Data-pattern tomography of entangled states

9.4 MB·

by  Vadim Reut

Data-pattern tomography of entangled states
Vadim Reut,1,2 Alexander Mikhalychev,1 and Dmitri Mogilevtsev1
1 B. I. Stepanov Institute of Physics, National Academy of Science of Belarus, Nezavisimosti Ave. 68, Minsk 220072 Belarus
2Department of Theoretical Physics and Astrophysics,
Belarusian State University, Nezavisimosty Ave. 4, Minsk 220030 Belarus
(Dated: January 19, 2017)
Wediscussthedata-patterntomographyforreconstructionofentangledstatesoflight. Weshow
that for a moderate number of probe coherent states it is possible to achieve high accuracy of
representation not only for single-mode states but also for two-mode entangled states. We analyze
the stability of these representations to the noise and demonstrate the conservation of the purity
and entanglement. Simulating the probe and signal measurements, we show that systematic error
inherentforrepresentationofrealisticsignalresponsewithfinitesetsofresponsesfromprobestates
still allows one to infer reliably the signal states preserving entanglement.
I. INTRODUCTION of basis states. If the observer believes that the signal
state is very likely residing in some operator subspace,
7 he or she can make use of this insight to define the set
Quantum tomography as a way of inferring a quan-
1 of probe states that spansthis subspace for data-pattern
tum state is potentially the most precise measuring tool
0 reconstruction [13, 14]. Naturally, the accuracy of the
2 available to a physicist [1–3]. However, this tool requires
method depends crucially on the accuracy of the signal
rather precise tuning. One needs to know characteris-
n representation. Systematic error intrinsic in the method
tics of the measurement setup which necessarily involves
a is unavoidably amplified in the process of the signal in-
J a calibration of it. Generally, it is quite a nontrivial
ference since both probe and signal patterns are subject
task equivalent to quantum process tomography of the
7 to statistical errors.
1 detecting system. Provided that one can describe the
setup with the few-parameter model (such as efficiency Here we show that such an amplification of errors

] and dark count rate of detectors, etc.), it is possible in would not lead to breaking of essentially quantum fea-
h
somecasestoperformthecalibrationwithoutacomplete tures of the state. Such a fragile feature as the entan-
p
- set of known probes by trading some information about glement survives the inference procedure, and for suffi-
t the probe for knowledge about the detector. For exam- ciently high accuracy of the representation the fidelity
n
a ple, with a twin-photon state, one can find the absolute of the signal reconstructions for the large number of sig-
u value of the detecting setup efficiency [4, 5]; entangle- nal state copies tends to the values close to the fidelity
q ment also makes possible “self-testing” or “blind tomog- given by the representation. Our simulation shows that
[
raphy”[6,7]. Tradingknowledgeofprobes(preferablyof this number of copies stands well within the region of

1 the most general nature, such as Gaussianity) for infor- experimental feasibility. Moreover, entanglement is not
v mation about the measurement gave rise to the concept broken even for a comparatively low number of signal
1 of self-sufficient, or self-calibrating tomography [8–12]. copies, when infidelity is several times higher than the
9 However, there is a possibility of skipping the cali- infidelity of the representation (which is rather remark-
8
4 bration stage altogether. This possibility is given by able if one takes into account the main feature of the
0 the data-pattern tomography [13, 14]. The idea of this data-pattern approach: the density matrix of the signal
. methodissomewhatsimilartothatofopticalimageanal- isapproximatedbythemixtureofnonorthogonalprojec-
1
ysis with a known optical response function [15]. An ob- tors). So the data-pattern scheme is quite feasible and
0
7 servermeasuresresponses(thedatapatterns)forasetof reliable also for the reconstruction of multimodal states.
1 knownquantumprobestatesandmatchesthemwiththe Also, we show that the representation of the signal in
: response obtained from the unknown signal of interest. terms of the classical probe basis is quite robust with
v
The data-pattern tomography can also be understood as respect to the noise of the representation weights.
