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Implementing Non-Projective Measurements via Linear Optics:
an Approach Based on Optimal Quantum State Discrimination
Peter van Loock1, Kae Nemoto1, William J. Munro1,2, Philippe Raynal3, and Norbert Lu¨tkenhaus3
1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
2Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom
3Quantum Information Theory Group, Institute of Theoretical Physics I,
Institute of Optics, Information and Photonics (Max-Planck Forschungsgruppe),
Universita¨t Erlangen-Nu¨rnberg, Staudtstr.7, D-91058 Erlangen, Germany
We discuss the problem of implementing generalized measurements (POVMs) with linear optics,
either based upon a static linear array or including conditional dynamics. In our approach, a given
POVM shall be identified as a solution to an optimization problem for a chosen cost function.
We formulate a general principle: the implementation is only possible if a linear-optics circuit
exists for which the quantum mechanical optimum (minimum) is still attainable after dephasing
the corresponding quantum states. The general principle enables us, for instance, to derive a set of
necessary conditions for the linear-optics implementation of the POVM that realizes the quantum
mechanically optimal unambiguous discrimination of two pure nonorthogonal states. This extends
our previous results on projection measurements and the exact discrimination of orthogonal states.
PACS numbers: 03.67.Hk, 42.25.Hz, 42.50.Dv
I. INTRODUCTION
The implementation of positive operator-valued mea-
sures (POVMs) for photonic quantum state signals is an
essential task in many quantum information protocols.
In general, in order to implement such measurements, a
nonlinear interaction of the signal states, described by
a Hamiltonian at least cubic in the optical mode op-
erators [1], is needed. With current technologies, how-
ever, these nonlinear effects are hard to obtain on the
level of single photons. Apart from hybrid schemes based
on weak nonlinearities and strong coherent probe pulses
[2], an alternative approach for inducing a nonlinear el-
ement is to exploit the effective nonlinearity associated
with a measurement. In particular, for photonic-qubit
states, universal quantum gates and hence any POVM
can be realized deterministically or asymptotically (near-
deterministically), using linear optics, photon counting,
entangled auxiliary photon states, and conditional dy-
namics (feedforward) [3, 4, 5, 6]. Moreover, cheaper
resources may suffice for the implementation of non-
deterministic gates and POVMs, using feedforward [3, 7]
or a static array of linear optics [3, 8, 9, 10, 11, 12, 13].
Here we will focus on the implementation of POVMs
using either static linear optics or feedforward and, in
particular, photon counting. Although there are some
specific results on this issue [14], a general and practical
solution to the problem as to whether a given POVM
can be implemented by linear optics is not known. Only
for the special class of projective measurements, a set of
simple criteria has been derived [15].
In the special case of a projection measurement, the
“signal states” to be distinguished (i.e., the basis that
spans the space to be projected on) are orthogonal. In
this case, quantum mechanically, an exact discrimination
with unit probability for a conclusive result is possible.
However, if the implementation of the projection mea-
surement is restricted to a limited class of transforma-
tions such as passive linear optics or Gaussian transfor-
mations, unit probability might be unattainable [15, 16].
The prime example to which such a no-go statement ap-
plies is the Bell measurement for polarization-encoded
photonic qubit states [15, 16, 17, 18]. Of course, such
a no-go statement for exact state discrimination does
not rule out the possibility for near-deterministic or non-
deterministic implementations. For example, the sim-
plest approximation to the single-photon qubit Bell mea-
surement only requires a symmetric beam splitter and
photon counting. This scheme achieves a success proba-
bility of one half, thus attaining the upper bound when
using linear optics and photon counting, but neither aux-
iliary photons nor feedforward [19].
A hierarchy of simple criteria for the exact discrimina-
tion of orthogonal states can be derived via a dephasing
approach [15]. The idea of this approach is to simulate
the actual detection, for instance, in the photon number
basis through a dephasing of the linearly transformed
states, turning them into mixtures diagonal in the Fock
basis. Any term in these mixtures represents a possi-
ble detection pattern for a given input state and a given
linear-optics circuit. The requirement for an exact dis-
crimination of the signal states is then that the overlap of
the dephased density operators vanishes, corresponding
to the non-existence of any coinciding patterns. Express-
ing the overlap in terms of the fidelity, this means that
the fidelity of the orthogonal states must remain zero af-
ter the linear transformation and the dephasing operation
have been applied to the states.
In order to extend the analysis of projection measure-
ments [15] to generalized measurements, the first obvious
approach is to consider von Neumann measurements in
a larger Hilbert space. Suitably chosen, these are then
equivalent to the POVM in the smaller signal space. In
fact, any POVM can be expressed in such a way via

2the Naimark extension. For signal states having only
one photon, already the Naimark extension approach re-
veals that any POVM can be implemented with linear
optics. A demonstration of this can be found in App. A.
Recent theoretical work on linear-optical implementa-
tions of one-photon POVMs and Kraus operators can be
found in Refs. [20, 21]. Previously, one-photon POVMs
via linear optics, in particular, for quantum state dis-
crimination were considered in Refs. [22, 23, 24]. There
have been also several experimental linear-optics realiza-
tions of a non-projective one-photon POVM, namely that
for unambiguous state discrimination [25, 26, 27] (a re-
view of experimental state discrimination can be found
in Ref. [28]). In general, however, for signal states with
arbitrarily many photons, to decide whether an exact im-
plementation of a given POVM is, in principle, possible
with linear optics is a non-trivial problem. Nevertheless,
approximate two-photon POVMs have been implemented
already via linear optics, for instance, for realizing a
“nonlocal measurement” on a two-photon state [29] us-
ing a non-deterministic two-photon controlled-NOT gate
[30].
Here, in order to address the question of the imple-
mentability of a general multi-photon POVM with linear
optics, we refer to a fundamental principle, independent
of the Naimark extension. In order to apply this princi-
ple, first, the POVM shall be identified as a solution of
an optimization problem for some cost function. In terms
of this cost function, the principle then states that the
implementation is only possible if a linear-optics circuit
exists for which the quantum mechanical optimum (min-
imum) is still attainable after dephasing the correspond-
ing quantum states. Whether linear optics or more gen-
eral linear transformations including multi-mode squeez-
ing are sufficient to implement the corresponding POVM
depends on the ability of these tools to obey the above
general rule. Applying this rule to the fidelity of two
nonorthogonal states will enable us to derive a set of nec-
essary conditions for the implementation of the quantum
mechanically optimal unambiguous state discimination
(USD), extending our analysis of discriminating orthog-
onal states [15]. The USD of non-orthogonal states is a
simple example for a non-projective POVM, where some
measurement results are inconclusive, but the remaining
results correctly identify the signal state.
The plan of the paper is as follows. First, in Sec. II,
we are going to explain how the effect of the detection
behind a linear-optics circuit can be described via de-
phasing. This enables us to present the main result of
the paper, a general principle for the implementation of
POVMs with linear optics. In Sec. III, we briefly review
how the known criteria for linear-optics projection mea-
surements follow from this general principle as a simple
special case. Finally, we turn to the implementation of
non-projective POVMs in Sec. IV, where our main focus
will be on the unambiguous discrimination of two pure
non-orthogonal states.
II. THE DEPHASING APPROACH TO POVMS
Given a general non-projective POVM, via the
Naimark extension approach it is pretty hard to decide
whether the POVM can be implemented with linear op-
tics. Here we propose an alternative strategy indepen-
dent of the Naimark extension, based upon a dephasing
approach. The dephasing effect will be used to mimic the
projection of the individual modes onto the detection ba-
sis. Let us first introduce the dephasing formalism.
The dephasing basis is determined by the detection
mechanism of the implementation. This might be either
the discrete photon number basis (photon counting) or
the continuous quadrature eigenstate basis (homodyne
detection). In the following, we will use the Fock basis as
the dephasing basis. This basis can be easily substituted
by other appropriate bases [15].
If the signal states ρˆ are linearly transformed into the
states ρˆH , and the photon number of the modes will be
detected, the corresponding dephasing effect can be de-
scribed as
ρˆH → ρˆ′H =
1
(2π)N
∫
dφNe−i~a†D~aρˆHei~a
†D~a . (1)
Here, we used dφN ≡ dφ1dφ2...dφN , the diagonal N ×
N matrix D, (D)ij = δijφi, and the vectors ~a =
(aˆ1, aˆ2, ..., aˆN )T and ~a† = (aˆ†1, aˆ†2, ..., aˆ†N ), representing
the annihilation and creation operators of all the elec-
tromagnetic modes involved.
