidelity of replication of U(φ) erated in the process of spontaneous parametric down-
isgivenbyEq. (9),whereφisreplacedwithφ+θ. After conversioninanonlinearcrystalpumpedbyalaserdiode.
averaging over the random uniformly distributed θ we Thetwosignalqubitsareencodedintothespatialandpo-
find that the fidelity becomes equal to the mean fidelity larizationdegreesoffreedomofthesignalphoton[24–27],
for any φ. respectively, while the auxiliary qubit is represented by
The mean fidelity F¯UU exceeds the fidelity 21 which is polarizationstateoftheidlerphoton[13,27]. Thespatial
achievable by applying the transformation U(φ) to one qubitissupportedbyaninherentlystableMach-Zehnder
of the qubits while no operation is applied to the second interferometerformedbytwocalcitebeam-displacersBD
qubit. Mean fidelity of 5/8 can be also achieved by a whichintroduceatransversalspatialoffsetbetweenverti-
measure-and-prepare scheme, where U(φ) is applied to a callyandhorizontallypolarizedbeam. Arbitraryproduct
fixed probe state + , optimal phase estimation is per- input state of the three qubits can be prepared with the
| (cid:105)
formed on the output U(φ)+ , and the estimated phase use of quarter- and half-wave plates (QWP and HWP),
| (cid:105)
φE determines the operation U(φE) U(φE) applied to and the output states can be measured in an arbitrary
⊗
thetwosignalqubits. However,thislatterprocedurecon- product basis with the help of a combination of wave-
ceptually differs from the generic superreplication proto- plates, polarizing beam splitters and single-photon de-
col and adds noise to the output state. tectors. The quantum Toffoli gate is implemented by a
Remarkably, Eq. (8) shows that the quantum circuit two-photon interference on a partially polarizing beam
depictedinFig. 1(b)canbealsointerpretedasascheme splitter PPBS that fully transmits horizontally polar-
1
for deterministic conversion of an arbitrary single-qubit ized photons and partially reflects vertically polarized
unitary phase gate U(φ) into a two-qubit controlled uni- photons [13, 28–31]. Similarly to other linear optical
tary gate CU(φ) [10–12]. While adding control to an
arbitrary unknown single-qubit operation U is known to
be impossible [23], this task becomes feasible provided
that one of the eigenvalues and eigenstates of the uni-
taryoperationisknown[10],whichispreciselythecaseof
thesingle-qubitphasegates(1)consideredinthepresent
work.
An appealing feature of the superreplication protocol
is that it is conceptually simple and universally applica-
ble to any N and M. Nevertheless, for specific N and
M onecanfurtheroptimizethisschemetomaximizethe
replication fidelity. In particular, the optimal scheme for
1 2 cloning of single-qubit phase gates was recently
→
derived by Bisio et al. [6] and is depicted in Fig. 2.
Average fidelity of this optimal cloning procedure reads
F = (3+2√2)/8 0.729. It is instructive to compare
≈
this scheme with the circuit in Fig. 1(b). Both circuits
FIG.3: (Coloronline)Experimentalsetup. HWP-half-wave
contain a Toffoli gate followed by application of U(φ) to
plate, QWP - quarter-wave plate, PPBS - partially polariz-
ancilla qubit at their cores, but the optimal cloning also
ing beam splitter with reflectances R = 2/3 and R = 0
V H
requires two additional controlled-Hadamard (CH) gates for vertical and horizontal polarizations, respectively, PBS -
on the input qubits and two controlled-NOT gates on polarizing beam splitter, BD - calcite beam displacer, APD
the output qubits instead of a single Toffoli gate. The - single-photon detector. Inset (a) shows the implemented
latter two CNOT gates can be replaced by a measure- quantum circuit and inset (b) depicts the single-photon de-
ment of the ancilla qubit in the superposition basis tection block DB.
|±(cid:105)
4
FIG. 4: (Color online) Real and imaginary parts of the experimentally determined quantum process matrices χ representing
the two-qubit operations R implemented on the two signal qubits are plotted for four values of the phase shift: φ = 0 (a),
φ=π/2 (b), φ=π (c), and φ=3π/2 (d). All process matrices are normalized such that Tr(χ)=1.
quantum gates [14], this Toffoli gate operates in the co- ing quantum process matrix χ = (Φ Φ), where
incidence basis [32] and its success is indicated by simul- Φ = 1(cid:80)1 jk jk denotes aIm⊗axRim|al(cid:105)ly(cid:104) e|ntangled
taneousdetectionofasinglephotonateachoutputport. s|ta(cid:105)te of2foujr,kq=u0b|its(cid:105),|an(cid:105)d represents a two-qubit iden-
More details about the experimental setup can be found tity operation, c.f. also EIq. (5). For each φ, the pro-
in Refs. [13, 33]. cess matrix χ was reconstructed from the experimen-
The phase gate U(φ) on the polarization state of the tal data using a Maximum Likelihood estimation [38].