i
X asearchfortheoptimalstateestimatoroverthesubspace The outline of the article is as follows. In Sec. II we
r that is spanned by the probe states. This approach is review the basics of the data-pattern scheme and discuss
a
naturallyinsensitivetoimperfectionsofthemeasurement the selection criteria of optimal basis sets. After that,
setupsincealldeviceimperfectionsareautomaticallyin- we analyze different basis sets of coherent states for the
corporated into and accounted for by the measured data representation of single-mode states as the initial simple
patterns. The data-pattern scheme was recently suc- problem and entangled double-mode states with a small
cessfully realized with few-photon signals and coherent average number of photons. Next, in Sec. III, the qual-
probes and was shown to be quite robust [16, 17]. ity of such expansions based on the sets with optimal
Theefficiencyofdata-patterntomographydependses- parameters is discussed. We analyze the stability of this
sentially on the choice of the basis set of probe states. proceduretothenoiseandevaluateentanglementandthe
It is highly desirable to use the smallest possible number purity of the represented states. Last, but not least, in
2
Sec. IV we present simulations of the procedure of data- Measurement
patternreconstructionusingoptimalsetsofcoherentpro- system
jectorsforsingle-modeandentangleddouble-modestates
Probe states
and demonstrate the survival of entanglement.
II. OPTIMAL BASIS SETS
Unknown state
In this section, we consider the representation of opti-
calquantumstatesbasedonthediscretebasissetindata-
pattern tomography. From a practical point of view, it
FIG. 1: Measurement scheme of data-pattern approach.
is essential to find optimal basis sets with the minimum
possible number of coherent projectors for the represen-
tation. However, to our knowledge, the optimization of
coefficients x can be found by fitting the signal data
thebasischoicehasnotyetbeenfullydiscussedandana- ξ
pattern with probe data patterns:
lyzed (here one can point to only two preliminary recent
works [14, 18]). In experiments [16, 17] the reconstruc- f(ρ) ≈(cid:88)x f(ξ), (3)
tion was done by considering only a set of probe states jk ξ jk
that was a priori deemed sufficiently large (from 48 to ξ
150). We analyze the applicability of different sets of
taking into account the physical constraints imposed on
probe states for data-pattern tomography. For the cho- the density operator ρAppr [ρAppr =(ρAppr)†, TrρAppr =
sen basis set, the efficiency of the reconstruction could 1, ρAppr ≥0]. These constraints imply fulfillment of the
be enhanced using, for example, the adaptive Bayesian
following conditions for estimated coefficients:
procedure [19–22].
(cid:88) (cid:88)
x =x∗, x =1, x σ ≥0. (4)
ξ ξ ξ ξ ξ
ξ ξ
A. General principles
Thepossibilityofaccuratedata-patternreconstruction
Firstofall,letusconsiderthegeneralprinciplesofthe is closely related to the representation (1). Having pre-
data-pattern scheme. We assume that there is an appro- liminaryinformationormakingareasonableguessabout
priatelychosenfinitesetofprobestateswhichcanbede- theclassofplausiblesignalstates(suchastheupperlimit
scribedbythedensityoperatorsσ , whereξ =1,...,M. on the average photon number) allows one to make the
ξ
We would like to reconstruct the true signal state de- reconstructionusinganappropriatelychosensetofprobe
scribed by the density operator ρ. The key point of the states spanning required subspace. The problem of the
discussionisthepossibilitytofitthesignalρwithamix- optimal basis-state selection naturally arises. To solve
ture of probes, this problem our research presented here follows certain
selection criteria imposed on the basis states {σ }.
ξ
M First of all, for practical purposes, it is advisable to
(cid:88)
ρ≈ρAppr = xξσξ, (1) use probe states that can be easily generated in the lab-
ξ=1 oratoryandprovideanaccuraterepresentationofanun-
known signal state. It was shown in previous works on
where x are real coefficients. In order to proceed with
ξ thedata-patternschemeinthesingle-modecasethatthe
the reconstruction, an observer carries out a number of
usualcoherentstatessatisfythesepracticalrequirements
somemeasurementsontheunknownsignalstateρanda
rather well for a wide class of signal states [13, 14]. No-
predefined set of probe states σ . The outcome k in the
ξ tice that some time ago considerable attention was paid
j-thmeasurementcanbedescribedbypositiveoperator-
to representing nonclassical quantum states in terms of
valued measures (POVM) Π ,
jk thenonorthogonalbasisofpurequantumstates(hereone
canmention,forexample,classicalworks[23,24]). More
p(jξk) =Tr(Πjkσξ), (2) recently, a number of works on representing entangled
p(ρ) =Tr(Π ρ), statesusingsuchcoherentbaseshasappeared(forexam-
jk jk ple,[25,26]). Inthecurrentwork,weareimplementinga
quite different approach: a representation of the density
where p(ξ,ρ) are probabilities for the probe ξ or signal ρ.
jk matrixofthesignalintermsofcoherent-stateprojectors.
