A goal of the emerging field of quantum control is to develop methods for
quantum technologies to function robustly in the presence of noise. Central
issues are the fundamental limitations on the available information about
quantum systems and the disturbance they suffer in the process of measurement.
In the context of a simple quantum control scenario--the stabilization of
non-orthogonal states of a qubit against dephasing--we experimentally explore
the use of weak measurements in feedback control. We find that, despite the
intrinsic difficultly of implementing them, weak measurements allow us to
control the qubit better in practice than is even theoretically possible
without them. Our work shows that these more general quantum measurements can
play an important role for feedback control of quantum systems.Comment: 4 pages, 3 figures. v2 Added extra citation, journal reference and
  DOI. Minor typographic correction

Almeida, M. P.

Barbieri, M.

Bartlett, S. D.

Dalton, R. B.

Gillett, G. G.

Lanyon, B. P.

O'Brien, J. L.

Pryde, G. J.

Resch, K. J.

White, A. G.

English

arXiv

A goal of the emerging field of quantum control is to develop methods for quantum technologies to function robustly in the presence of noise. Central issues are the fundamental limitations on the available information about quantum systems and the disturbance they suffer in the process of measurement. In the context of a simple quantum control scenario-the stabilization of nonorthogonal states of a qubit against dephasing-we experimentally explore the use of weak measurements in feedback control. We find that, despite the intrinsic difficultly of implementing them, weak measurements allow us to control the qubit better in practice than is even theoretically possible without them. Our work shows that these more general quantum measurements can play an important role for feedback control of quantum systems