idler photon is implemented using a sequence of suitably The experimentally determined matrices χ are plotted
rotated quarter- and half-wave plates, see Fig. 3. At the in Fig. 4 for four values of the phase shift φ: 0, π, π,
2
output, the polarization of the idler photon is measured and 3π. AdditionalresultsareprovidedintheAppendix
2
in the superposition basis , and we accept only those which for comparison also contains plots of the corre-
|±(cid:105)
events where it is projected onto state + . This ensures sponding theoretical matrices χ of the ideal two-qubit
| (cid:105) th
that we do not have to apply any feed-forward operation unitary operations (8). We quantify the performance
onto the signal qubits. While such conditioning reduces of our experimental scheme by the quantum process fi-
the success rate of the protocol by a factor of 2, it does delity F of each implemented operation with the cor-
CU
not represent a fundamental modification, because the responding unitary operation CU(φ) that would be im-
linear optical Toffoli gate utilized in our experiment is plemented by our setup under ideal experimental condi-
probabilistic in any case. Since both signal qubits are tions. Recallthatfidelityofageneralcompletelypositive
encoded into a state of a single photon, a real-time elec- map χ with a unitary operation U [39, 40] is defined as
trooptical feed-forward scheme [34–37] could be in prin- F = Φ χΦ /Tr[χ]. The fidelity F (φ) calculated
U U CU
ciple exploited to apply the controlled-Z gate to signal from t(cid:104)he |ex|peri(cid:105)mentally reconstructed process matrices
qubits if the auxiliary idler qubit is projected onto state χ is plotted in Fig. 5(a). We find that F is in the
CU
. range of 0.829 F 0.897 and the mean fidelity
|−(cid:105) reads F¯ = 0.8≤72 wCUhic≤h indicates high-quality perfor-
CU
mance of our setup for all φ.
IV. EXPERIMENTAL RESULTS The experimentally determined fidelities F are con-
CU
sistent with the fidelity of the Toffoli gate, F = 0.894,
T
We have performed a full tomographic characteriza- whichwasdeterminedinourearlierwork[33]. Themain
tion of the experimentally implemented quantum gate experimental limitations that reduce the fidelity of the
replication protocol. The measurements were performed protocolincludeimperfectionsofthecentralpartiallypo-
for eight different values of the phase shift, φ = kπ, larizingbeamsplitterPPBS ,whichexhibitsR =0.660
8 1 V
k = 0,1, ,7, and for each φ we have reconstructed and R = 0.017 instead of R = 2/3 and R = 0,
H V H
···
the resulting quantum operation on the two signal limited visibility of two-photon interference = 0.958,
R V
qubits. We make use of the Choi-Jamiolkowski iso- and residual phase fluctuations in the Mach-Zehnder in-
morphism [15, 16] and represent by its correspond- terferometer formed by the two calcite beam displacers
R
5
errors. The maximum statistical uncertainty of fidelity
that we obtain reads 0.0013 (one standard deviation).
V. SUMMARY
In summary, we have successfully experimentally
demonstrated the underlying physical mechanism of su-
perreplication of quantum phase gates. Specifically, we
have imprinted information about the structure of the
input state of signal qubits onto an auxiliary qubit via a
suitablecontrolledunitaryoperation(herequantumTof-
foli gate), applied the operation which should be repli-
cated to the auxiliary qubit, and then disentangled the
auxiliaryqubitfromthesignalqubitsbyasuitablequan-
tum measurement. Intriguingly, this procedure converts
the replicated single-qubit phase gate to a two-qubit
controlled-phasegatesoitaddsacontroltoanarbitrary
FIG. 5: (Color online) Experimentally determined quantum
unknown phase gate U(φ).
gate fidelities F (φ) (a) and F (φ) (b) are plotted for the
CU UU
eight values of φ that were probed experimentally. The solid Our work paves the way towards experimental imple-
line in panel (b) represents the ideal theoretical dependence mentations of even more advanced schemes for quantum
(9) and the dashed line is the least-square fit of the form gate replication. The demonstrated approach based on
A+Bcosφ, with A=0.543 and B =0.322. Statistical error the Toffoli gate can be extended to arbitrary number of
bars are smaller than the markers. signal qubits M and copies of unitary phase operation
N. For any M and N, the transformation (3) could be
implementedbyasequenceofseveral(M+N)-qubitgen-
[33]. Additional factors that may influence the experi-
eralizedToffoligates,whereeachgeneralizedTofolligate
mentincludeimperfectionsofthewaveplatesandpolar-
would induce nontrivial bit flips on the subspace of aux-
izing beam splitters that serve for state preparation and
iliary qubits for one particular basis state of the signal
analysis. A more detailed discussion of these effects and
qubits and would behave as an identity operation for all
theirinfluenceonperformanceoftheexperimentalsetup
other basis states of the signal qubits. However, for the
can be found in Ref. [33].
relevant scenario N √M the number of gates would
We now turn our attention to characterization of gate ≥
growexponentiallywiththenumberofqubits. Therefore,
replication. For this purpose, we calculate the fidelity
itislikelythattheconstituentgeneralizedquantumTof-
F (φ)oftheimplementedoperationswithU(φ) U(φ).