The effect of the dephasing is that it turns the lin-
early transformed states into a Fock-diagonal density
matrix. This mixture contains all possible photon num-
ber patterns for a given input state and a given linear-
optics circuit. The weights of the different terms in
this mixture are determined by the probabilities for
obtaining the corresponding pattern via photon detec-
tion. Thus, the dephasing formalism is an equivalent
description for the effect of the detection after the linear-
optics transformation (see Fig. 1). An example for a
pure signal state ρˆ = |χ〉〈χ|, and hence a pure trans-
formed state ρˆH ≡ |χH〉〈χH |, would be |χH〉 = α|110〉+
β|101〉+γ|002〉. In this case, the dephased state becomes
ρˆ′H = |α|2|110〉〈110|+ |β|2|101〉〈101|+ |γ|2|002〉〈002|, cor-
responding to the possible detection patterns 110, 101,
and 002.
The advantage of the dephasing formalism is that the
effect of the detection is described on the level of the state
transformations and the final states become as classical
as they can get. These close-to-classical states can then
be analyzed with respect to a given quantum informa-
tion task. One may also consider only partially dephased
states which are Fock-diagonal only with respect to the
dephased modes. Partial dephasing mimics those proto-
cols where only a subset of the modes is detected and
a subsequent linear-optics transformation is applied to
the remaining modes conditioned upon the measurement
outcomes (conditional dynamics).

3FIG. 1: (Color online) An equivalent description of the detec-
tion mechanism after a unitary state transformation via de-
phasing. In the case of photon counting, the dephased density
matrix is a mixture of all possible photon number patterns for
a given input state and a given state transformation. Here,
we are mainly concerned about linear-optics transformations.
Using the dephasing formalism, we now propose the
following strategy in order to decide whether a given
POVM can be implemented via linear optics. First, the
POVM shall be identified as a unique solution to an opti-
mization problem. For the cost function to be optimized,
we then refer to a general principle: the implementa-
tion is only possible if a linear-optics circuit exists for
which the quantum mechanical optimum (minimum) is
still attainable after dephasing the corresponding quan-
tum states. Thus, optimizing the cost function for the
dephased states must yield the same minimum as for the
original signal states. The linear-optics circuit must be
chosen such that
Coptimallinear optics, dephasing
!= Coptimalquantum mechanics , (2)
where the symbol C denotes the corresponding cost func-
tions [31]. This general criterion is a necessary and suf-
ficient condition for the possibility of implementing the
corresponding POVM. The sufficiency here is due to the
close-to-classical character of the totally dephased output
states which are directly linked to the click patterns of the
implementation. In the case of only partially dephased
states corresponding to a conditional-dynamics protocol,
the statement in Eq. (2) is no longer sufficient but only
necessary for the implementability of the POVM. Simi-
larly, if the POVM is not a unique solution to the opti-
mization problem, the condition in Eq. (2) is only neces-
sary.
In general, it will be highly non-trivial to find the quan-
tum mechanical optimum of the corresponding cost func-
tions. In many cases, neither for pure states, as typically
given before the dephasing, nor, in particular, for mixed
states, as obtained after dephasing, a closed expression
for the optimum exists.
However, for instance, for the non-projective POVM
that is an optimal solution to the unambiguous discrimi-
nation of two pure non-orthogonal states, the correspond-
ing cost function is the failure probability and its opti-
mum/minimum before dephasing is simply the overlap
(fidelity) of the states. For the mixed states after de-
phasing, in this case, at least a lower bound for the cost
function can also be given in terms of the fidelity of the
states. It is then possible to derive a relatively simple
set of necessary conditions for the implementability of
the corresponding POVM. Later we will discuss this ex-
ample in detail. However, before applying the general
principle in Eq. (2) to non-projective POVMs, let us first
review how the known criteria for projection measure-
ments follow from this principle as a simple special case.
III. PROJECTION MEASUREMENTS
Following the approach of the preceding section, given
a projection measurement, we shall consider this mea-
surement as the optimal solution to the discrimination
of orthogonal states. A suitable cost function for an
error-free state discrimination is the failure probability,
i.e., the probability for obtaining an inconclusive result.
Now the optimal strategy in order to discriminate states
within an orthogonal set is to do a projection measure-
ment on the space spanned by these orthogonal states.
This strategy will always lead to a conclusive error-free
result. Since this implies zero cost for discriminating or-
thogonal states, Coptimalquantum mechanics = 0, a linear-optics
implementation of exact state discrimination means that
Coptimallinear optics, dephasing
!= 0 according to Eq. (2).
In order to discriminate any two pure orthogonal states
from the projection measurement basis, the quantum me-
chanically optimal/minimal failure probability is given
by the overlap of the states to be discriminated. Ex-
pressing the overlap in terms of the fidelity, F (ρˆ1, ρˆ2) ≡
(
Tr
√√ρˆ1ρˆ2
√ρˆ1
)2
, for two pure orthogonal signal states,
+ and −, of course, we have F (ρˆ+, ρˆ−) = 0. Hence
after dephasing, the minimal failure probability must
not become nonzero, in order to satisfy our principle
in Eq. (2). Since in any mixed-state discrimination
scheme, the squared failure probability is lower bounded
by the fidelity of the mixed states [32], the condition
for implementing the exact state discrimination becomes
F (ρˆ′+,H , ρˆ′−,H)
!= 0. Thus, we have Tr(ρˆ′+,H ρˆ′−,H) = 0,
since always 0 ≤ Tr(ρˆ1 ρˆ2) ≤ F (ρˆ1, ρˆ2). Using the de-
phasing integral from Eq. (1), one can then derive a hi-
erarchy of simple conditions for the exact discrimination
of two or even more states [15]. These conditions are
necessary and sufficient for the possibility of exactly im-
plementing the corresponding projection measurement.
For a two-dimensional projection measurement, corre-
sponding to the discrimination of two orthogonal states
|χ+〉 and |χ−〉, the necessary and sufficient conditions
for an exact implementation via linear optics and, for

4instance, photon counting, are given by [15],
〈χ+|cˆ†j cˆj |χ−〉 = 0 , ∀j , (3)
〈χ+|cˆ†j cˆ
†
j′ cˆj cˆj′ |χ−〉 = 0 , ∀j, j′ ,
〈χ+|cˆ†j cˆ
†
j′ cˆ
†
j′′ cˆj cˆj′ cˆj′′ |χ−〉 = 0 , ∀j, j′, j′′ ,
... =
...
Here, the mode operators cˆj = Uˆ †aˆjUˆ =
∑
i Ujiaˆi are
those corresponding to the output modes of the linear-
optics circuit. In the remainder of this section, we will
add some new and useful observations to the results of
Ref. [15] on projection measurements.
Assuming signal states with a fixed number of photons
(say N photons), there is an obvious interpretation for
the highest order conditions (i.e., the Nth order condi-
tions), because for these we have
〈χ+|cˆ†j cˆ
†
j′ cˆ
†
j′′ · · · cˆj cˆj′ cˆj′′ · · · |χ−〉 =
〈χ+,H |aˆ†j aˆ
†
j′ aˆ
†
j′′ · · · aˆjaˆj′ aˆj′′ · · · |χ−,H〉 ∝
Ψ∗(j, j′, j′′, ...|+)×Ψ(j, j′, j′′, ...|−) , (4)
where Ψ(j, j′, j′′, ...|±) is the probability amplitude for
detecting a photon in mode j and another photon in
mode j′, etc., when the input was the + or − state. Thus
Ψ(j, j′, j′′, ...|±) represents the probability amplitude for
any possible pattern to be detected at the output.