University of Queensland eSpace

Experimental Feedback Control of Quantum Systems Using Weak Measurements
G.G. Gillett,1,* R. B. Dalton,1,† B. P. Lanyon,1 M. P. Almeida,1 M. Barbieri,1,5 G. J. Pryde,2 J. L. O’Brien,3 K. J. Resch,4
S. D. Bartlett,6 and A.G. White1
1Department of Physics and Centre for Quantum Computer Technology, The University of Queensland, Brisbane 4072, Australia
2Centre for Quantum Dynamics and Centre for Quantum Computer Technology, Griffith University, Brisbane 4111, Australia
3Centre for Quantum Photonics, H.H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering,
University of Bristol, Bristol BS8 1UB, United Kingdom
4Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario, N2L 3G1 Canada
5Laboratoire Charles Fabry, Institut d’Optique, Palaiseau 91127, France
6School of Physics, The University of Sydney, Sydney 2006, Australia
(Received 24 November 2009; published 26 February 2010)
A goal of the emerging field of quantum control is to develop methods for quantum technologies to
function robustly in the presence of noise. Central issues are the fundamental limitations on the available
information about quantum systems and the disturbance they suffer in the process of measurement. In the
context of a simple quantum control scenario—the stabilization of nonorthogonal states of a qubit against
dephasing—we experimentally explore the use of weak measurements in feedback control. We find that,
despite the intrinsic difficultly of implementing them, weak measurements allow us to control the qubit
better in practice than is even theoretically possible without them. Our work shows that these more general
quantum measurements can play an important role for feedback control of quantum systems.
DOI: 10.1103/PhysRevLett.104.080503 PACS numbers: 03.67.Pp, 02.30.Yy, 03.65.Ta, 42.50.Ex
Quantum technologies, such as quantum computing or
quantum key distribution, offer many advantages over their
classical counterparts, but it is a major challenge to make
these new technologies function robustly in the presence of
noise. The field of quantum control—the application of
control theory to quantum systems—offers a spectrum of
techniques to develop robust quantum technologies [1,2].
Control strategies can be efficient for driving quantum
systems to a known target state, with applications ranging
from phase estimation [3,4] and state discrimination [5,6]
to controlled state evolution in cavity QED [7] to the cool-
ing of single atoms [8] and macroscopic oscillators [9–13].
One such technique from control theory—feedback con-
trol—monitors the system and feeds back corrections onto
it. In classical feedback control strategies, it is always
beneficial to acquire as much information about the system
as possible, in order to identify the best correction.
However, this approach is not generally appropriate for
the control of quantum systems, where two features of
quantum measurement become important. First,
Heisenberg’s uncertainty principle imposes fundamental
limits on the amount of information that can be obtained
about a quantum system, even in principle. Second, the act
of measurement necessarily disturbs the quantum system
in an unpredictable way. These fundamental features of
quantum mechanics require a reevaluation of conventional
methods and techniques from control theory when devel-
oping the theory of quantum control. In particular, it sug-
gests the use of variable-strength or ‘‘weak’’ measurements
[14], which balance the trade-off between information gain
and disturbance in quantum systems.
In this Letter, we experimentally investigate a control
scheme of the smallest possible quantum system—a qu-
bit—for which weak measurements are necessary to
achieve optimal control. Motivated by the proposal of
Branczyk et al. [15], we demonstrate the stabilization of
a single qubit, prepared in one of two nonorthogonal states,
against dephasing noise [16]. Using weak measurements
and active feedback, we achieve an improvement over even
the theoretical best performance when using either strong
projective measurements or schemes without any measure-
ment. In contrast to control schemes conditioned on one
outcome of a weak measurement (and failure for the other
outcome) [17], our scheme uses active feedback and suc-
ceeds for any weak measurement outcome.
We choose two nonorthogonal states because they
serve as the simplest set of inputs demonstrating the limi-
tation imposed by quantum measurement: due to their
nonorthogonality, it is impossible to design a control pro-
cedure that can perfectly discriminate the input state [1]
and subsequently control the resulting (known) input
against noise. Nonorthogonal qubit states are particularly
important in the well-known B92 quantum key distribution
protocol [18].
Consider the stabilization of a qubit, which we will call
the ‘‘signal,’’ prepared in one of two nonorthogonal states
jci ¼ cos2 jþi  sin2 ji [19], where 2 represents the
angle between these two states on the Bloch sphere. The
signal qubit is transmitted through a noisy quantum chan-
