Measuring non-linear functionals of quantum harmonic oscillator states

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Measuring non-linear functionals of quantum harmonic oscillator states.
K. L. Pregnell∗
Centre for Quantum Computer Technology, Department of Physics
University of Queensland, Brisbane, QLD 4072, Australia
(Dated: February 1, 2008)
Using only linear interactions and a local parity measurement we show how entanglement can be
detected between two harmonic oscillators. The scheme generalizes to measure both linear and non-
linear functionals of an arbitrary oscillator state. This leads to many applications including purity
tests, eigenvalue estimation, entropy and distance measures - all without the need for non-linear
interactions or complete state reconstruction. Remarkably, experimental realization of the proposed
scheme is already within the reach of current technology with linear optics.
In the context of quantum communication and com-
puting protocols, measures such as purity, fidelity and
entanglement characterize the performance and non-
classical resources in a physical experiment. Such mea-
sures not only provide a link with theoretical models,
but also provide standards for defining benchmarks [1].
With continuing technical developments in these areas,
it is becoming increasingly necessary to identify practical
and efficient schemes to measure such quantities.
One obvious method is to first reconstruct the com-
plete density matrix from a series of measurements us-
ing, for example, the well-know procedure of quantum
state tomography [2, 3, 4]. From this reconstructed den-
sity matrix the desired measure can then be computed.
Although this technique is realizable, it is not efficient
as much more information about the quantum state is
obtained than is actually needed.
A more direct method was recently proposed by Filip
[5], and expanded upon by others [6, 7, 8]. They pro-
posed a specific quantum circuit capable of measuring
the non-linear functional Tr(ρkaρ
l
b) of the density matri-
ces ρa and ρb. The scheme requires as inputs, k and
l copies of the states ρa and ρb, respectively. It was
independently shown by Brun [9] that any polynomial
function of a state up to degree q can be estimated by a
joint measurement on q copies of the system. Amazingly,
when the systems are entangled the measurement dou-
bles as an entanglement witness [10], giving a negative
value for some entangled states while ensuring a positive
value for all separable states.
The circuit, although elegant, unfortunately requires a
nontrivial interaction between a control qubit and the k
and l copies of each system - the targets. In its simplest
form with two targets, k + l = 2, the interaction must
generate a controlled-swap operation between the con-
trol and the two targets. Such an operation does nothing
to the targets if the control is in the logical zero state
and applies the unitary swap operation
V |φ1〉|φ2〉 = |φ2〉|φ1〉 (1)
to the two targets if the control is in the logical one
state, where here |φ1〉 and |φ2〉 are arbitrary target states.
Implementing this gate is a challenging requirement for
any experimental architecture, especially when the tar-
get states are of arbitrary dimension. In the case of a
harmonic oscillator where the dimension of the Hilbert
space is infinite, implementing this gate using, for exam-
ple, non-deterministic KLM-type gates [11, 12] is only
possible when there is at most one quanta of energy in
each oscillator.
In this paper we propose an alternative to the Filip
scheme for harmonic oscillators which, remarkably, re-
quires only linear coupling between different oscillators
making it realizable with current technology. Like the
Filip proposal, the scheme can measure both linear and
non-linear functionals of the density matrix, while for
entangled oscillators the scheme acts as an entanglement
witness. We begin by first reviewing some of the prop-
erties of the swap operator [6, 7, 8, 9], along with its
generalization to multi-particles, and show how it relates
to a host of relevant information measures.
For the swap operator defined in Eqn. (1) it is straight-
forward to show that for two separable states ρa and ρb,
Tr(ρaρb) = Trab(ρa ⊗ ρbV ). (2)
Although this seems like a trivial relation, it suggests
that a direct way to measure the overlap between two
unknown states is to measure the expectation value of the
swap operator 〈V 〉. In the case where both systems are
in the same state ρi, the expectation value is equivalent
to the purity of the system, 〈V 〉 = Tr(ρ2i ). From the
purity and the overlap one could obtain, for example,
the Hilbert-Schmidt distance Tr[(ρa − ρb)2] .
In the case where the state is not separable Eqn. (2) is
no longer valid. To illustrate this take, for example, the
entangled state (|α〉|β〉 − |β〉|α〉)/
√
2, where |α〉 and |β〉
are orthonormal. From (1) the expectation value of the
swap operator is −1 which, in contrast to (2), is negative.