Now it becomes clear why any highest order must
vanish for exact state discrimination. Only those pat-
terns that do not occur at all and the successful pat-
terns that can be triggered only by one of the two states
lead to Ψ∗(j, j′, j′′, ...|+)× Ψ(j, j′, j′′, ...|−) = 0. In con-
trast, for any failure pattern, the product of the proba-
bility amplitudes becomes nonzero, Ψ∗(j, j′, j′′, ...|+) ×
Ψ(j, j′, j′′, ...|−) 6= 0. As a result, the highest order
conditions alone are necessary and sufficient for exact
state discrimination. Fulfilling all the highest order con-
ditions then implies that all lower order conditions are
satisfied as well. However, note that the converse does
not hold. The lower order conditions are only necessary,
but not sufficient for exact state discrimination. Thus, if
the lower order conditions are satisfied, the highest or-
der conditions may well be violated. As the lower orders
are easier to calculate than the higher orders, one would
normally start by computing the lowest orders. In order
to rule out the possibility of exact state discrimination,
it is then sufficient to find a violation of any lower or-
der condition (“no-go” statement). However, for a “go”
statement, the lower orders alone do not suffice. In this
case, for verifying that exact state discrimination is pos-
sible, one has to either calculate the higher orders as well
or directly check a possible solution inferred from the
lower orders. All these observations also indicate that for
unambiguously discriminating two nonorthogonal signal
states of fixed photon number, there must be at least one
highest order condition that is violated (corresponding to
the existence of at least one failure pattern and hence a
nonzero failure probability).
Let us now consider non-projective POVMs including
the optimal unambiguous discrimination of nonorthogo-
nal states via linear optics.
IV. NON-PROJECTIVE POVMS
Our goal is now, similar to the criteria for projection
measurements, to derive relatively simple conditions for
the implementation of a given non-projective POVM.
Our approach shall be based upon the general principle
expressed in Eq. (2).
We have seen already that there are state estimation
problems with trivial optimal POVM solutions. For in-
stance, discriminating orthogonal states optimally means
to perform the corresponding projection measurement.
A very natural way to optimally discriminate quantum
states drawn from a set of linearly independent states is
to perform a POVM that minimizes the probabiltity of
identifying the wrong states. This so-called minimum er-
ror discrimination (MED) can always be described by a
projection measurement onto a suitably chosen basis in
the signal Hilbert space [33]. Therefore, in order to decide
whether for a given set of quantum states MED can be
implemented via linear optics, we can also directly apply
the conditions for projection measurements. An exam-
ple for this is the MED of two symmetric coherent states
| ±α〉 which cannot be accomplished via non-asymptotic
linear-optics schemes [34].
Another trivial example is the optimal estimation of
an unknown qubit state. In this case, the optimal mean
fidelity F¯ optimalquantum mechanics = 2/3 [31] can be attained by
randomly choosing an arbitrary qubit basis, measuring in
this basis, and estimating the state via the basis vector
that corresponds to the outcome of the measurement.
Thus, trivially, the optimal estimation of a completely
unknown qubit state α|0¯〉+β|1¯〉 in photonic dual-rail en-
coding, |0¯〉 ≡ |10〉, |1¯〉 ≡ |01〉, can be implemented by di-
rectly detecting the photons in the two modes. In fact, in
order to satisfy our general principle in Eq. (2), we need
to fulfil 1−F¯ optimallinear optics,dephasing = 1−F¯
optimal
quantum mechanics =
1/3; this can be accomplished by directly dephasing the
input state [35].
An example that leads to highly non-trivial POVM
solutions is the calculation of the accessible information
in quantum communication, involving an extremely dif-
ficult optimization problem. Although our general prin-
ciple expressed in Eq. (2) applies to this problem as well,
here we are not going to attempt to treat a linear-optics
implementation of the accessible information gain.
By contrast, a relatively simple optimization leads
to the optimal unambiguous discrimination of quantum
states, i.e., a scheme that either identifies the signal state
correctly or it yields an inconclusive result with the small-
est probability allowed by quantum theory. In this case,
the cost function is the probability for obtaining an in-
conclusive result. Now it has been shown that in general,
this failure probability squared has a lower bound deter-

5mined by the fidelity of the signal states, Prob2fail ≥ F
[32]. For two pure non-orthogonal signal states, the min-
imal failure probability squared exactly coincides with
the overlap (fidelity) of the two states (assuming equal a
priori probabilities [36, 37, 38]). Thus, as for implement-
ing optimal unambiguous state discrimination (USD), we
can directly apply our general principle to the fidelities of
the states before and after dephasing. The corresponding
optimal POVM solution is a non-trivial non-projective
POVM, consisting of two POVM elements for the cor-
rect identification of the states and one that describes
the inconclusive result [39, 40, 41]. Let us now consider
the question whether this optimal USD of two pure non-
orthogonal states can be implemented with linear optics.
A. Optimal unambiguous state discrimination
The optimal unambiguous state discrimination (USD)
of two pure non-orthogonal states
|χ+〉 = α|0¯〉+ β|1¯〉 ,
|χ−〉 = α|0¯〉 − β|1¯〉 , (5)
where α > β are assumed to be real and {|0¯〉, |1¯〉} are
two basis states, corresponds to a projection onto the
orthogonal set (see also App. A),
|wµ〉 = |uµ〉+ |Nµ〉 , (6)
in an extended Hilbert space. Here, the {|uµ〉} are state
vectors in a Hilbert space K such that
Eˆµ = |uµ〉〈uµ| (7)
are the POVM operators of a three-valued POVM, µ =
1, 2, 3, with ∑µ Eˆµ = 1 . The vectors {|Nµ〉} are defined
in the complementary space K⊥ orthogonal to K, with
the total Hilbert space H = K ⊕ K⊥. For the optimal
USD, one can show that
|u1/2〉 =
1√
2
(β
α |0¯〉 ± |1¯〉
)
,
|N1/2〉 =
1√
2
√
1− β
2
α2 |2¯〉 ,
|u3〉 =
√
1− β
2
α2 |0¯〉 , |N3〉 = −
β
α |2¯〉 , (8)
and 〈2¯|0¯〉 = 〈2¯|1¯〉 = 0. The first two POVM elements
(µ = 1, 2) here refer to the two signal states, whereas the
third POVM element (µ = 3) corresponds to the incon-
clusive result. To make the discrimination unambiguous,
we have indeed Tr(Eˆ1|χ−〉〈χ−|) = Tr(Eˆ2|χ+〉〈χ+|) = 0
with Eˆµ from Eq. (7) and Eq. (8). To make it optimal,
we have
Probsucc = Tr(Eˆ1|χ+〉〈χ+|)/2 + Tr(Eˆ2|χ−〉〈χ−|)/2
= 1− Probfail
= 1− Tr(Eˆ3|χ+〉〈χ+|)/2− Tr(Eˆ3|χ−〉〈χ−|)/2
= 1− |〈χ+|χ−〉| = 1− (α2 − β2) = 2 β2 .
(9)
As for the linear-optical implementation, using one-
photon signal states and multiple-rail encoding, |0¯〉 ≡
|100〉, |1¯〉 ≡ |010〉, |2¯〉 ≡ |001〉, one can directly imple-
ment the corresponding POVM for the optimal USD,
as described in App. A for general single-photon based
POVMs [Eq. (A3) and Eq. (A4)]. In this case, the out-
put states after the linear-optics circuit, |100〉, |010〉, and
|001〉, uniquely refer to one of the three orthogonal states
|wµ〉, and hence identify the signal states |χ+〉 and |χ−〉
with the best possible probability. However, in general,
for arbitrary signal states, it turns out to be very hard
to decide whether the optimal USD can be implemented,
because of the infinite number of possible Naimark ex-
tensions. In the following, we will investigate the opti-
mal USD of two pure states independent of the Naimark
extension, using the general principle introduced in the
preceding sections and expressed in Eq. (2).
A suitable cost function for the USD of two pure non-
orthogonal states is the failure probability. When opti-
mized over all possible POVMs, the minimal failure prob-
ability corresponds to the overlap of the states. Thus,
according to Eq. (2) and since after dephasing the mixed-
state USD failure probability is bounded from below by
the fidelity [32], we obtain the condition
F (ρˆ′+,H , ρˆ′−,H)
!= F (ρˆ+, ρˆ−) , (10)
where F (ρˆ+, ρˆ−) is the fidelity of the input states and
F (ρˆ′+,H , ρˆ′−,H) is the fidelity after linear optics and de-
phasing. Note that the fidelity of the dephased density
matrices only yields a lower bound on the failure proba-
bility and the optimal failure probability may well exceed
this bound. Thus, even for a fixed array of linear optics
[42], described by totally dephased density matrices, the
criterion in Eq. (10) is, in general, only a necessary con-
dition for optimal USD. As a result, for the optimal USD
of two pure states via linear optics and subsequent pho-
ton counting (of all modes after static linear optics or
only one first mode in a conditional-dynamics scheme),
we have the following rule: the linear-optics circuit must
be chosen such that the overlap of the two states in terms
of the fidelity is the same before and after dephasing.