nel that causes dephasing [16]; i.e., with probability p a
phase-flip operator Z (where Zj0i ¼ j0i, Zj1i ¼ j1i) is
applied to the system (where 0  p  1=2). This process
PRL 104, 080503 (2010) P HY S I CA L R EV I EW LE T T E R S
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0031-9007=10=104(8)=080503(4) 080503-1  2010 The American Physical Society
can be described by a quantum operation  ! 0 on the
initial state  ¼ jcihcj, given by
0 ¼ ð1 pÞ þ pZZ: (1)
We seek a control procedure, described by a map C, that
can return the signal as close as possible to its initial
noiseless state, independent of which initial state was
prepared; see Fig. 1(a). To quantify the performance of C
we use the average fidelity F between the noiseless input
state and the corrected output state,
F ¼ 1
2
½hcþjCð0þÞjcþi þ hcjCð0Þjci (2)
¼ 1
2
ðFcþ þ FcÞ: (3)
A general quantum feedback control scheme allows for a
variable-strength, nondestructive, measurement of the qu-
bit state. The result of this measurement then determines
the sense of a correction rotation. We now develop an
alternate scheme [20] to the one presented in Ref. [15].
In particular, we choose a family of weak measurement
that, in addition to allowing for optimal control, exhibits
interesting limiting cases with which to compare perform-
ance. Consider a measurement in the logical basis fj0i; j1ig,
with measurement operators [16]
Mþ ¼ cosð=2Þj0ih0j þ sinð=2Þj1ih1j; (4)
M ¼ sinð=2Þj0ih0j þ cosð=2Þj1ih1j; (5)
where  ranges from 0 to =2. The corresponding positive
operator-valued measurement operators [16] are given by
 ¼ MyM ¼ ½1 cosðÞZ=2, where 1 is the identity
operator. Defining the measurement strength as cos, we
have  ¼ =2 ( ¼ 0) corresponding to no measurement
(projective measurement). Based on the weak measure-
ment outcome , we then perform a correction rotation
Y by an angle around the y axis of the Bloch sphere;
i.e., Y ¼ expðiYÞ [21].
The average fidelity of this scheme can be optimized
(analytically) over the correction angle, yielding
Fð; p; Þ ¼ 1
2
þ 1
2
½f1 ð1 ð1 2pÞ sinÞcos2g2
þ cos2cos21=2; (6)
for a correction angle
optð; p; Þ ¼ tan1 cos cos
1 ð1 ð1 2pÞ sinÞcos2 : (7)
In the case of zero strength ( ¼ =2), no measurement is
performed, and Eq. (7) yields an optimal correction rota-
tion of  ¼ 0. We call this the ‘‘do-nothing’’ (DN) control
scheme. In the case of maximum strength projective mea-
surement ( ¼ 0), there is in general a nonzero optimal
correction rotation angle. This measurement in this scheme
is the well-known Helstrom measurement [22] that
achieves the highest probability of discriminating between
the two nonorthogonal states 0, and so we refer to this as
the ‘‘Helstrom’’ (H) scheme.
In general, neither the DN scheme nor the H scheme
correspond to the optimal control protocol. Maximizing
Eq. (6) with respect to  yields
optð; pÞ ¼ sin1 ð1 2pÞsin
2
1 ð1 2pÞ2cos2 : (8)
Figure 2(a) shows a contour plot of the optimal measure-
(a)
(b)
FIG. 1 (color). (a) Schematic of a generic quantum feedback
control procedure. A ‘‘signal’’ qubit passes through a noisy
channel, and is subsequently measured and corrected based on
the result. Our scheme is shown schematically in the boxes.
(b) Experimental diagram. Qubits are encoded in the polarization
of single photons (horizontal jHi  j1i, vertical jVi  j0i) and
manipulated with half wave plates (HWP) and quarter wave
plates (QWP). Optical modes are overlapped on a partially
polarizing beam splitter, with reflectivities RH ¼ 1=3 (RV ¼ 1)
for horizontally (vertically) polarized light. Conditional on there
being only one photon in each input and output mode, the gate
ideally applies the operation j0ih0j  1þ j1ih1j  Z [23–25].
The outcome of a projective measurement of the meter in the
þ= basis [19], using a polarizing beam splitter (RH ¼ 0, RV ¼
1), determines a correction rotation on the signal qubit. Wave
plate set A corrects for unwanted polarization rotations induced
by the fiber delay. A coincident detection event between either
(Mþ and S1) or (M and S1) signals a successful run. Photon
pairs are generated via spontaneous parametric down-conversion
(SPDC) of a type I BiBO crystal cut to produce degenerate
820 nm photon pairs. The crystal is pumped by a 100 mW
beam at 410 nm produced from second harmonic generation of
an 820 nm mode-locked Ti:sapphire laser (repetition rate
80 MHz). We spectrally filter the SPDC using 1 nm interfer-
ence filters centered at 820 nm, then couple into two single mode
fibers. Pump power is kept low to reduce the presence of multi-
pair emission from SPDC [30,31]. P CELL is a Pockels cell.
PRL 104, 080503 (2010) P HY S I CA L R EV I EW LE T T E R S
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080503-2
ment strength cosopt. Except for limiting cases of the
initial conditions ( ¼ f0; =2g, p ¼ f0; 0:5g) neither the
H nor DN schemes are optimal; i.e., in general there is
always a nontrivial measurement strength which optimizes
control performance. Substituting Eq. (8) into Eq. (6)
yields a final expression for the optimum fidelity:
F optð;pÞ¼12þ
1
2