Such measurements under a finite number of signal and
Second, it is desirable to use the smallest possible
probe copies result in frequency distributions fj(kρ) and number of probe states in order to minimize computa-
f(ξ) (see Fig. 1), which represent the data patterns for tional resources for reconstruction of an unknown sig-
jk
the signal state and the probe states, respectively. So nal state. In subsequent research we use the fidelity
(cid:112)
from Eqs. (1) and (2) one has that the representation F(ρ,ρAppr)= ρ1/2ρApprρ1/2 as the measure of quality
3
of representation (1) [27]. Aiming for practical applica-
1 ,0 0 0 0
tions,werequiretheaccuracyofrepresentation(1)tobe
greater than experimental measurement precision. It is
not possible to evaluate the required accuracy precisely,
bscuhtemereromrsayinbreetcaeknetnwaosrkthse[1r6e,fe1r7en]coenptohinetdfaotrao-puartetsetrin- lity0 ,9 9 9 9
e
mations. Havingrelativeexperimentalerrorsonthescale id
F
ofseveralpercent,wedeterminetherequiredprecisionof
the expansion (1) to be an order of magnitude greater. 0 ,9 9 9 8
Thus, thecriterionoftherequiredaccuracyoftherepre-
0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0
sentation can be expressed as F(ρ,ρAppr) ≥ 0.999. No-
ticethatstatesthatarecloseintermsoffidelitymayhave p
rather different physical properties, as has been shown (a)
theoreticallyandexperimentally,forexample,in[28–30].
1 ,0 0 0 0
So, when judging the quality of the state representation,
we estimate also purity and the entanglement.
Third,thesetofbasisstatesmustbesuitableforrepre-
spernetcaistiioonn.ofInawthiidsepraapnegre,owfequcaonntsuidmersttahteesbwroitahdrcelqausisreodf lity0 ,9 9 9 5
e
states with small average number of photons which are id
F
widely used and applied in quantum cryptography and
quantum computing [2, 31]. 0 ,9 9 9 0
Next, we shall analyze the optimal basis sets of co-
0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0
herent projectors for the single-mode case as the initial
simple problem. After that, based on the results ob- p
tained, we shall consider the case of entangled double- (b)
mode states.
FIG. 2: The fidelities of the representation of mixed states
ρ=p|0(cid:105)(cid:104)0|+(1−p)|1(cid:105)(cid:104)1| (p ∈ [0,1]) using coherent projec-
tors with amplitudes selected in (a) the square lattice on the
B. Single-mode case phase plain with N ×N =6×6, d=0.1 and (b) the helical
grid with N =17, ∆r=0.016, ∆ϕ=π/4 (b).
First, we consider the selection of the optimal basis
sets for the single-mode case as the initial problem. For
this case we analyze the expansion for the single-photon number of nodes along each axis N and grid pitch d.
state, the coherent state with amplitude α = 0.5, the The increase in the accuracy of representation is pro-
even coherent state (the so-called “Schr¨odinger’s kitten” videdbytheincreaseinthenumberofnodesN×N (the
state)ψ ∝|α=0.5(cid:105)+|α=−0.5(cid:105),andthesuperposition number of coherent projectors) and the decrease in grid
of the vacuum and the single-photon states. We assume pitch d since the recognition of small-scale details on the
the basis sets {σ = |α (cid:105)(cid:104)α |}, where α are amplitudes phase plane requires advanced resolution. On the other
ξ ξ ξ ξ
of coherent projectors. To solve the formulated problem hand, filling a large area of the phase plain with the lat-
we use cvx for matlab, a package for specifying and tice with small d requires an excessive number of basis
solving convex programs [32, 33]. The disciplined con- states. Thismeansthattherearesomeoptimalvaluesof
vex programming methodology is implemented in this the number of nodes along each axis N and grid pitch d.
system. Itisassumedthatonefollowscertainrulesspec- ThedependenciesofthefidelityontheparametersN and
ifying a problem. If we take into account joint concavity d enable us to find the optimal parameters of the square
of the fidelity F in two arguments [2], it can be verified lattice taking into account the criterion of the required
that the problem of expansion formulated above satis- accuracyoftherepresentation(F ≥0.999). Theanalysis
fies the whole set of rules. To improve convergence, we of the representation (1) for the above-mentioned signal
impose additional constraints on the absolute values of statesenablesustodetermineoptimalparametersinthis
the coefficients x : |x | ≤ const. It is most natural to case: N ×N =6×6, d=0.05−0.15.