This statement, as expressed by Eq. (10), extends the
exact discrimination of orthogonal states to the more gen-
eral scenario for optimal discrimination of nonorthogonal
states. Whether linear optics or, more generally, linear
transformations including multi-mode squeezing (corre-
sponding to arbitrary quadratic interactions) are suffi-
cient to implement optimal USD depends on the ability
of these tools to obey the above rule. When focusing

6on the special case of USD, a more direct derivation of
the fidelity criterion in Eq. (10) is possible and given in
App. B. Let us now examine the statement in Eq. (10)
in more detail for a fixed array of linear optics.
B. Optimal USD via a fixed linear network
For a fixed array of linear optics, all output modes
will be detected at once. Therefore, Eq. (10) refers to
totally dephased density matrices. In order to check the
criterion in Eq. (10), we find that the fidelity before and
after linear optics becomes
F (ρˆ+, ρˆ−) = F (ρˆ+,H , ρˆ−,H) =
∑
m,n
α∗mαnβmβ∗n , (11)
because after the linear-optics transformation, the output
states will always take on the following form,
|χ+,H〉 =
∑
k
αk|{k} 〉+
∑
m
αm|{m} 〉,
|χ−,H〉 =
∑
l
βl|{l} 〉+
∑
m
βm|{m} 〉 , (12)
where the coefficients depend on the linear-optics circuit
chosen in a particular implementation. The indices k
and l denote photon number patterns, i.e., N -mode Fock
states, that exclusively occur in the expansion of |χ+,H〉
and |χ−,H〉, respectively. Hence these patterns unam-
biguously refer to the + state or to the − state. How-
ever, because of the finite overlap of the input states,
we must include patterns that occur in the expansion of
both states. These ambiguous patterns are denoted by
the index m. In general, the amplitudes of the ambigu-
ous N -mode Fock states in the expansions, and hence
the probabilities for the corresponding patterns to be de-
tected, may be different for the + and the − state.
After dephasing, the output states take on the follow-
ing form
ρˆ′+,H =
∑
k
P+k |{k} 〉〈{k}|+
∑
m
P+m|{m} 〉〈{m}|,
ρˆ′−,H =
∑
l
P−l |{l} 〉〈{l}|+
∑
m
P−m|{m} 〉〈{m}|,
(13)
corresponding to a dephasing of the states in Eq. (12)
with the probabilities given by P+k = |αk|2, P−l = |βl|2,
P+m = |αm|2, and P−m = |βm|2.
The fidelity after linear optics and dephasing is now
given by
F (ρˆ′+,H , ρˆ′−,H) =
(
Tr
√
ρˆ′+,H ρˆ′−,H
)2
=
(
∑
m
√
P+mP−m
)2
. (14)
Thus, the fidelity criterion from Eq. (10) can be expressed
by
∑
m,n
√
P+mP+nP−mP−n
!=
∑
m,n
α∗mαnβmβ∗n , (15)
using Eq. (14) and Eq. (11). This, however, implies that
∑
m,n
|αm||αn||βm||βn| !=
∑
m,n
|αm||αn||βm||βn|
× ei(φ−m−φ−n +φ+n−φ+m) , (16)
where αm = |αm|eiφ
+
m and βm = |βm|eiφ
−
m , etc. The only
possible way to satisfy Eq. (16) is for ei(φ−m−φ−n +φ+n−φ+m) =
1, ∀m,n. Thus, we have φ−m − φ+m = φ, ∀m. A direct
consequence of this result is that the overlap of the input
states can be written as
|〈χ+|χ−〉| = |〈χ+,H |χ−,H〉|
=
∣
∣
∣
∣
∣
∑
m
α∗mβm
∣
∣
∣
∣
∣
=
∑
m
|αm||βm|. (17)
Let us now look at the first-order expression from the
conditions in Eq. (3). We obtain
〈χ+|cˆ†j cˆj |χ−〉 = 〈χ+,H |aˆ
†
j aˆj |χ−,H〉 (18)
= eiφ
∑
m
|αm||βm|〈{m} |aˆ†j aˆj|{m} 〉 ,
because annihilating a photon in the jth mode of both
states only leads to nonzero contributions from coincid-
ing patterns. Using Eq. (17) and Eq. (18), there are
two observations we can make. First, the modulus of
any first order expression 〈χ+|cˆ†j cˆj |χ−〉 is bounded from
above such that
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
≤ N |〈χ+|χ−〉| , ∀j , (19)
where N is the maximum photon number in the states.
In addition, we have
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
=
〈χ+|cˆ†j′ cˆj′ |χ−〉
∣
∣
∣
〈χ+|cˆ†j′ cˆj′ |χ−〉
∣
∣
∣
, ∀j, j′ , (20)
provided that 〈χ+|cˆ†j cˆj |χ−〉 and 〈χ+|cˆ
†
j′ cˆj′ |χ−〉 are both
nonzero. Similarly, for the second-order expressions, we
obtain
〈χ+|cˆ†j cˆ
†
j′ cˆj cˆj′ |χ−〉 = 〈χ+,H |aˆ
†
j aˆ
†
j′ aˆj aˆj′ |χ−,H〉 (21)
= eiφ
∑
m
|αm||βm|
×〈{m} |aˆ†jaˆ
†
j′ aˆj aˆj′ |{m} 〉.
This leads to
∣
∣
∣
〈χ+|cˆ†j cˆ
†
j′ cˆj cˆj′ |χ−〉
∣
∣
∣
≤ N(N − 1) |〈χ+|χ−〉| , ∀j, j′,
(22)

7and
〈χ+|cˆ†j cˆ
†
i cˆj cˆi|χ−〉
∣
∣
∣
〈χ+|cˆ†j cˆ
†
i cˆj cˆi|χ−〉
∣
∣
∣
=
〈χ+|cˆ†j′ cˆ
†
i′ cˆj′ cˆi′ |χ−〉
∣
∣
∣
〈χ+|cˆ†j′ cˆ
†
i′ cˆj′ cˆi′ |χ−〉
∣
∣
∣
, ∀j, i, j′, i′ ,
(23)
provided that 〈χ+|cˆ†j cˆ
†
i cˆj cˆi|χ−〉 and 〈χ+|cˆ
†
j′ cˆ
†
i′ cˆj′ cˆi′ |χ−〉
are both nonzero. Moreover, for all non-vanishing expres-
sions, also the phases of different orders must coincide.
As a result, we have proven the following theorem for the
implementability of optimal USD of two pure nonorthog-
onal states with linear optics.
Theorem: it is a necessary (but, in general, not suf-
ficient) criterion for the possibility of implementing the
optimal USD of two pure nonorthogonal states |χ±〉 via
static linear optics and photon counting that the hierar-
chies of conditions,
〈χ+|χ−〉
|〈χ+|χ−〉|
=
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
=
〈χ+|cˆ†j′ cˆj′ |χ−〉
∣
∣
∣
〈χ+|cˆ†j′ cˆj′ |χ−〉
∣
∣
∣
=
〈χ+|cˆ†j cˆ
†
i cˆj cˆi|χ−〉
∣
∣
∣
〈χ+|cˆ†j cˆ
†
i cˆj cˆi|χ−〉
∣
∣
∣
=
〈χ+|cˆ†j′ cˆ
†
i′ cˆj′ cˆi′ |χ−〉
∣
∣
∣
〈χ+|cˆ†j′ cˆ
†
i′ cˆj′ cˆi′ |χ−〉
∣
∣
∣
, etc. ,
∀j, i, j′, i′, etc. , (24)
for any non-vanishing orders, and
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
≤ N |〈χ+|χ−〉| , ∀j ,
∣
∣
∣
〈χ+|cˆ†j cˆ
†
j′ cˆj cˆj′ |χ−〉
∣
∣
∣
≤ N(N − 1) |〈χ+|χ−〉| , ∀j, j′,
... ≤
... (25)
∣
∣
∣
〈χ+|cˆ†j cˆ
†
j′ cˆ
†
j′′ · · · cˆj cˆj′ cˆj′′ · · · |χ−〉
∣
∣
∣
≤
N ! |〈χ+|χ−〉| , ∀j, j′, j′′, ...,
are satisfied, where N is the maximum photon number in
the states. For the existence of a linear-optics solution to
the POVM that realizes the optimal USD, output mode
operators cˆj must be found such that a unitary matrix
U can be constructed with cˆj =
∑
i Ujiaˆi and the hierar-
chies of conditions are satisfied for these operators.