This is a example of the separability criteria of Horodecki,
Horodecki and Horodecki [13] which states that a density
matrix ρ is entangled iff there exists a Hermitian operator
H , an entanglement witness, such that
Tr(ρH) < 0, (3)
while for all separable states ρsep
Tr(ρsepH) ≥ 0. (4)

2From Eqn. (2) and by explicit example we see that the
swap operator is an example of an entanglement witness.
A measurement of 〈V 〉 is said to have witnessed the en-
tanglement when the outcome 〈V 〉 < 0. In fact, it can
be shown that the swap operator is an optimal entan-
glement witness in the sense that it forms a hyper-plane
that is tangential to the convex set of separable states
[14, 15]. That is a separable state ρ′sep exist such that
Tr[ρ′sepV ] = 0. Such a state is |α〉 ⊗ |β〉.
A generalization of the swap operator to multiple sys-
tems is defined as
VN |φ1〉|φ2〉 . . . |φN 〉 = |φN 〉|φ1〉 . . . |φN−1〉, (5)
which is not Hermitian for N ≥ 3. For separable states
ρsep = ρ1 ⊗ ρ2 ⊗ . . . ρN Eqn. (2) generalizes to
Tr[ρsepVN ] = Tr[ρaρb . . . ρN ]. (6)
For N identical copies of a state, abbreviated as ρ⊗N ,
this becomes
Tr[ρ⊗NVN ] = Tr(ρN ) =
∑
i
λNi (7)
where λi is the ith eigenvalue of ρ. We note that for a set
of N identical pure states the expectation value is unity,
while if the set is not homogeneous the expectation value
will be less than unity. This serves as a practical state
discrimination test of multiple systems [23].
For a d−dimensional system the spectrum of ρ can
be obtained from the (d − 1) values of Tr(ρN ) for N =
2, 3, . . . , d [10]. Once the spectrum is known any nonlin-
ear functional of the general form Tr[f(ρ)] can be com-
puted through the corresponding function of the eigen-
values
∑
i f(λi). This follows from the fact that the trace
is independent of the basis in which the density matrix
is expressed. In the case when the system is entangled,
knowledge of the spectrum of both the entangled state
and the reduced subsystems can be used to test for en-
tanglement through the majorization condition of Nielsen
[16, 17].
In all of the above cases the desired measure was ob-
tained from a single value: the expectation value of the
swap operator. We now introduce a simple experiment
to measure 〈VN 〉 for a system of N harmonic oscillators
in an arbitrary state ρ.
The experiment, illustrated in Fig. (1), is conducted
in two stages. First the N oscillators evolve under the
action of the unitary operator Ω. Following that a mea-
surement is performed in the energy eigenbasis of each
oscillator and the expectation value of an operator D is
measured. The specific form of Ω and D is such that
Tr[Ω ρΩ†D] = 〈VN 〉 (8)
for all states ρ. Using the cyclic property of the trace
this implies
VN = Ω†DΩ. (9)
!
1 2 3 N
1 2 3 N
"
FIG. 1: Apparatus to measure Tr(ρVN). The evolution Ω gen-
erates a discrete Fourier transformation of the N oscillators.
Following that an energy measurement of N − 1 oscillators is
performed from which Tr(ρVN) is estimated.
Since, in general, VN is not Hermitian, it is not strictly
possible to associate D in Eqns (8) and (9) with an ob-
servable. Nevertheless, it is possible to express a general
operator as a weighted sum of POVM elements as was
done in [18]. From the linearity of the trace, the ex-
pectation value of the operator corresponds to the same
weighted sum of measured probabilities.
From the action of VN on the basis state in Eqn. (5)
it can be shown that any operator Oj acting solely on
the jth oscillator transforms as VNOjV †N → Oj+1 (mod
N). Of particular importance are the creation and anni-
hilation operators, a†j and aj respectively, which satisfy
the canonical commutation relation [ai, a
†
j] = δi,j . Using
the Kronecker delta function the transformation of the
creation operators can be written as
VNa
†
jV
†
N =
∑
i
a†iδi,j+1. (10)
Since the set of creation operators {a†j} generates an or-
thonormal basis from the ground state |0〉⊗|0〉⊗ . . .⊗|0〉,
Eqn. (10), along with the condition
VN |0〉 ⊗ |0〉 ⊗ . . .⊗ |0〉 = |0〉 ⊗ |0〉 ⊗ . . .⊗ |0〉, (11)
is an equivalent definition of VN . This is a simple but
important result as it shows that the swap operation gen-
erates a linear and unitary transformation of the set of
creation and annihilation operators. We now seek a so-
lution to Eqn. (9) where the operators Ω and D are of
the same kind. Specifically, we require
Ωa†jΩ
† =
∑
i
a†iΩij (12)
Da†jD
† =
∑
i
a†iDij , (13)
where the coefficients Ωij and Dij , to be determined
later, are elements of the unitary matrices Ω and D re-
spectively.