ξ ξ
selectthediscretesetsofcoherentstates{σ =|α (cid:105)(cid:104)α |} Notice that choosing probes not on the simple square
ξ ξ ξ
by constructing the square lattice near the origin on the grid but in a more sophisticated way may, in fact, lead
phase plane with the axes representing the values of the to better accuracy with a smaller number of probes. For
real and imaginary parts of complex amplitudes {α }. the signal with a small average number of photons we
ξ
Let us discuss now the possibility of representing single- also consider the sets of coherent states chosen in the
mode quantum states based on these states and find the helical grid on the phase plane [18]. All probe states are
optimal sets, which meet the above-mentioned criteria. equidistant in radius and angle; we optimize the number
The optimization parameters of these basis sets are the of nodes N, the step of the radial distance ∆r, and the
4
1 ,0 0 0 0 0
0 ,9 9 9 9 9
y y
Fidelit Fidelit idelity0 ,9 9 9 9 8
F
0 ,9 9 9 9 7
0 ,9 9 9 9 6
0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0
p
(a) (b)
FIG.4: Thefidelitiesoftherepresentationofentangledmixed
FIG. 3: The fidelity of the representation of entangled states ρ= 1−p(cid:0)|0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) (cid:1)(cid:0)(cid:104)0| (cid:104)1| +(cid:104)1| (cid:104)0| (cid:1) +
double-mode states against the grid pitch d of the square p(cid:0)|0(cid:105) |0(cid:105) +|1(cid:105)2|1(cid:105) (cid:1)1(cid:0)(cid:104)0|2(cid:104)0| +1(cid:104)1|2(cid:104)1| (cid:1)1(p2∈ [0,11])2using
lattice of coherent projectors: (a) the number of nodes 2 1 2 1 2 1 2 1 2
coherent projectors with amplitudes selected in the square
along each axis N = 6; (b) the number of nodes along
lattice on the phase plain with N =7, d=0.05.
each axis N = 7. The predefined double-mode states
(cid:0) (cid:1)
are ψ=const× |α(cid:105) |−α(cid:105) +|−α(cid:105) |α(cid:105) with α = 0.5
1 2 1 √2
(cid:0) (cid:1)
(black bars), ψ= |0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) / 2 (dark gray bars),
(cid:0) 1(cid:1) √2 1 2 d for the two modes.
ψ= |0(cid:105) |0(cid:105) +|1(cid:105) |1(cid:105) / 2 (light gray bars).
1 2 1 2
Accurate representation (1) of states that are the ten-
sor product of pure single-mode states is possible for
the parameters found in the previous section (N = 6,
step of the angle ∆ϕ = 2π(m/n), with m,n ∈ N. The
d = 0.05 − 0.15) since F(ρ ⊗ρ ,ρAppr⊗ρAppr) =
optimization procedure for this set of probes gives the 1 2 1 2
following optimal parameters: N =17, ∆ϕ=π/4, ∆r = F(ρ1,ρA1ppr)F(ρ2,ρA2ppr) [27]. Intuitively, in the two-
0.009−0.016. One can see that the use of the helical mode case it is natural to expect some nontrivial com-
grid with optimal parameters requires indeed a smaller plication due to the presence of entanglement. We an-
number of states in comparison with the square lattice. alyze the expansions (1) for the pure entangled states
mentioned above in this section. The results presented
Theanalysisoftherepresentation(1)givenaboveisfor
in Fig. 3 indicate that, indeed, the representation of en-
pure states. However, obviously, the expansion (1) holds
tangled double-mode states requires a greater number of
for mixed states as well. Let us demonstrate how the ac-
nodes along each axis N than in the single-mode case.
curacy of the representation fares with the mixed states.
This figure demonstrates that a representation with fi-
For this purpose, we analyze the fidelity of the repre-
delityF ≥0.999ispossibleforthesetswiththefollowing
sentation for the mixed states ρ=p|0(cid:105)(cid:104)0|+(1−p)|1(cid:105)(cid:104)1|
optimal parameters: N =7, d=0.05−0.20.