Note that for fixed photon number, the first set of con-
ditions of the theorem [that for the phases in Eq. (24)]
implies the second one [that for the absolute values in
Eq. (25)], however, the converse does not hold. The rea-
son is that in any linear-optics scheme, when the input
states have a fixed number of photons N , the total sum
of a given order satisfies a relation similar to
〈χ+|
∑
j
cˆ†j cˆj |χ−〉 = N 〈χ+|χ−〉 . (26)
Here, the sum of the first-order expressions leads to the
total photon number operator for the output modes, and
the relation in Eq. (26) follows from the photon number
conservation property of linear optics. Analogous condi-
tions can be found for the total sum of the higher order
expressions. Now for the sum of the first orders, for in-
stance, according to Eq. (26), we obtain
∣
∣
∣
∣
∣
∣
∑
j
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
∣
∣
∣
= N |〈χ+|χ−〉| , (27)
and, provided the phases of all non-vanishing first orders
coincide as required to obey Eq. (24),
∣
∣
∣
∣
∣
∣
∑
j
〈χ+|cˆ†j cˆj|χ−〉
∣
∣
∣
∣
∣
∣
=
∑
j
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
. (28)
The two relations of Eq. (27) and Eq. (28) together imply
that the first-order conditions in Eq. (25) are automat-
ically satisfied by any linear-optics transformation that
fulfils the first-order conditions in Eq. (24).
The hierarchies of conditions in Eq. (24) and Eq. (25)
are necessary for optimal USD of two nonorthogonal
states. In words, these criteria mean that for optimal
USD, the phases of all non-vanishing orders must coin-
cide. For example, any real orders must have the same
sign in optimal USD. In the special case of orthogonal
states, 〈χ+|χ−〉 = 0, one can easily see in Eq. (25) that
the hierarchy for exact state discrimination from Eq. (3)
can be retrieved. An alternative derivation of the condi-
tions in Eq. (24) and Eq. (25), independent of the fidelity
criterion in Eq. (10), is given in App. C.
Let us now look at an example of two nonorthogonal
states with only two photons (which is the simplest non-
trivial extension to the trivial case of one-photon states).
We are going to consider the two-photon toy model state
α|20〉 ± β|11〉, where, without loss of generality, α and β
are assumed to be real. For the special case of an orthog-
onal pair α = β, it is know that there is no linear-optics
solution for optimally and hence exactly discriminating
these two states, including feedforward and arbitrary
auxiliary states [15]. Here we consider the nonorthogo-
nal case without auxiliary photons, but arbitrarily many
additional vacuum modes.
Defining Uj1 ≡ ν1 and Uj2 ≡ ν2 for the elements of
the jth row of the unitary matrix in cˆj =
∑
i Ujiaˆi, the
first-order expression for mode j then becomes
〈χ+|cˆ†j cˆj |χ−〉 = 2α2|ν1|2 − β2(|ν1|2 + |ν2|2)
+
√
2αβ(ν1ν∗2 − c.c.) . (29)
Now assuming a fixed array of linear optics, the condi-
tions in Eq. (24) are necessary for optimal USD. In order
to satisfy the first-order conditions there for any modes
j, j′, etc., the expression in Eq. (29) must become real for
any j, j′, etc., because the 0th order 〈χ+|χ−〉 = α2 − β2
is real. The same argument applies to the second-order

8expressions,
〈χ+|(cˆ†j)2cˆ2j |χ−〉 = 2|ν1|2
[
α2|ν1|2 − 2β2|ν2|2
+
√
2αβ(ν1ν∗2 − c.c.)
]
. (30)
Knowing that all these expressions must be real, let us
evaluate the first-order and second-order conditions for
positive and negative 0th order 〈χ+|χ−〉, α2 > β2 or
α2 < β2, respectively. According to Eq. (24), we obtain
~ν ~a ≥ 0 , and ~ν ~b ≥ 0 , (31)
for α2 > β2, and
~ν ~a ≤ 0 , and ~ν ~b ≤ 0 , (32)
for α2 < β2, where
~ν ≡
(
|ν1|2
|ν2|2
)
, ~a ≡
(
2α2 − β2
−β2
)
, ~b ≡
(
α2
−2β2
)
.
(33)
In Eq. (31), ~ν ~b ≥ 0 implies that α2 ≥ 2β2, because
otherwise, for α2 < 2β2, the only way to prevent ~ν ~b
from becoming negative is to have |ν1|2 > |ν2|2 for any
modes j, j′, etc., according to Eq. (24). However, no
unitary matrix can be constructed, where all the elements
j, j′, etc., in the first two columns satisfy |ν1|2 > |ν2|2.
Similarly, in Eq. (32), ~ν ~a ≤ 0 leads to α2 ≤ β2 (which
is also simply given by the 0th order). Thus, there is a
regime, β2 ≤ α2 < 2β2 (including the orthogonal case
α2 = β2), where optimal USD is impossible for a fixed
array of linear optics and without auxiliary photons.
For α2 = 2β2, the optimal solution is a simple 50/50
beam splitter, |ν1|2 = |ν2|2 = 1/2 for modes j = 1, 2.
In this case, in agreement with Eq. (31), we obtain
~ν ~a = β2 > 0 and ~ν ~b = 0. The orthogonal set of the
corresponding von Neumann measurement becomes
|w1/2〉 =
1√
2
[
1√
2
(|20〉+ |02〉)± |11〉
]
,
|w3〉 =
1√
2
(|20〉 − |02〉) , (34)
choosing for the Naimark extension |2¯〉 ≡ |02〉 [see
Eqs. (5)-(8)]. A symmetric beam splitter turns these
three states into |20〉, |02〉, and |11〉, respectively, which
via photon counting uniquely refer to the three different
POVM elements. Thus, optimal USD for α =
√
2β can
be achieved with a simple beam splitter. The two signal
states α|20〉 ± β|11〉 are transformed by the symmetric
beam splitter into
√
2
3 |20〉+ 1√3 |11〉 and
√
2
3 |02〉+ 1√3 |11〉,
respectively. Indeed, we have Probsucc = 2/3 = 2β2.
C. Optimal USD via conditional dynamics
We may also apply the general fidelity criterion to a
more sophisticated linear-optics implementation of state
discrimination, namely one that includes conditional dy-
namics (feedforward): instead of detecting all output
modes after the linear-optics circuit, one may select only
one mode for detection. After this first measurement,
one can then send the conditional state of the remain-
ing modes through another linear-optics circuit which de-
pends on the measurement outcome. In the most general
approach, one can include as many feedforward steps as
modes are in the signal states, or even more by adding
auxiliary states.
The extension from a static linear-optics scheme to
a scheme that may include conditional dynamics is
straightforward for projection measurements [15]. In this
case, simply the subset of the fixed-array conditions, re-
ferring only to a particular mode operator cˆj , is nec-
essary for the exact state discrimination after detect-
ing a first mode j. For instance, for implementing a
two-dimensional projection measurement, corresponding
to the discrimination of two orthogonal states |χ+〉 and
|χ−〉, we have the subset of the conditions in Eq. (3),
〈χ+|
(
cˆ†j
)n
(cˆj)n |χ−〉 = 0 , ∀n ≥ 0 . (35)
These criteria express the necessary requirement for ex-
act state discimination that the detection of one mode
must be either conclusive or the orthogonality of the sig-
nal states must be preserved in the conditional states of
the remaining modes [15]. The non-existence of some cˆj
fulfilling Eq. (35) means that as soon as one output mode
is selected and measured, this will make exact discrimi-
nation of the states impossible.