cos2þ sin
4
1ð12pÞ2cos2

1=2
: (9)
In Ref. [15], it was shown that this average fidelity is the
optimal performance that can be achieved by any feedback
control scheme (all possible physical maps).
To quantify the improvement offered by variable-
strength measurements we consider the quantity Fdiff ¼
Fopt maxf FH; FDNg, which is graphed in Fig. 2(b). The
largest improvement, Fdiff ¼ 0:026, occurs at  ¼ 0:715
and p ¼ 0:115.
We implement the general control strategy in a photonic
architecture, encoding qubits into the polarization of single
photons; see Fig. 1(b). The required variable-strength
quantum nondestructive measurement on the signal qubit
is realized by entangling it to another ‘‘meter’’ qubit (pho-
ton) using a nondeterministic linear optic controlled-Z (CZ)
gate [23–25], then performing a full strength projective
measurement on the meter. This implements a measure-
ment on the signal [Eqs. (4) and (5)] with a strength
determined by the input meter state ji ¼ cos2 jþi þ
sin2 ji, as described in [26].
The sign of the correction rotation on the signal photon,
 of Eq. (8), is determined by the outcome of the meter
measurement, i.e., whether detector Mþ or M fires [see
Fig. 1(b)]. We implement this correction using a 3 mm
rubidium titanyl phosphate (RTP) Pockels cell with a half
wave voltage of 700 V at 820 nm. A similar scheme to
perform classical feedforward based on the outcome of a
measurement was implemented in [27], which allowed for
a minimum-disturbance measurement of polarization en-
coded qubits. We adjust the correction  by varying the
applied voltage. Rather than requiring the Pockels cell to
implement two different rotations () we use fixed wave
plates [labeled signal analysis in Fig. 1(b)] to always rotate
the signal by , and the Pockels cell to rotate by þ2
conditional on the meter outcome Mþ. To allow time for
the outcome of the meter measurement to be processed and
to trigger the Pockels cell, we send the signal photon
through 50 m of optical fiber. The birefringence of this
fiber causes unwanted polarization rotations which are
removed using additional wave plates; see Fig. 1(b).
Noise on the signal state is implemented by making use
of the decomposition of Eq. (1) into an ensemble of pure
states f; ZZg, weighted by the noise probability. For
each measurement strength, we perform state tomography
on the signal output for each input state  and ZZ
separately. The count rates are then combined, weighted by
the noise probability p, and the noisy density matrix is
reconstructed via a linear inversion [28,29]. At the output
of the optical circuit we observe a count rate of approxi-
mately 100 coincident photon pairs per second.
We implement the feedback scheme at  ¼ 0:715, p ¼
0:145 [a point at which our theoretical analysis predicts
close to the largest improvement Fdiff , Fig. 2(b)] and three
measurement strengths: zero ( cos ¼ 0, the DN scheme);
the theoretical optimal value ( cosopt); and maximum
( cos ¼ 1, the H scheme). Figure 3(a) compares the
experimental results with the ideal and a model that in-
cludes experimental imperfections. This model allows for
measured imperfections in the reflectivities of the central
beam splitter in the entangling (CZ) gate [23–25], Fig. 1(b),
which deviated from ideal by around a percent (RH ¼
0:345, RV ¼ 0:995). The close fit between our experimen-
tal model and results shows that the main limiting factors
in our experiment are imperfections in the properties of
manufactured optical elements. We attribute the remaining
differences between the model and data to thermal drift
during experiments.
The key result is that our experimental control protocol
performed best (on average and for each individual state)
when employing an intermediate strength measurement:
the highest measured average fidelity achieved is Fopt ¼
0:947 0:001 at a measurement strength of cos ¼ 0:93.
Furthermore, this result is higher than even the theoretical
best performance of the limiting schemes (FH ¼
0:9344>FDN); i.e., despite experimental imperfections,
the use of intermediate strength measurements produce
higher fidelity than a perfect experiment constrained to
using the DN or H strategies. Figure 3(b) presents the six
states, from which the fidelities in Fig. 3(a) were deduced,
as vectors on the Bloch sphere.
In conclusion, we have demonstrated quantum feedback
control on a single quantum system—a photonic polariza-
00 π8
π
8
π
4
π
4
3π
8
3π
8
π
2
π
2
θθ
p
(b)(a)
0.00.0
0.10.1
0.20.2
0.30.3
0.40.4
0.50.5
0.01
0.02
0.9
0.99
0.999
0.9999
FIG. 2. (a) Contour plot of the optimal measurement strength
cosopt as a function of p (the noise probability) and  (half the
angle between the two nonorthogonal qubit states, on the Bloch
sphere). (b) Contour plot of Fdiff representing the improvement
to the control scheme achieved by allowing for variable-strength
measurements; see text. The dashed line shows the cases where
FDN ¼ FH . Below this line FDN > FH, i.e., the Helstrom
scheme (H) performs worse than doing nothing at all. This
clearly emphasizes that strong measurements are not generally
appropriate for the control of quantum systems.
PRL 104, 080503 (2010) P HY S I CA L R EV I EW LE T T E R S
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080503-3
tion qubit. We demonstrated the use of weak measurement
in obtaining the optimal trade-off between information
gain and measurement back-action. This motivates the
investigation of generalized measurements for control pro-
tocols in a range of quantum systems, where quantum
control will ultimately be used for realizing quantum
technologies.
We acknowledge Agata Bran´czyk, Andrew Doherty, and
Alexei Gilchrist for valuable discussions. This research
was supported by the Australian Research Council.
*Corresponding author.
gillett@physics.uq.edu.au
†Corresponding author.
rohan.dalton@gmail.com
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[19] We have defined ji ¼ 1ffiffi
2
p ðj0i  j1iÞ, where j0i and j1i
represent the computational basis of the qubit.
[20] J. Combes, and R. Blume-Kohout (private communica-
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[21] With our choice of noise and subsequent weak measure-
ment, the Bloch vector is at all times in the x-z plane. A Y
rotation is therefore a natural candidate for the correction,
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0 (DN)
0 (DN)
0 (DN) 0.25 0.50 0.75 1 (H)
0.90
0.92
0.94
0.96
Fi
de
lit
y
cos( ) (Measurement strength)
Ideal
Model Data
o
pt
im
al
1 (H)
1 (H)
optimal
optimal
| 〉 |−〉
| 0 〉
|1〉
(a) (b)
FIG. 3 (color). Experimental results of our quantum feedback control procedure. (a) Fidelities between input and output states
[Eqs. (2) and (3)], as a function of measurement strength ( cos). The model is discussed in the text. Error bars are calculated from
Poissonian photon counting statistics. (b) Corresponding output states presented as vectors on the Bloch sphere. Light gray vectors are
those predicted by the experimental model, as in (a). The colored vectors correspond to colored points in (a), and the corresponding
measurement strength is shown next to each vector.
PRL 104, 080503 (2010) P HY S I CA L R EV I EW LE T T E R S
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080503-4