(p∈[0,1]). Figure 2 shows that the fidelities of the rep-
resentationofmixedstatesexceedtheminimumrequired It is also useful to demonstrate that the method
accuracy for optimal basis sets of coherent projectors in considered works properly for mixed states in
the cases of the square lattice and the helical grid. This this case. For this reason, we consider the ex-
analysis confirms that the method considered works ap- pansion of the state with the density operator
propriately for mixed states as well. ρ= 1−2p(cid:0)|0(cid:105)1|1(cid:105)2+|1(cid:105)1|0(cid:105)2(cid:1)(cid:0)(cid:104)0|1(cid:104)1|2+(cid:104)1|1(cid:104)0|2(cid:1) +
+ p(cid:0)|0(cid:105) |0(cid:105) +|1(cid:105) |1(cid:105) (cid:1)(cid:0)(cid:104)0| (cid:104)0| +(cid:104)1| (cid:104)1| (cid:1) (p ∈ [0,1]).
2 1 2 1 2 1 2 1 2
An analysis of the fidelities of the representation of
these states confirms that optimal basis sets of coherent
C. Double-mode case
projectors are appropriate for the accurate expansion of
mixed states as well (see Fig. 4).
Now let us move to the consideration of the
We close this section by noting that the basis sets of
multimode case. Since we are aiming, primar-
coherent projectors found are applicable for the essen-
ily, to show how the entanglement survives the
tially accurate representation of single-mode and entan-
data-pattern inference, we restrict ourselves to the
gled double-mode states with a small average number
double-mode case. The representations are analyzed
√ of photons. We note that the achieved precision of the
(cid:0) (cid:1)
for the following states:√ψ = |0(cid:105)1|1(cid:105)2+|1(cid:105)1|0(cid:105)2 / 2, representation (1), F(ρ,ρAppr) ≥ 0.999, indicates that
(cid:0) (cid:1) (cid:0)
ψ = |0(cid:105)1|0(cid:105)2+|1(cid:105)1|1(cid:105)2 / 2, ψ =const× |α(cid:105)1|−α(cid:105)2+ these optimal sets of probe states can reliably represent
(cid:1)
| − α(cid:105) |α(cid:105) (α=0.5). We choose the basis sets a wide class of signal states. This circumstance allows
1 2
{σ =|α (cid:105) (cid:104)α |⊗|α (cid:105) (cid:104)α |} as the tensor product of us to expect that these sets of coherent projectors may
ξ ξ1 1 ξ1 ξ2 2 ξ2
the coherent states selected in the nodes of the square besuccessfullyimplementedindata-patterntomography
lattice on the phase plane. The parameters to optimize of single-mode and entangled double-mode states effec-
arethenumberofnodesalongeachaxisN andgridpitch tively.
5

1,00 1,00 the coherent state with amplitude α =0.5 in the single-
√
(cid:0) (cid:1)
mode case and for state ψ = |0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) / 2 in
1 2 1 2
0,95 0,95 the double-mode case are presented in Fig. 5. One can
lity lity seethatthefidelityremainsratherhighoveraquitelarge
ide0,90 ide0,90 range of the rms amplitude error.
F F
0,85 0,85
0,00 0,02 0,04 0,06 0,08 0,10 0,0 0,1 0,2 0,3 0,4
s s
(a) (b)
1,00 B. Entanglement estimation
0,95
It is not obvious that the expansion (1) based on the
idelity0,90 setofcoherentprojectorsconservestheentanglement. In
F order to demonstrate this, we estimate entanglement for
the same states represented using the square lattice of
0,85
0,00 0,01 0,02 0,03 basis states.
s
We use an entanglement witness (EW) to determine
(c)
whetherastateisseparableornot. Adensityoperatorρ
FIG. 5: The fidelities of the representation (5) against the describes an entangled state iff there exists a Hermitian
rms amplitude error. (a) The fidelities of the representation operatorW (calledEW)whichdetectsitsentanglement,
of the coherent state with amplitude α = 0.5 for the square i.e., Tr(Wρ)<0 and Tr(Wσ )>0 for all σ separa-
sep sep
lattice of coherent projectors with N =6, d=0.15. (b) The ble [34, 35]. We calculate the entanglement witness us-
fidelities of the representation of coherent state with ampli-
ing a method proposed in Ref. [36]. For the calculation
tude α = 0.5 for the helical grid with N = 17, ∆r = 0.016, of the operator W we use cvx [32, 33]. Figure 6 shows
∆ϕ=π/4. (c)Thefidelitiesoftherepresentationofthestate
(cid:0) (cid:1) √ the results of these calculations for the square lattice of
ψ= |0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) / 2usingcoherentprojectorsinthe
1 2 1 2 basis states with N = 6 and N = 7. One can see that
pitches of the square lattice with N =7, d=0.15.
all values of the trace Tr(Wρ) estimated are negative.