For the general case, including non-projective POVMs,
the extension from static linear optics to conditional dy-
namics is slightly more subtle. A complete derivation of
the conditions for implementing optimal USD via linear
optics and feedforward can be found in App. D. The
resulting conditions necessary for optimal USD when de-
tecting a first mode j are again simply the subset of the
fixed-array conditions in Eq. (24) and Eq. (25) referring
to this one mode; thus, we obtain
〈χ+|χ−〉
|〈χ+|χ−〉|
=
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
=
〈χ+|(cˆ†j)2cˆ2j |χ−〉
∣
∣
∣
〈χ+|(cˆ†j)2cˆ2j |χ−〉
∣
∣
∣
= · · · =
〈χ+|(cˆ†j)ncˆnj |χ−〉
∣
∣
∣
〈χ+|(cˆ†j)ncˆnj |χ−〉
∣
∣
∣
, etc. , (36)
for any non-vanishing orders, and
∣
∣
∣
〈χ+|cˆ†j cˆj |χ−〉
∣
∣
∣
≤ N |〈χ+|χ−〉| ,
∣
∣
∣
〈χ+|(cˆ†j)2cˆ2j |χ−〉
∣
∣
∣
≤ N(N − 1) |〈χ+|χ−〉| , etc. (37)
These conditions are also a direct consequence of the
optimal-USD fidelity criterion in Eq. (10). However, this

9time, only partially dephased density operators, corre-
sponding to the detection of only one mode, must be
considered. These partially dephased density matrices
are, in general, no longer diagonal in the Fock basis (for
details, see App. D).
In the next section, we examine whether our new set of
conditions enables us to make general statements about
the use of auxiliary photons for the optimal USD of two
nonorthogonal states.
D. Auxiliary photons for optimal USD
Let us consider the following question: can the use of
an auxiliary state make optimal USD via linear optics
possible when it is impossible without an ancilla state.
In this case, the input states to be discriminated be-
come |χ±〉 = |s±〉 ⊗ |ψaux〉, where |s±〉 represent the
signal states and |ψaux〉 is the auxiliary state. The aux-
iliary state contains optical modes in addition to the
signal modes, and these extra modes may be occupied
by additional photons. For the special case of projec-
tive POVMs, it has been shown already, using the crite-
ria for projection measurements, that if the orthogonal
signal states contain a fixed number of photons, adding
an ancilla state (including extra photons or not) never
helps [15]. This can be seen by splitting the input modes
into a set of signal and a set of auxiliary modes, thus
decomposing the output mode operator cˆj =
∑
i Ujiaˆi
into two corresponding parts as (dropping the index j)
cˆ = bscˆs + bauxcˆaux, with real coefficients bs and baux,
where cˆs acts only upon the signal modes and cˆaux only
on the auxiliary modes. Using these output mode op-
erators and assuming orthogonal signal states with fixed
photon number, the criteria for exact state discrimination
are the same with or without arbitrary ancilla states [15].
Now for the case of the non-projective POVM for op-
timal USD of two pure nonorthogonal states, the same
approach as described in the preceding paragraph will
not enable us to make a general statement. Inserting the
output mode operator cˆ = bscˆs + bauxcˆaux into, for exam-
ple, the first-order expression 〈χ+|cˆ†cˆ|χ−〉 yields a result
that, in general, for 〈s+|s−〉 6= 0 (with either a fixed or
an undetermined photon number), still depends on the
auxiliary state.
In general, we cannot rule out the possibility that
adding an ancilla helps to satisfy the conditions for opti-
mal USD when they cannot be fulfilled without ancilla.
Moreover, adding only auxiliary vacuum modes without
extra photons might also be useful and necessary in or-
der to build up a unitary matrix for the mode operators
cˆj =
∑
i Ujiaˆi. In fact, in the one-photon example dis-
cussed after Eq. (9), adding an extra vacuum mode is
essential in order to extend the two-dimensional signal
Hilbert space to an at least three-dimensional space re-
quired for the POVM and to construct the corresponding
unitary matrix.
There are also known examples, where adding extra
FIG. 2: Implementing the optimal unambiguous discrimi-
nation of two symmetric coherent states via a simple 50/50
beam splitter and an auxiliary coherent state of the same am-
plitude.
photons makes the optimal USD of the two signal states
via linear optics possible. One such example for the case
of infinite-dimensional signal and auxiliary states, both
with an undetermined and unbounded photon number, is
the optimal USD of so-called binary coherent states. In
this case, the optimal USD can be easily achieved using
a 50/50 beam splitter and an ancilla coherent state (see
Fig. 2). In our notation, one has |s±〉 ≡ |±α〉 (α assumed
to be real), |ψaux〉 ≡ |α〉, and |χ±〉 = |s±〉 ⊗ |ψaux〉. This
two-mode state is now transformed by the 50/50 beam
splitter into
|χ+,H〉 = |
√
2α〉 ⊗ |0〉,
|χ−,H〉 = |0〉 ⊗ | −
√
2α〉 . (38)
For these states, a detector click in mode 1 can only be
triggered by the + state, whereas a click in mode 2 unam-
biguously refers to the − state. However, there are incon-
clusive “events” corresponding to the two-mode vacuum
state, |ψinconcl〉 = e−α
2 |00〉, using Eq. (C2) from App. C
with φ = 0. Since the failure probability is then given by
Problin.opt.fail = (e−2α
2
+ e−2α2)/2 = e−2α2 = 〈+α| − α〉,
this scheme turns out to be optimal. Thus, we ex-
pect that the corresponding solution satisfies our criteria
for optimal USD. For a particular mode j, again using
Uj1 ≡ ν1 and Uj2 ≡ ν2 for the elements of the jth row
of the unitary matrix in cˆj =
∑
i Ujiaˆi, we obtain the
nth-order condition,
〈χ+|(cˆ†j)ncˆnj |χ−〉 = 〈+α| − α〉
(
|ν1|2α2 (39)
−|ν2|2 2〈ψaux|aˆ†2aˆ2|ψaux〉2
)n
.
Apparently, for any mode j = 1, 2, any order n ≥ 1
can be set to zero by choosing a 50/50 beam splitter,
|ν1|2 = |ν2|2 = 1/2, and the appropriate ancilla state,
|ψaux〉 ≡ |α〉. This solution is indeed in agreement with
the conditions that we derived for optimal USD. The ob-
vious reason, why all nonzero orders vanish in this exam-
ple, is that the only failure pattern here is |00〉 which al-
ways vanishes upon applying annihilation operators [see,

10
e.g., Eq. (18)]. From this observation follows that also
any cross orders for modes 1 and 2 will vanish with the
above solution. Let us emphasize again that in this ex-
ample, neither the signal nor the auxiliary state contain
a fixed number of photons. For such a scenario, even
in the case of projective POVMs [15], adding auxiliary
photons may indeed help. However, conversely, even in-
cluding non-projective POVMs such as the optimal USD
of two pure nonorthogonal states, we are not aware of
any example of a POVM for signal states with a fixed
number of photons where it helps to add extra photons.
Of course, this statement does not apply to asymptotic
schemes [3] for which it is known that auxiliary photons
are, in general, a useful and necessary extra resource.
Let us finally note that for the optimal USD of more
than two coherent states, symmetrically distributed in
phase space, the optimal USD [40] cannot be achieved as
easily as for the binary case. However, there are asymp-
totic linear-optics solutions including the use of feedfor-
ward [43].
V. CONCLUSIONS
We considered the problem of implementing general-
ized measurements (POVMs) with linear optics. Such
an implementation may either be based upon a static
array of linear optics or it may include conditional dy-
namics (feedforward). Extending our previous results
on projective measurements, we focused, in particular,
on non-projective measurements. Our approach to this
problem can be formulated as a general principle in the
following way. We start by identifying a given POVM
as a solution to an optimization problem for a chosen
cost function. The implementation is then only possi-
ble if a linear-optics circuit exists for which the quan-
tum mechanical optimum is still attainable after dephas-
ing the corresponding quantum states. As an example
for applying this principle to the problem of implement-
ing a non-projective POVM, we discussed in detail the
optimal unambiguous state discrimination (USD) of two
pure nonorthogonal states. In order to implement the
POVM that realizes the quantum mechanically optimal
USD with linear optics, according to the general princi-
ple, the linear-optics circuit must be chosen such that the
overlap of the states, in terms of the fidelity, is the same
before and after dephasing. This statement extends the
exact discrimination of orthogonal states to the more gen-
eral scenario for optimal discrimination of nonorthogonal
states. Using the fidelity criterion, we derived hierarchies
of necessary conditions for the possibility of implement-
ing the optimal USD of two pure nonorthogonal states via
linear optics and photon counting. The resulting con-
ditions are a generalization of our previous criteria for
projection measurements and the exact discrimination of
orthogonal states.