3In general, any such unitary operator U which trans-
forms a set of N creation operators linearly and unitarily
Ua†jU
† =
∑
i
a†iUij (14)
is an element of the U(N) Lie group [19]. A property
of the Lie group is that there corresponds a Hermitian
operator H of the general form
H =
∑
ij
λija
†
iaj , (15)
with λij = λ∗ji, which generates the group element U =
exp(−iH). Noting that an arbitrary generator H acting
on the ground state is identically zero, we can Taylor
expand U as powers of H and show that Eqn. (11) is
satisfied for any unitary U of the Lie group, including
the swap operator.
The physical importance of Eqn (12) is that the evo-
lution operator Ω can be implemented with a linear
coupling Hamiltonian of the form given in Eqn. (15).
Reck et al. have shown how any such multi-particle cou-
pling Hamiltonian can be implemented using a sequence
of two particle interactions [20]. In an optics context
this corresponds to an array of beam-splitters and phase-
shifters. As for the measurement, our requirement that
D be an element of U(N) as-well-as correspond to a mea-
surement in the energy basis implies that it must be gen-
erated by an operator of the form HD =
∑
j θja
†
jaj . Ex-
plicitly, we require
D = exp[−iθ1a†1a1]⊗ exp[−iθ2a†2a2]⊗ . . .
. . .⊗ exp[−iθNa†NaN ]. (16)
where θj are free parameters yet to be determined.
To derive the specific form of the matrix elements Ωij
and the coefficients θj we substitute Eqn. (9) into (10)
and derive, with the help of (12) and (13), the matrix
equation
VN = Ω†DΩ (17)
where [VN ]ij := δi,j+1. We note that the commutation
relation between the creation and annihilation operators
implies
exp(−iφ a†jaj) a
†
k exp(iφ a
†
jaj) = a
†
k exp(−iφ δj,k), (18)
from which it is straightforward to show from (16) and
(13) that the matrix D is diagonal with elements Djj =
exp(−iθj). The matrix Ω in (17) then is such that it
diagonalisesVN . Using standard techniques we can solve
Eqn. (17) to give
Ωij = ωij/
√
N (19)
Djj = ωj−1 (20)
where ω = exp(i2pi/N) is the N th root of unity. We see
that the matrix Ω is a discrete Fourier transformation
and defines the action of the unitary operator Ω. The
unknown phases in (16) are found from (20) to be
θj = 2pi(j − 1)/N, j = 1, 2, . . . , N (21)
and characterize the operator D in (16). To express this
in terms of POVM elements we rewrite the energy op-
erator a†jaj in the energy basis of the j
th oscillator as
∑
nj nj |nj〉〈nj |. With this (16) becomes
D =
∑
{ni}
w1(n1) . . . wN (nN )|n1 . . . nN 〉〈n1 . . . nN | (22)
where
wj(nj) = exp(−iθjnj) (23)
is a complex weighting coefficient. In the context
of a measurement the projector |n1 . . . nN 〉〈n1 . . . nN |
is associated with the joint measurement outcome
(n1, n2, . . . , nN), which is interpreted as the outcome n1
occurring at the first oscillator, n2 at the second oscil-
lator and so on. The probability of observing the event
(n1, n2, . . . , nN) is given by the overlap of the state with
the associated projector. For the state ΩρΩ† the joint
probability is
Pr(n1, n2, . . . , nN) = Tr(ΩρΩ†|n1 . . . nN 〉〈n1 . . . nN |).
(24)
From (22) and (24) the expectation value Tr(ΩρΩ†D) can
be expressed as a (complex) weighted sum of measured
probabilities,
Tr(ΩρΩ†D) (25)
=
∑
{ni}
w1(n1)w2(n2) . . . wN (nN )Pr(n1, n2, . . . , nN )
which is, by definition, the expectation value 〈VN 〉. Inter-
estingly, θ1 = 0 which means that the weighting function
w1(n1) = 1 and is independent of the measurement out-
come n1 of oscillator one. Accordingly, no information
about the state of oscillator one (after the interaction)
is used in the estimate of 〈VN 〉. To simplify matters ex-
perimentally, no measurement need be performed on this
oscillator.
To illustrate the practicality of this apparatus we will
briefly discuss the simplest case which is when N = 2.