Experimental feedback control of quantum systems using weak measurements

Gillett, GG

Dalton, RB

Lanyon, BP

Almeida, MP

Barbieri, M

Pryde, GJ

O'Brien, JL

Resch, KJ

Bartlett, SD

Explore Bristol Research

A goal of the emerging field of quantum control is to develop methods for quantum technologies to  function robustly in the presence of noise. Central issues are the fundamental limitations on the available  information about quantum systems and the disturbance they suffer in the process of measurement. In the  context of a simple quantum control scenario-the stabilization of nonorthogonal states of a qubit against  dephasing-we experimentally explore the use of weak measurements in feedback control. We find that,  despite the intrinsic difficultly of implementing them, weak measurements allow us to control the qubit  better in practice than is even theoretically possible without them. Our work shows that these more general  quantum measurements can play an important role for feedback control of quantum systems.Full Tex

Gillett, G.

Dalton, R.

Lanyon, B.

Almeida, M.

Pryde, Geoff

O'Brien, J.

Resch, K.

Bartlett, S.

White, A.

Griffith Research Online

Experimental feedback control of quantum systems using weak measurementsG. G. Gillett1†, R. B. Dalton1†, B. P. Lanyon1, M. P. Almeida1, M. Barbieri1,5,G. J. Pryde2, J. L. O’Brien3, K. J. Resch4, S. D. Bartlett6 and A. G. White11 Department of Physics and Centre for Quantum Computer Technology,the University of Queensland, Brisbane 4072, Australia2 Centre for Quantum Dynamics and Centre for Quantum Computer Technology, Griffith University, Brisbane 4111, Australia3 Centre for Quantum Photonics, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering,University of Bristol, Bristol BS8 1UB, United Kingdom.4 Institute for Quantum Computing and Department of Physics and Astronomy,University of Waterloo, Waterloo, Ontario, N2L 3G1 Canada5 Laboratoire Charles Fabry, Institut d’Optique, Palaiseau 91127, France.6 School of Physics, The University of Sydney, Sydney 2006, Australia and†These authors contributed equally to this work(Dated: March 1, 2010)A goal of the emerging field of quantum control is to develop methods for quantum technologiesto function robustly in the presence of noise. Central issues are the fundamental limitations on theavailable information about quantum systems and the disturbance they suffer in the process of mea-surement. In the context of a simple quantum control scenario—the stabilization of non-orthogonalstates of a qubit against dephasing—we experimentally explore the use of weak measurements infeedback control. We find that, despite the intrinsic difficultly of implementing them, weak mea-surements allow us to control the qubit better in practice than is even theoretically possible withoutthem. Our work shows that these more general quantum measurements can play an important rolefor feedback control of quantum systems.PACS numbers: 02.30.Yy, 03.67.Ac, 42.50.Ex, 03.65.Tq, 03.65.-wQuantum technologies, such as quantum computing orquantum key distribution, offer many advantages overtheir classical counterparts, but it is a major challengeto make these new technologies function robustly in thepresence of noise. The field of quantum control—the ap-plication of control theory to quantum systems—offers aspectrum of techniques to develop robust quantum tech-nologies [1, 2]. Control strategies can be efficient for driv-ing quantum systems to a known target state, with ap-plications ranging from phase estimation[3, 4] and statediscrimination [5, 6], to controlled state evolution in cav-ity QED [7], to the cooling of single atoms [8] and macro-scopic oscillators [9–13].One such technique from control theory—feedbackcontrol—monitors the system and feeds back correctionsonto it. In classical feedback control strategies, it is al-ways beneficial to acquire as much information about thesystem as possible, in order to identify the best correc-tion. However, this approach is not generally appropriatefor the control of quantum systems, where two features ofquantum measurement become important. First, Heisen-berg’s uncertainty principle imposes fundamental limitson the amount of information that can be obtained abouta quantum system, even in principle. Second, the act ofmeasurement necessarily disturbs the quantum systemin an unpredictable way. These fundamental featuresof quantum mechanics require a reevaluation of conven-tional methods and techniques from control theory whendeveloping the theory of quantum