As for the detection mechanism, here we only stud-
ied the case of photon counting which leads to dephased
states diagonal in the Fock basis. Potential extensions of
our results may include different detection mechanisms
such as homodyne detection, as we discussed previously
already in the context of projective measurements. More-
over, apart from passive linear-optics circuits, our criteria
can also be applied to arbitrary linear mode transforma-
tions, including multi-mode squeezing. When analyzing
those POVMs that realize unambiguous state discrimi-
nation, one may also consider the USD of sets of three or
more linearly independent states. Finally, let us empha-
size that our approach of choosing suitable cost functions
and applying them to the dephased quantum states might
be as well useful for finding bounds on the efficiency of
implementing POVMs with linear optics.
Acknowledgments
WM, PR, and NL acknowledge the support from the EU
project RAMBOQ. PvL and KN acknowledge funding
from MIC in Japan. PR and NL acknowledge the support
of the DFG under the Emmy-Noether programme. This
work was also supported by the network of competence
QIP of the state of Bavaria (A8).
APPENDIX A: ONE-PHOTON SIGNAL STATES
Let us consider all those POVMs where the signal
states contain only one photon. In this typical and im-
portant case, any unitary operation (gate) can be ac-
complished with linear optics [44]. This statement ap-
plies to arbitrary qudit states, where each basis vector of
the qudit is described by one photon occupying one of d
modes, aˆ†i |0〉, i = 1...d (“multiple-rail encoding”). Simi-
larly, any POVM can be implemented solely by means of
linear optics for these one-photon signal states. This can
be understood by looking at the corresponding Naimark
extension of the POVM. The POVM is then described
by a von Neumann measurement onto the orthogonal set
|wµ〉 = |uµ〉+ |Nµ〉 , (A1)
in a Hilbert space larger than the original signal
space. Here, the {|uµ〉} are (unnormalized, potentially
nonorthogonal) state vectors in a Hilbert space K such
that
Eˆµ = |uµ〉〈uµ| (A2)
are the POVM operators of an N -valued POVM, µ =
1...N , with ∑µ Eˆµ = 1 . The vectors {|Nµ〉} are defined
in the complementary space K⊥ orthogonal to K, with
the total Hilbert space H = K⊕K⊥. If the dimension of
the signal space is n, we have |Nµ〉 =
∑N
i=n+1 bµi|vi〉 with
some complex coefficients bµi and {|vi〉} a basis in K⊥.

11
In the multiple-rail encoding, this leads to an orthogonal
set of vectors
|wµ〉 =
N
∑
j=1
Uµj aˆ†j |0〉 , (A3)
with a unitary N × N matrix U having elements Uµj .
The application of a linear-optics transformation V to
this set (in order to project onto it) can be written as
|wµ〉 −→ |w′µ〉 =
N
∑
j,k=1
UµjV ∗kj aˆ†k|0〉
=
N
∑
k=1
δµkaˆ†k|0〉 = aˆ†µ|0〉 , (A4)
choosing V ≡ U . As a result, when detecting the out-
going state, for every one-photon click in mode µ, one
can unambiguously identify the input state |wµ〉. This
is why it is no surprise that any POVM for one-photon
states can be implemented via linear optics with unit suc-
cess probability (there is also an extension of this result
for one-photon implementations from any POVM to any
Kraus operator [20, 21]).
For states other than one-photon states, it is a priori
not clear whether a given POVM can be implemented
with linear optics. A possible approach to deciding this
would be to apply the criteria for projective measure-
ments [15] to the orthogonal set in Eq. (6). The main
difficulty then is that one must consider any possible
Naimark extension vectors {|Nµ〉} in order to be able to
decide whether the POVM can be implemented or not. In
particular, the extension of the signal Hilbert space can
be arbitrarily large. Therefore, in an optical implementa-
tion, arbitrary ancilla states must be taken into account,
including arbitrarily many extra modes and photons. It
seems that, in general, more complicated approaches are
required to deal with the potentially infinite-dimensional
problem of adding arbitrary auxiliary states [13] (how-
ever, see [8]). In this paper, we propose a dephasing
approach to the problem of implementing POVMs via
linear optics, independent of the Naimark extension.
APPENDIX B: ALTERNATIVE DERIVATION OF
THE OPTIMAL-USD FIDELITY CRITERION
Without referring to the general principle in Eq. (2) for
arbitrary cost functions and POVMs, here we directly de-
rive the corresponding (necessary) criterion for the spe-
cial case of optimal USD in terms of fidelities.
In general, for any state discrimination scheme based
on static linear optics, we have the following fidelity
bounds,
F (ρˆ+, ρˆ−) ≤ F (ρˆ′+,H , ρˆ′−,H) ≤
(
Problin.opt.fail
)2
. (B1)
In words, the fidelity of the linearly transformed and de-
phased output states is lower bounded by the fidelity of
the input states and upper bounded by the squared fail-
ure probability in the linear-optics implementation of un-
ambiguous state discrimination. The lower bound here
corresponds to the general rule that the fidelity of two
density matrices cannot decrease under CPTP maps [45].
As for the upper bound, we may note that in any scheme,
the linearly transformed and dephased output states take
on the form of Eq. (13) corresponding to a total dephas-
ing of the states in Eq. (12). Since the two density ma-
trices in Eq. (13) are diagonal in the Fock basis and com-
mute, we have the relation in Eq. (14). Now the failure
probability is given by Problin.opt.fail =
∑
m(P
+
m + P
−
m)/2.
However, we also have (P+m + P
−
m)/2 ≥
√
P+mP−m, ∀m,
thus proving the upper bound in Eq. (B1).
According to the fidelity bounds in Eq. (B1), we obtain
Eq. (10) as a necessary condition for the optimal USD
of two states via static linear optics and photon count-
ing, because optimal USD requires
(
Problin.opt.fail
)2
=
|〈χ+|χ−〉|2 = F (ρˆ+, ρˆ−).
One can now further exploit the fact that the bounds
in Eq. (B1) also hold for partially dephased density ma-
trices, corresponding to schemes that include conditional
dynamics. In particular, the upper bound in Eq. (B1)
holds for the partially dephased density matrices as well,
because in any mixed-state discrimination scheme, the
squared failure probability is lower bounded by the fi-
delity of the mixed states [32].
APPENDIX C: ALTERNATIVE DERIVATION OF
THE USD CONDITIONS FOR A FIXED ARRAY
Let us consider the optimal USD of two pure
nonorthogonal states using a fixed array of linear optics.
We will give an alternative derivation of the conditions
in Eq. (24) and Eq. (25), independent of the fidelity cri-
terion in Eq. (10).
After the linear-optics transformation, the output
states will always take on the form of Eq. (12), for con-
venience, written again here,
|χ+,H〉 =
∑
k
αk|{k} 〉+
∑
m
αm|{m} 〉,
|χ−,H〉 =
∑
l
βl|{l} 〉+
∑
m
βm|{m} 〉 . (C1)
The patterns labeled by k and l are those that unam-
biguously refer to the + state and to the − state, respec-
tively. Because of the finite overlap of the input states,
we must include patterns that occur in the expansion of
both states. These ambiguous patterns are denoted by
the index m. In general, the amplitudes of the ambigu-
ous N -mode Fock states in the expansions, and hence
the probabilities for the corresponding patterns to be de-
tected, may be different for the + and the − state. In

12
the following, we will first prove that in any optimal USD
scheme, the modulus of the amplitudes of any failure pat-
tern must indeed be equal for both states. Further, we
will show that for optimal USD, any relative phases in
the expansion of the failure patterns are reduced to a
single global phase. As a result, the output states after
linear optics in optimal USD must be describable in a
three-dimensional vector space such that
|χ+,H〉 = |ψ+concl〉+ |ψinconcl〉,
|χ−,H〉 = |ψ−concl〉+ eiφ |ψinconcl〉 . (C2)
Here the states |ψ+concl〉, |ψ−concl〉, and |ψinconcl〉 are all mu-
tually orthogonal. They represent the vectors of all con-
clusive patterns for the + state, of those for the − state,
and the vector of all inconclusive patterns, respectively.