Thus, the representation (1) with these discrete sets of
coherent projectors conserves entanglement. Figure 6(b)
III. EXPANSION ANALYSIS
demonstratesthatthevaluesofthetraceTr(WρAppr)for
the expansions using optimal basis sets are very close to
In this section we consider the quality of the expan-
the values found for the precise density matrices, which
sion (1) using discrete sets of coherent projectors. Since
indicates the closeness of their entanglements.
any real measurements are connected with the noise of
differentsources, itisessentialtoanalyzethestabilityof
the representation using the basis sets considered. Af-
ter that, we shall analyze the conservation of entangle-
mentfortheexpansionbasedontheoptimaldiscretesets.
Last, the purity of the expansions is investigated based C. Purity
on the analysis of their eigenvalues.
At the end of this section we analyze the purity
of the density matrices ρAppr, which is defined as
A. Stability µ[ρAppr]=Tr(ρAppr)2 =(cid:80) λ2 (λ are the eigenvalues
k k k
of the density matrix ρAppr). We analyze the purity of
Analyzingthestabilityofthereconstructionsusingthe the density matrices ρAppr of the states considered ear-
optimal basis sets, we compare quantum state ρ with lier. The calculations for the expansions using the op-
the result of the expansion containing fluctuations with timal basis sets show that the purity is well conserved
normal distribution N(0,σ2) in the coefficients: in the single-mode case. Figure 7 shows the dependence
of the purity of the representation of entangled double-
N
(cid:88) mode states against the grid pitch d of the square lattice
ρAppr(cid:48) = (x +{noise})σ . (5)
ξ ξ ofcoherentprojectorswithN =6andN =7. According
ξ=1 to Fig. 7(b) the expansions using the optimal basis sets
are found to conserve the purity with great precision.
Notice that we enforce semipositivity and unit trace of
ρAppr(cid:48). Plots of the fidelity F(cid:0)ρ,ρAppr(cid:48)(cid:1) against the rms Inconclusion,wecanassertthattheoptimalbasissets
amplitude error indicate that the representations using considered in this section are applicable for a highly ac-
the optimal sets of coherent states are quite stable for curate representation of entangled states with a small
both the single- and double-mode states. The plots for average number of photons.
6
r
(a) (b)
(a)
FIG.7: Thepurityoftherepresentationofentangleddouble-
mode states against the grid pitch d of the square lattice of
coherentprojectors: (a)thenumberofnodesalongeachaxis
N =6(fidelitylevelofrepresentationρAppr is0.994−0.999);
(b) thenumber ofnodes alongeach axisN =7(fidelity level
r ofrepresentationρAppr is0.999−0.99999). Thestatesconsid-
(cid:0) (cid:1)
ered are ψ=const× |α(cid:105) |−α(cid:105) +|−α(cid:105) |α(cid:105) with α=0.5
1 2 √1 2
(cid:0) (cid:1)
(black bars), ψ= |0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) / 2 (dark gray bars),
1 √2 1 2
(cid:0) (cid:1)
ψ= |0(cid:105) |0(cid:105) +|1(cid:105) |1(cid:105) / 2 (light gray bars).