control. In particular,it suggests the use of variable strength or ‘weak’ mea-surements [14], which balance the trade-off between in-formation gain and disturbance in quantum systems.In this Letter, we experimentally investigate a con-trol scheme of the smallest possible quantum system—aqubit—for which weak measurements are necessary toachieve optimal control. Motivated by the proposal ofBranczyk et al. [15], we demonstrate the stabilization ofa single qubit, prepared in one of two non-orthogonalstates, against dephasing noise [16]. Using weak mea-surements and active feedback, we achieve an improve-ment over even the theoretical best performance whenusing either strong projective measurements, or schemeswithout any measurement. In contrast to control schemesconditioned on one outcome of a weak measurement (andfailure for the other outcome) [17], our scheme uses ac-tive feedback and succeeds for any weak measurementoutcome.We choose two non-orthogonal states because theyserve as the simplest set of inputs demonstrating thelimitation imposed by quantum measurement: due totheir non-orthogonality, it is impossible to design a con-trol procedure that can perfectly discriminate the inputstate [1] and subsequently control the resulting (known)input against noise. Non-orthogonal qubit states are par-ticularly important in the well-known B92 quantum keydistribution protocol [18].Consider the stabilization of a qubit, which we willcall the ‘signal’, prepared in one of two non-orthogonalstates |ψ±〉= cos θ2 |+〉± sin θ2 |−〉 [19], where 2θ representsthe angle between these two states on the Bloch sphere.arXiv:0911.3698v2  [quant-ph]  1 Mar 20102(a)P-CELLFiber delayMeter prepSignal prepQWP HWPFiber CouplerRH=1/3RV=1measurementcorrectionM+M-S1A Signal analysisPathlength controlnoisemeasurement correction|ψ〉  ρ|φ〉USignalMeterEntangling Gatecontrol scheme (C)compare|ψ〉|φ〉RH=0RV=1Photonsource(b)FIG. 1: (a) Schematic of a generic quantum feedback controlprocedure. A ‘signal’ qubit passes through a noisy channel,is subsequently measured and corrected based on the result.Our scheme is shown schematically in the boxes. (b) Exper-imental diagram. Qubits are encoded in the polarisation ofsingle photons, (Horizontal |H〉≡|1〉, Vertical |V 〉≡|0〉) andmanipulated with half waveplates (HWP) and quarter wave-plates (QWP). Optical modes are overlapped on a partiallypolarising beam splitter. with reflectivities RH = 1/3 (RV =1) for horizontally (vertically) polarised light . Conditionalon there being only 1 photon in each input and output mode,the gate ideally applies the operation |0〉〈0|⊗1+|1〉〈1|⊗Z [23–25]. The outcome of a projective measurement of the meter inthe +/- basis [19], using a polarising beamsplitter (RH = 0,RV = 1), determines a correction rotation on the signal qubit.Wave plate set A corrects for unwanted polarisation rotationsinduced by the fiber delay. A coincident detection event be-tween either (M+ & S1) or (M− & S1) signals a successfulrun. Photon pairs are generated via spontaneous parametricdownconversion (spdc) of a type I BiBO crystal cut to pro-duce degenerate 820nm photon pairs. The crystal is pumpedby a ≈100mW beam at 410nm produced from second har-monic generation of an 820nm mode-locked Ti:Sa laser (rep.rate 80MHz). We spectrally filter the spdc using ±1nm inter-ference filters centered at 820nm, then couple into two singlemode fibres. Pump power is kept low so to reduce the pres-ence of multi-pair emission from spdc [30, 31]. p-cell is aPockels cell.The signal qubit is transmitted through a noisy quantumchannel that causes dephasing [16], i.e., with probabilityp a phase-flip operator Z (where Z|0〉=|0〉, Z|1〉=−|1〉) isapplied to the system (where 0≤p≤1/2). This processcan be described by a quantum operation ρ±→ρ′± on theinitial state ρ±=|ψ±〉〈ψ±|, given byρ′± = (1− p)ρ± + pZρ±Z. (1)We seek a control procedure, described by a map C,that can return the signal as close as possible to its ini-tial noiseless state, independent of which initial state wasprepared; see Fig. 1a. To quantify the performance of Cwe use the average fidelity F between the noiseless inputstate and the corrected output state.F = 12 [〈ψ+|C(ρ′+)|ψ+〉+ 〈ψ−|C(ρ′−)|ψ−〉] (2)= 12 (Fψ+ + Fψ−) . (3)A general quantum feedback control scheme allows fora variable-strength, non-destructive, measurement of thequbit state. The result of this measurement then de-termines the sense of a correction rotation. We nowdevelop an alternate scheme [20] to the one presentedin Ref. [15]. In particular, we choose a family of weakmeasurement