UU ⊗ foli gates would have to exhibit fidelities exponentially
The results are plotted in Fig. 5(b) and we can see that
closeto1toensurethedesiredhigh-fidelityperformance
F exhibits the expected periodic dependence on φ as
UU of the superreplication protocol.
predicted by the theoretical formula (9). Due to various
imperfections,theobservedfidelitiesaresmallerthanthe
theoretical prediction, and the dashed line in Fig. 5(b)
Acknowledgments
shows the least-square fit of the form A+Bcosφ to the
data, which well describes the observed dependence of
F (φ) on φ. The mean fidelity reads F¯ = 0.543, This work was supported by the Czech Science Foun-
UU UU
whichexceedsthefidelity0.5achievablebyanaiveproto- dation (GA13-20319S).
colwhereU(φ)isappliedtoonequbitwhilenooperation
is performed on the other qubit.
Statisticaluncertaintiesoffidelitieswereestimatedas- Appendix: Plots of all reconstructed quantum
sumingPoissonianstatisticsofthemeasuredtwo-photon process matrices
coincidence counts. Since the fidelities are estimated in-
directly from the reconstructed quantum process matri- This Appendix contains two figures, Fig. 6 and Fig. 7,
ces χ, we have performed repeated Monte Carlo simula- thatshowalleightreconstructedquantumprocessmatri-
tions of the experiment, and for each run of the simula- cesχwhichcharacterizethequantumgatesimplemented
tion we have determined the quantum process matrix χ on the two signal qubits for φ = kπ, k = 0,...,7. For
8
and the fidelities F and F . This yielded an ensem- comparison, quantum process matrices χ of the corre-
UU CU th
bleoffidelitieswhichwasusedtocalculatethestatistical sponding ideal unitary gates CU(φ) are also shown.
6
(a)
(b)
(c)
(d)
FIG. 6: (Color online) Real and imaginary parts of the experimentally determined quantum process matrix χ representing
the quantum operation implemented on the two signal qubits (first and second column), and real and imaginary parts of the
quantum process matrix χ of the corresponding ideal unitary operation CU(φ) (third and fourth column), are plotted for
th
fourvaluesofthephaseshift: φ=0(a),φ=π/4(b),φ=π/2(c),andφ=3π/4(d). Allprocessmatricesarenormalizedsuch
that Tr(χ)=1.
[1] W.-K.WoottersandW.-H.Zurek,Nature(London)299, [5] G. Chiribella, G.M. D’Ariano, and P. Perinotti, Phys.
802 (1982). Rev. Lett. 101, 180504 (2008).
[2] D. Dieks, Phys. Lett. A 92, 271 (1982). [6] A. Bisio, G.M. D’Ariano, P. Perinotti, and M. Sedla´k,
[3] V. Scarani, H. Bechmann-Pasquinucci, N.J. Cerf, M. Phys. Lett. A 378, 1797 (2014).
Duˇsek, N. Lu¨tkenhaus, and M. Peev, Rev. Mod. Phys. [7] W.Du¨r,P.Sekatski,andM.Skotiniotis,Phys.Rev.Lett.
81, 1301 (2009). 114, 120503 (2015).
[4] V.Scarani,S.Iblisdir,N.Gisin,andA.Ac´ın,Rev.Mod. [8] G. Chiribella, Y. Yang, and C. Huang, Phys. Rev. Lett.
Phys. 77, 1225 (2005). 114, 120504 (2015).
7
(a)
(b)
(c)
(d)
FIG. 7: (Color online) The same as Fig. 6, φ=π (a), φ=5π/4 (b), φ=3π/2 (c), and φ=7π/4 (d).
[9] G.Chiribella,Y.Yang,andA.C.-C.Yao,Nat.Commun. [16] M.-D. Choi, Linear Algebra Appl. 10, 285 (1975).
4, 2915 (2013). [17] M. A. Nielsen and I. L. Chuang, Quantum Computation
[10] A. Y. Kitaev, quant-ph/9511026. andQuantumInformation (CambridgeUniversityPress,
[11] B.P.Lanyon,M.Barbieri,M.P.Almeida,T.Jennewein, Cambridge, UK, 2000).