1 2 1 2
tomography in the simplest and most straightforward
(b) way, let us take the intended measurements to be pro-
jectionsontocoherentstatesforsingle-anddouble-mode
FIG. 6: Plots of the trace Tr(Wρ) against the grid pitch
cases. These measurements are described by the POVM
d of the square lattice on the phase plane: (a) the num-
elements Π =|β (cid:105)(cid:104)β | and Π =|β (cid:105) (cid:104)β |⊗|β (cid:105) (cid:104)β |
ber of nodes along each axis N = 6 (fidelity level of repre- j j j j j1 j j1 j2 j j2
sentation ρAppr is 0.994−0.999); (b) the number of nodes (j = 1,...,K) with amplitudes βj, βj1, βj2. For the as-
along each axis N = 7 (fidelity level of representation ρAppr sumed ideal lossless detection, the probabilities p(ξ) and
j
is 0.999−0.99999). The entanglement witnesses W are ob- p(ρ)ofobservingthepositiveoutcomeinthejthmeasure-
tained solving the optimization problem for the precise den- j
ment for a probe state σ and a signal state ρ are given
sity matrices. Filled bars represent the results for the den- ξ
sityoperatorsρAppr beingtherepresentationofthefollowing byEqs. (2). Amplitudesofthecoherentprojectorsform-
states: ψ=const×(cid:0)|α(cid:105)1|−α(cid:105)2+|−α(cid:105)√1|α(cid:105)2(cid:1) with α = 0.5 ingthePOVMelements{Πj}areselectedasequidistant
(cid:0) (cid:1) phase-space points that form a square lattice in our sim-
(black bars), ψ= |0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) / 2 (dark gray bars)
1 2 √1 2
(cid:0) (cid:1) ulations. We assume that the probabilities of observing
and ψ= |0(cid:105) |0(cid:105) +|1(cid:105) |1(cid:105) / 2 (light gray bars). White
1 2 1 2
every coherent-state setting j for the probe state σ and
bars represent the trace Tr(Wρ) for the precise density ma- ξ
trices of the corresponding states. the signal state ρ are measured with a finite number of
statecopiesN . Theexperimentalfrequenciesf(ξ) and
rep f
f(ρ) aresimulatedusingabinomialdistributionwithpa-
IV. RECONSTRUCTION USING THE f
OPTIMAL BASIS SETS rameters Nrep, p(jξ) and Nrep, p(jρ) for the sets of probe
states and the signal state, respectively.
Inthissection,wedemonstratethepossibilityofaccu- Oneisabletocarryoutthereconstructionminimizing
rate data-pattern reconstruction of single-mode and en- the distance
tangled double-mode states using the optimal basis sets
of the coherent projectors found. To this end, we shall
specify and simulate the set of measurements for this E[x ]=(cid:88)(cid:16)f(ρ)−(cid:88)x f(ξ)(cid:17)2. (6)
scheme. ξ j ξ j
j ξ
A. Set of measurements Minimization of the functional (6) subject to the con-
straints imposed on the coefficients {x } represents the
ξ
As a means of demonstrating the applicability of the semidefiniteconvexproblem[37]andcanbesolvedusing
discrete basis sets of coherent projectors in data-pattern the package cvx [32, 33].
7

1 ,0 0 1 ,0 0
0 ,9 8
0 ,9 5
0 ,9 6
lity lity
e e
id id
F0 ,9 4 F
0 ,9 0
0 ,9 2
1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 1 0 9
N N
re p re p
FIG. 8: The fidelities of the reconstruction of single-mode FIG. 9: The fidelities of the reconstruction of entangled
states against the number of state copies N . The ampli- double-mode states against the number of state copies N .
rep rep
tudes of the probe states and projectors {Π } are chosen in The amplitudes of the probe states and projectors {Π }
j j
the nodes of the square lattice with the optimal parameters are chosen in the nodes of the square lattice with the op-
N =6, d=0.15. The reconstruction process is simulated for timal parameters N = 7, d = 0.15. The reconstruc-
(cid:0) (cid:1)
the single-photon state (rhombuses), the coherent state with tion is done for ψ=const× |α(cid:105) |−α(cid:105) +|−α(cid:105) |α(cid:105) with
1 2 √ 1 2
amplitude α = 0.5 (squares), the even coherent state with α = 0.5 (circles), ψ=(cid:0)|0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) (cid:1)/ 2 (squares),
√ 1 2 1 2
amplitude α=0.5 (triangles), and the superposition of the (cid:0) (cid:1)
ψ= |0(cid:105) |0(cid:105) +|1(cid:105) |1(cid:105) / 2 (triangles).
1 2 1 2
vacuum and the single-photon states (circles).
B. Results
increasing the number of state copies N . So for suf-
rep
ficiently high accuracy of the representation the fidelity
We are now ready to demonstrate the possibility of
of the signal reconstructions for a large number of signal
accuratedata-patternreconstructionusingthediscussed
statecopiesindeedtendstothevaluesclosetothefidelity
optimal sets for the reconstruction of single-mode and
given by the representation. Furthermore, this number
entangled double-mode states. Typically, the number of
of copies stands well within the region of experimental
employed measurement settings K is not equal to the
feasibility.
number of probe states M. However, for the sake of
simplicity we let K = M and take the square lattice for
Finally, we analyze the conservation of entanglement
the POVM elements {Π } to be the same as that for the
j for the signal states reconstructed with different num-
set of basis states {σ }.