that, in addition to allowing for optimalcontrol, exhibits interesting limiting cases with which tocompare performance. Consider a measurement in thelogical basis {|0〉, |1〉}, with measurement operators [16]M+ = cos(χ/2)|0〉〈0|+ sin(χ/2)|1〉〈1| , (4)M− = sin(χ/2)|0〉〈0|+ cos(χ/2)|1〉〈1| , (5)where χ ranges from 0 to pi/2. The correspondingpositive operator-valued measurement operators [16] aregiven by Π±=M†±M±=[1 ± cos (χ)Z]/2, where 1 is theidentity operator. Defining the measurement strength ascosχ, we have χ=pi/2 (χ=0) corresponding to no mea-surement (projective measurement). Based on the weakmeasurement outcome ±, we then perform a correctionrotation Y±η by an angle ±η around the y-axis of theBloch sphere, i.e., Y±η = exp(±iηY ) [21].The average fidelity of this scheme can be optimized(analytically) over the correction angle, yieldingF (θ, p, χ) = 12 +12[{1−(1−(1−2p) sinχ) cos2 θ}2+ cos2 χ cos2 θ]1/2, (6)for a correction angleηopt(θ, p, χ) = tan−1 cosχ cos θ1−(1−(1−2p) sinχ) cos2 θ . (7)In the case of zero strength (χ=pi/2), no measurementis performed, and Eqn. (7) yields an optimal correctionrotation of η=0. We call this the ‘do-nothing’ (DN) con-trol scheme. In the case of maximum strength projec-tive measurement (χ=0), there is in general a non-zerooptimal correction rotation angle. This measurement inthis scheme is the well-known Helstrom measurement [22]that achieves the highest probability of discriminatingbetween the two non-orthogonal states ρ′±, and so werefer to this as the ‘Helstrom’ (H) scheme.In general, neither the DN scheme nor the H schemecorrespond to the optimal control protocol. Maximizing3 00 pi8pi8pi4pi43pi83pi8pi2pi2θθp(a) (b)0.00.00.10.10.20.20.30.30.40.40.50.50.010.020.90.990.9990.9999FIG. 2: (a) Contour plot of the optimal measurementstrength cosχopt as a function of p (the noise probability)and θ (half the angle between the two non-orthogonal qubitstates, on the Bloch sphere). (b) Contour plot of F diff rep-resenting the improvement to the control scheme achieved byallowing for variable strength measurements, see text. Thedashed line shows the cases where FDN=FH . Below thisline FDN>FH i.e. the Helstrom scheme (H) performs worsethan doing nothing at all. This clearly emphasises that strongmeasurements are not generally appropriate for the control ofquantum systems.Eqn. (6) with respect to χ yieldsχopt(θ, p) = sin−1 (1− 2p) sin2 θ1− (1− 2p)2 cos2 θ . (8)Fig. 2a shows a contour plot of the optimal measurementstrength cosχopt. Except for limiting cases of the initialconditions (θ={0, pi/2}, p={0, 0.5}) neither the H nor DNschemes are optimal, i.e., in general there is always anon-trivial measurement strength which optimises controlperformance. Substituting Eqn. (8) into Eqn. (6) yieldsa final expression for the optimum fidelity:F opt(θ, p) =12 +12[cos2 θ +sin4 θ1−(1−2p)2 cos2 θ]1/2. (9)In Ref. [15], it was shown that this average fidelity isthe optimal performance that can be achieved by anyfeedback control scheme (all possible physical maps).To quantify the improvement offered by vari-able strength measurements we consider the quantityF diff=F opt−max{FH , FDN}, which is graphed in Fig 2b.The largest improvement, Fdiff=0.026, occurs at θ=0.715and p=0.115.We implement the general control strategy in a pho-tonic architecture, encoding qubits into the polarisationof single photons, see Fig. 1b. The required variablestrength quantum non-destructive measurement on thesignal qubit is realised by entangling it to another ‘me-ter’ qubit (photon) using a non-deterministic linear op-tic controlled-z (cz) gate [23–25], then performing a fullstrength projective measurement on the meter. This im-plements a measurement on the signal (Eqns. (4)-(5))with a strength determined by the input meter state|φ〉= cos χ2 |+〉+ sin χ2 |−〉, as described in [26].The sign of the correction rotation on the signal pho-ton, ±η of Eqn. (8), is determined by the outcome of themeter measurement, i.e., whether detector M+ or M−fires (see Fig. 1b). We implement this correction usinga 3mm Rubidium Titanyl Phosphate (RTP) Pockels cellwith a half-wave voltage of ∼700 V at 820 nm. A sim-ilar scheme to perform classical feed-forward based onthe outcome of a measurement was implemented in [27]which allowed for a minimum-disturbance measurementof polarisation encoded qubits. We adjust the correctionη by varying the applied voltage. Rather than requir-ing the Pockels cell to implement two different rotations(±η) we use fixed wave plates (labelled signal analysis inFig. 