T.C.Ralph,K.J.Resch,G.J.Pryde,J.L.O’Brien,A. [18] D.G.Cory,M.D.Price,W.Maas,E.Knill,R.Laflamme,
Gilchrist, and A. G. White, Nature Phys. 5, 134 (2009). W.H.Zurek,T.F.Havel,andS.S.Somaroo,Phys.Rev.
[12] X.-Q. Zhou, T.-C. Ralph, P. Kalasuwan, M. Zhang, A. Lett. 81, 2152 (1998).
Peruzzo, B.P. Lanyon, and J.L. O’Brien, Nat. Commun. [19] T. Monz, K. Kim, W. Ha¨nsel, M. Riebe, A. S. Villar, P.
2, 413 (2011). Schindler,M.Chwalla,M.Hennrich,andR.Blatt,Phys.
[13] M. Miˇcuda, M. Sedla´k, I. Straka, M. Mikova´, M. Duˇsek, Rev. Lett. 102, 040501 (2009).
M.Jeˇzek,andJ.Fiura´ˇsek,Phys.Rev.Lett.111,160407 [20] A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and
(2013). A. Wallraff, Nature 481, 170 (2012).
[14] P.Kok,W.J.Munro,K.Nemoto,T.C.Ralph,J.P.Dowl- [21] M.D.Reed,L.DiCarlo,S.E.Nigg,L.Sun,L.Frunzio,S.
ing, and G.J. Milburn, Rev. Mod. Phys. 79, 135 (2007). M.Girvin,andR.J.Schoelkopf,Nature482,382(2012).
[15] A. Jamiol(cid:32)kowski, Rep. Math. Phys. 3, 275 (1972). [22] C.H.Bennett,G.Brassard,S.Popescu,B.Schumacher,
8
J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, [32] T.C.Ralph,N.K.Langford,T.B.Bell,andA.G.White,
722 (1996). Phys. Rev. A 65, 062324 (2002).
[23] M. Arau´jo, A. Feix, F. Costa, and Cˇ. Brukner, New J. [33] M. Miˇcuda, M. Mikova´, I. Straka, M. Sedla´k, M. Duˇsek,
Phys. 16, 093026 (2014). M. Jeˇzek, and J. Fiura´ˇsek, Phys. Rev. A 92, 032312
[24] M. Fiorentino and F.N.C. Wong, Phys. Rev. Lett. 93, (2015).
070502 (2004). [34] T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys.
[25] W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gu¨hne, A. Rev. A 66, 052305 (2002).
Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. [35] S.Giacomini,F.Sciarrino,E.Lombardi,andF.DeMar-
Pan, Nature Phys. 6, 331 (2010) tini Phys. Rev. A 66, 030302(R) (2002).
[26] C. Vitelli, N. Spagnolo, L. Aparo, F. Sciarrino, E. San- [36] R. Prevedel, P. Walther, F. Tiefenbacher, P. Bhi, R.
tamato, and L. Marrucci, Nature Phot. 7, 521 (2013). Kaltenbaek,T.Jennewein,andA.Zeilinger,Nature445,
[27] L.A. Rozema, D.H. Mahler, A. Hayat, P.S. Turner, and 65-69 (2007).
A.M. Steinberg, Phys. Rev. Lett. 113, 160504 (2014). [37] M. Mikova´, H. Fikerova´, I. Straka, M. Miˇcuda, J.
[28] R.Okamoto,H.F.Hofmann,S.Takeuchi,andK.Sasaki, Fiura´ˇsek, M. Jeˇzek, and M. Duˇsek, Phys. Rev. A 85,
Phys. Rev. Lett. 95, 210506 (2005). 012305 (2012).
[29] N.K.Langford,T.J.Weinhold,R.Prevedel,K.J.Resch, [38] Z. Hradil, J. Rˇeha´ˇcek, J. Fiura´ˇsek, and M. Jeˇzek,
A.Gilchrist,J.L.O’Brien,G.J.Pryde,andA.G.White, Maximum-Likelihood Methods in Quantum Mechanics,
Phys. Rev. Lett. 95, 210504 (2005). Lect. Notes Phys. 649, 59 (2004).
[30] N.Kiesel,C.Schmid, U.Weber, R.Ursin,andH.Wein- [39] B.W. Schumacher, Phys. Rev. A 54, 2614 (1996).
furter, Phys. Rev. Lett. 95, 210505 (2005). [40] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.
[31] K.Lemr,A.Cˇernoch,J.Soubusta,K.Kieling,J.Eisert, Rev. A 60, 1888 (1999).
and M. Duˇsek, Phys. Rev. Lett. 106, 013602 (2011).