The circuit diagram of this experiment is illustrated in
Fig. 2. As previously mentioned, for separable states ρ1
and ρ2 the experiment estimates the overlap Tr(ρ1ρ2)
between two unknown states. By changing the states at
the input of the apparatus we can choose the quantity
we wish to measure, ranging from the purity when both
systems are in identical states ρi, through to the fidelity
〈α|ρ|α〉 if one of the states is pure |α〉. On the other

41
2
! 50/50
1
2
FIG. 2: Apparatus to measure Tr(ρV2). No measurement is
performed on oscillator one, while detector two measures the
average value of the parity. The two oscillators interact via a
50/50 coupling device.
hand, for entangled states the experiment acts like an
entanglement witness giving a negative value for some
entangled states while ensuring a positive value for all
separable states.
From Eqns (12) and (19) the required unitary trans-
formation is of the form
Ω
(
a†1
a†2
)
Ω† =
1√
2
(
1 1
1 −1
)(
a†1
a†2
)
. (26)
This specific unitary is generated by an interaction
Hamiltonian of the form iκ(a†1a2 − a1a†2) applied for a
time pi/(4κ), where κ is the interaction strength. In op-
tics, where the two oscillators are realized by different
spatial modes of the quantized field for example, this
can be achieved with a simple 50-50 beam-splitter. In
the case where the oscillators are distinguished by dif-
ferent polarization and/or temporal modes, then addi-
tional polarizing beam-splitters and/or time delays would
be required. Following the interaction a measurement is
performed in the energy basis of each oscillator and the
probability Pr(n1, n2) is observed after repeated trials.
To extract 〈V2〉 the distribution is weighted against the
coefficients w1(n1) = 1 and w2(n2) = (−1)n2 given by
(23) and (21). The result is
〈V2〉 =
∑
n2
(−1)n2Pr(n2) (27)
where Pr(n2) =
∑
n1
Pr(n1, n2), and is independent of
n1, the outcome of the measurement on oscillator one.
To simplify matters experimentally only the distribution
Pr+ =
∑
n2
Pr(n2) and Pr− = 1−Pr+ of the second oscil-
lator need be measured. The expectation value 〈V2〉 can
then be obtained from the difference Pr+ − Pr−, which
is known as the average parity and corresponds to the
zero of the Wigner function[21]. Similar simplifications
reside in the measurement of 〈VN 〉 for higher values of N .
It is a surprising result that such a range of meaningful
measures can be obtained from an experiment that re-
quires no non-linearity, no interferometers, just a linear
interaction and a single parity measurement.
We note that higher order moments of the swap oper-
ator 〈(VN )k〉 can also be obtained from the general ap-
paratus illustrated in Fig 1. This can be seen by writing
(VN )k = Ω†DkΩ. Physically, this corresponds to the
same evolution Ω of the state ρ followed by a measure-
ment of Dk which can be written as
Dk = exp[−ikθ1a†1a1]⊗ exp[−ikθ2a†2a2]⊗ . . .
. . .⊗ exp[−ikθNa†NaN ]. (28)
Repeating the calculation it is seen that this corre-
sponds to measuring the same probability distribution
Pr(n1, n2 . . . nN), however now the distribution is weight-
ing by the functions wj(nj , k) = [wj(nj)]k.
The measurement procedure introduced here gener-
alizes to measure the expectation value Tr(ρU) of any
unitary operator U that transforms the creation opera-
tors of N harmonic oscillators linearly. The key is that
the associated unitary matrix U can always be diagonal-
ized as Ω†DΩ where both Ω and D are associated with
the unitary evolution Ω and the measurement D through
Eqns (12) and (13) respectively. The expectation value
Tr(ρU) is then given by the weighted sum of probabilities
as in (25), where the specific values of the phases θj are
determined from the diagonal matrix D.
In conclusion, we have introduced a procedure to di-
rectly measure the quantity Tr(ρU) for any unitary op-
erator generated by a linear coupling Hamiltonian. The
procedure removes the experimentally challenging step of
entangling an ancilla to the N systems which was present
in previous proposals. The result is a more practical pro-
cedure involving only linear interactions between N oscil-
lators and local energy measurements. In the case where
the unitary operator is the N -particle swap operator, the
measurement corresponds to a range of useful measures
depending on the input state, including the purity, the
fidelity, the overlap, an entanglement witness and gener-
alized non-linear functionals.
∗ Electronic address: pregnell@physics.uq.edu.au; We
thank D. T. Pegg and J. Dodd for useful discussions relat-
ing to the manuscript and A. Lund for proof reading the
manuscript. This work was supported by the Australian
Research Council.
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