1b) to always rotate the signal by −η, and the Pock-els cell to rotate by +2η conditional on the meter outcomeM+. To allow time for the outcome of the meter mea-surement to be processed and to trigger the Pockels cell,we send the signal photon through 50m of optical fiber.The birefringence of this fiber causes unwanted polarisa-tion rotations which are removed using additional waveplates, see Fig. 1b.Noise on the signal state is implemented by making useof the decomposition of Eqn. (1) into an ensemble of purestates {ρ± , Zρ±Z}, weighted by the noise probability.For each measurement strength, we perform state tomog-raphy on the signal output for each input state ρ± andZρ±Z separately. The count rates are then combined,weighted by the noise probability p, and the noisy den-sity matrix is reconstructed via a linear inversion [28, 29].At the output of the optical circuit we observe a countrate of approximately 100 coincident photon pairs persecond.We implement the feedback scheme at θ=0.715,p=0.145 (a point at which our theoretical analysis pre-dicts close to the largest improvement F diff , Fig. 2b)and three measurement strengths: zero (cosχ=0, theDN scheme); the theoretical optimal value (cosχopt); andmaximum (cosχ=1, the H scheme). Fig 3a compares theexperimental results with the ideal and a model that in-cludes experimental imperfections. This model allowsfor measured imperfections in the reflectivities of thecentral beamsplitter in the entangling (cz) gate [23–25],Fig. 1b, which deviated from ideal by around a percent(RH=0.345, RV =0.995). The close fit between our exper-imental model and results shows that the main limitingfactors in our experiment are imperfections in the prop-erties of manufactured optical elements. We attributethe remaining differences between the model and data tothermal drift during experiments.The key result is that our experimental control pro-tocol performed best (on average and for each indi-vidual state) when employing an intermediate strengthmeasurement: the highest measured average fidelityachieved is F¯opt=0.947±0.001 at a measurement strength4 0 (DN)0 (DN)0 (DN) 0.25 0.50 0.75 1 (H)0.900.920.940.96Fidelitycos(χ) (Measurement strength)F¯ IdealF¯F¯HF¯DNModel DataFψ−Fψ+optimal1 (H)1 (H)optimaloptimal|+〉 |−〉|0〉|1〉|ψ+〉|ψ−〉(a) (b)FIG. 3: Experimental results of our quantum feedback control procedure. (a) Fidelities between input and output states(Eqns 2-3), as a function of measurement strength (cosχ). The model is discussed in the text. Error bars are calculated frompoissonian photon counting statistics. (b) Corresponding output states presented as vectors on the Bloch sphere. Light greyvectors are those predicted by the experimental model, as in panel a. of the figure. The coloured vectors correspond to colouredpoints in panel a., and the corresponding measurement strength is shown next to each vector.of cosχ=0.93. Furthermore, this result is higher thaneven the theoretical best performance of the limitingschemes (FH=0.9344>FDN ), i.e., despite experimentalimperfections, the use of intermediate strength measure-ments produce higher fidelity than a perfect experimentconstrained to using the DN or H strategies. Fig. 3bpresents the six states, from which the fidelities in Fig. 3awere deduced, as vectors on the Bloch sphere.In conclusion, we have demonstrated quantum feed-back control on a single quantum system - a photonicpolarization qubit. We demonstrated the use of weakmeasurement in obtaining the optimal tradeoff betweeninformation gain and measurement back-action. Thismotivates the investigation of generalized measurementsfor control protocols in a range of quantum systems,where quantum control will ultimately be used for re-alizing quantum technologies.We acknowledge Agata Bran´czyk, Andrew Dohertyand Alexei Gilchrist for valuable discussions. This re-search was supported by the Australian Research Coun-cil.[1] H. M. Wiseman and G. J. Milburn, Quantum Measure-ment and Control (Cambridge University Press, 2009).[2] H. Rabitz, New Journal of Physics 11, 105030 (2009).[3] M. A. Armen et al, Phys. Rev. Lett. 89, 133602 (2002).[4] B. L. Higgins et al, Nature 450, 393 (2007).[5] R. L. Cook, P. J. Martin, and J. M. Geremia, Nature446, 774 (2007).[6] B. L. Higgins et al, Phys. Rev. Lett. 103, 220503 (2009).[7] W. P. Smith et al, Phys. Rev. Lett. 89, 133601 (2002).[8] P. Bushev et al, Phys. Rev. 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Experimental feedback control of quantum systems using weak measurements

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