ξ bers of state copies N . For this purpose, we calculate
rep
In the single-mode case we select the amplitudes of the entanglement witness W using a method proposed
probe states and projectors {Πj =|βj(cid:105)(cid:104)βj|} to be in the in Ref. [36]. An entanglement witness operator calcu-
nodes of the square lattice with the optimal parameters lated for the precise density matrix ρ does not have to
N = 6, d = 0.15. Figure 8 shows examples of fideli- be EW for the states ρAppr reconstructed with a rela-
ties of the single-mode states reconstructed against the tively small number of state copies. Therefore, for the
number of state copies Nrep. In the same way we carry calculation of the entanglement witness W we solve the
out the reconstruction of entangled double-mode states problemofconvexoptimizationforthedensityoperators
by selecting the amplitudes of the discrete sets of probe ρAppr reconstructed. Figure10showstheresultsofthese
states and projectors {Πj = |βj1(cid:105)j(cid:104)βj1|⊗|βj2(cid:105)j(cid:104)βj2|} as calculationsforthesamesetsofprobestatesandprojec-
phase-space points that form the square lattice with op- tors as in Fig. 9 (we select them in the square lattice on
timal parameters N = 7, d = 0.15. Figure 9 demon- the phase plane with parameters N =7, d=0.15). Fig-
stratesthefidelitiesofthisreconstructionprocessforen- ure 10 demonstrates that the entanglement survives the
tangled double-mode states. Figures 8 and 9 show that inference procedure even for a reasonably small number
the fidelities F(ρ,ρAppr) for single-mode and entangled of state copies N .
rep
double-mode states are comparably close for reasonably
large numbers of state copies (N > 104). One can We conclude that using a discrete probe set of appro-
rep
seethatanobserverisabletoreconstructsingle-modeor priately chosen coherent projectors in data-pattern to-
entangleddouble-modestateswithafidelitythatisarbi- mographyenablesustoaccuratelyreconstructentangled
trarily close to the target fidelity F(ρ,ρAppr)=0.999 by optical quantum states.
8
V. CONCLUSIONS
We have discussed the data-pattern approach to
quantum tomography of entangled states. The efficiency
of this procedure depends essentially on the choice of
the basis set of the probe states. We substantiated
the choice of the discrete set of coherent states by
constructing a regular grid of basis states and finding
the optimal expansion for given quantum states based
on this basis set using the method considered in the
paper. Wehavedemonstrated the possibilityofaccurate
representationofentangledstatesbasedonthesediscrete
sets of coherent projectors, found the optimal ones for
entangled double-mode states as well as single-mode
states with a small average number of photons, and
demonstrated the robustness of the representation
with respect to added noise. The simulations of the
reconstruction process demonstrated the feasibility and
FIG. 10: Plot of the trace Tr(WApprρAppr) against the
theeffectivenessofdata-patternquantumtomographyof
number of state copies N for the following signal
rep
states: ψ=const×(cid:0)|α(cid:105) |−α(cid:105) +|−α(cid:105) |α(cid:105) (cid:1) with α = entangled states using the discrete basis set of coherent
1 2 1 √2
(cid:0) (cid:1) states. Theresultspresentedshowthatthedata-pattern
0.5 (black bars), ψ= |0(cid:105) |1(cid:105) +|1(cid:105) |0(cid:105) / 2 (dark gray
(cid:0) 1 2 (cid:1) √1 2 approach may become an efficient tool in experimental
bars), and ψ= |0(cid:105) |0(cid:105) +|1(cid:105) |1(cid:105) / 2 (light gray bars).
1 2 1 2
quantum-state reconstruction of entangled states.
The entanglement witnesses WAppr are obtained by solv-
ing the optimization problem for the reconstructed den-
sity matrices. The values of Tr(Wρ) by obtained ACKNOWLEDGMENTS
solving the optimization problem for the precise den-
sity matrices are approximately equal to −0.00089 for
(cid:0) (cid:1)
the signal state ψ=const× |α(cid:105)1|−α(cid:105)2+|−α(cid:105)1|α(cid:105)2 a√nd This work was supported by the National Academy of
(cid:0) (cid:1)
−0.00249 for the signal state√s ψ= |0(cid:105)1|1(cid:105)2+|1(cid:105)1|0(cid:105)2 / 2 Sciences of Belarus through the program Convergence
(cid:0) (cid:1)
and ψ= |0(cid:105)1|0(cid:105)2+|1(cid:105)1|1(cid:105)2 / 2. The fidelity level for the and the European Commission through the SUPER-
reconstructed states is 0.86−0.999. TWIN project (Contract No. 686731).
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