As for the proof, we exploit the fact that in opti-
mal USD, the failure probability equals the modulus of
the overlap of the states to be discriminated, Probfail =
|〈χ+|χ−〉| (assuming equal a priori probabilities). This
implies that a linear-optical implementation of optimal
USD must satisfy
Problin.opt.fail =
1
2
∑
m
(|αm|2 + |βm|2) != |〈χ+|χ−〉|
= |〈χ+,H |χ−,H〉| =
∣
∣
∣
∣
∣
∑
m
α∗mβm
∣
∣
∣
∣
∣
, (C3)
using Eq. (C1). The factor 1/2 in the first line of Eq. (C3)
corresponds to the a priori probabilities. Then, because
of |∑m α∗mβm| ≤
∑
m |α∗mβm|, we also have
Problin.opt.fail =
1
2
∑
m
(|αm|2 + |βm|2)
≤
∑
m
|α∗mβm| , (C4)
or,
∑
m
(|αm| − |βm|)2 ≤ 0 . (C5)
The last inequality proves that |αm| = |βm|, ∀m. More-
over, it implies that |
∑
m α∗mβm|
!=
∑
m |α∗mβm|, and
hence
∣
∣
∣
∣
∣
∑
m
|αm|2eiφm
∣
∣
∣
∣
∣
!=
∑
m
|αm|2 , (C6)
using βm = αmeiφm . However, Eq. (C6) can only be
satisfied for eiφm = eiφ, ∀m. This concludes the proof
of Eq. (C2). For the case of optimal USD, we can now
replace Eq. (C1) by
|χ+,H〉 =
∑
k
αk|{k} 〉+
∑
m
αm|{m} 〉,
|χ−,H〉 =
∑
l
βl|{l} 〉+ eiφ
∑
m
αm|{m} 〉 . (C7)
Let us now use this result in order to calculate the first-
order expression. Similar to Eq. (18), we obtain now
〈χ+|cˆ†j cˆj |χ−〉 = 〈χ+,H |aˆ
†
j aˆj|χ−,H〉 (C8)
= eiφ
∑
m
|αm|2〈{m} |aˆ†jaˆj |{m} 〉 .
Since analogous expressions can be found for all higher
orders, the same arguments as those in the discussion
after Eq. (18) apply here again. Thus, finally we obtain
the same hierarchies of necessary conditions as in Eq. (24)
and Eq. (25) for optimal USD using a fixed array of linear
optics.
For the special case of exact discrimination of two or-
thogonal states |χ+〉 and |χ−〉 via photon counting, the
linearly transformed states take on the form,
|χ+,H〉 =
∑
k
αk|{k} 〉,
|χ−,H〉 =
∑
l
βl|{l} 〉 . (C9)
Now there are no ambiguous patterns in the expansions.
Let us again examine the expression
〈χ+|cˆ†j cˆj |χ−〉 = 〈χ+,H |aˆ
†
j aˆj |χ−,H〉 . (C10)
According to Eq. (C9), the output states of the linear-
optics transformation in exact state discrimination must
satisfy 〈χ+,H |aˆ†j aˆj |χ−,H〉 = 0, because annihilating a
photon in the jth mode of the two states only yields
a nonzero overlap for coinciding patterns. Similarly, we
have
〈χ+|cˆ†j cˆ
†
j′ cˆ
†
j′′ · · · cˆj cˆj′ cˆj′′ · · · |χ−〉 = (C11)
〈χ+,H |aˆ†j aˆ
†
j′ aˆ
†
j′′ · · · aˆj aˆj′ aˆj′′ · · · |χ−,H〉 = 0 , ∀j, j′, j′′... ,
because annihilating a photon in the jth, j′th, j′′th, etc.,
mode of the two states also only yields a nonzero over-
lap for coinciding patterns. Thus, we end up having the
following set of conditions for exact state discrimination
〈χ+,H |aˆ†j aˆj |χ−,H〉 = 0 , ∀j , (C12)
〈χ+,H |aˆ†j aˆ
†
j′ aˆj aˆj′ |χ−,H〉 = 0 , ∀j, j′ ,
〈χ+,H |aˆ†j aˆ
†
j′ aˆ
†
j′′ aˆj aˆj′ aˆj′′ |χ−,H〉 = 0 , ∀j, j′, j′′ ,
... =
...
or, equivalently, as described in Eq. (3).
APPENDIX D: DERIVATION OF THE USD
CONDITIONS FOR CONDITIONAL DYNAMICS
We consider the optimal USD of two pure nonorthogo-
nal states via linear optics including conditional dynam-
ics. Let us assume, without loss of generality, that mode

13
1 is detected first, corresponding to a partial dephasing of
the states only with respect to that mode. Now instead
of writing the states after linear optics as in Eq. (12), we
use the following expressions,
|χ+,H〉 =
∑
k
αk|k〉1 ⊗ |γ˜+k 〉+
∑
m
αm|m〉1 ⊗ |γ˜+m〉,
|χ−,H〉 =
∑
l
βl|l〉1 ⊗ |γ˜−l 〉+
∑
m
βm|m〉1 ⊗ |γ˜−m〉 ,
(D1)
where this time, the states |k〉1 and |l〉1 represent those
number states of mode 1 which only occur in the ex-
pansion of the + and the − state, respectively. The
one-mode states |m〉1 lead to the ambiguous detection
events in mode 1. Finally, the states |γ˜+k 〉, etc., refer
to the corresponding conditional states of the remaining
modes (after normalization). Similarly, for the partially
dephased density operators, we obtain
ρˆ′+,H =
∑
k
P+k |k〉1〈k| ⊗ |γ˜+k 〉〈γ˜+k |
+
∑
m
P+m|m〉1〈m| ⊗ |γ˜+m〉〈γ˜+m|,
ρˆ′−,H =
∑
l
P−l |l〉1〈l| ⊗ |γ˜−l 〉〈γ˜−l |
+
∑
m
P−m|m〉1〈m| ⊗ |γ˜−m〉〈γ˜−m| . (D2)
Note that the partially dephased states are no longer di-
agonal in the Fock basis, i.e., the conditional density ma-
trices may contain off-diagonal terms. The corresponding
fidelities are now
F (ρˆ+, ρˆ−) = F (ρˆ+,H , ρˆ−,H) (D3)
=
∑
m,n
α∗mαnβmβ∗n 〈γ˜+m|γ˜−m〉〈γ˜+n |γ˜−n 〉∗ ,
and
F (ρˆ′+,H , ρˆ′−,H) =
(
∑
m
√
P+mP−m
∣
∣〈γ˜+m|γ˜−m〉
∣
∣
)2
. (D4)
Finally, we end up having the following condition due to
the fidelity criterion in Eq. (10),
∑
m,n
√
P+mP+nP−mP−n
∣
∣〈γ˜+m|γ˜−m〉
∣
∣
∣
∣〈γ˜+n |γ˜−n 〉
∣
∣
!=
∑
m,n
α∗mαnβmβ∗n〈γ˜+m|γ˜−m〉〈γ˜+n |γ˜−n 〉∗ , (D5)
or, in terms of the unnormalized conditional states,
∑
m,n
∣
∣〈γ+m|γ−m〉
∣
∣
∣
∣〈γ+n |γ−n 〉
∣
∣
!=
∑
m,n
〈γ+m|γ−m〉〈γ+n |γ−n 〉∗. (D6)
Now the only way to satisfy this condition is through
〈γ+m|γ−m〉
∣
∣〈γ+m|γ−m〉
∣
∣
!=
〈γ+n |γ−n 〉
∣
∣〈γ+n |γ−n 〉
∣
∣
, (D7)
for any nonzero overlaps labeled by m and n. In
other words, for any two inconclusive one-mode detec-
tion events, any non-vanishing overlaps of the conditional
states coming from the + signal and the − signal must
have equal phases. Finally, we can now again examine
the first-order condition of our criteria, however, here
only for the detected mode 1,
〈χ+|cˆ†1cˆ1|χ−〉 = 〈χ+,H |aˆ†1aˆ1|χ−,H〉 (D8)
=
∑
m
〈γ+m|γ−m〉 1〈m|aˆ†1aˆ1|m〉1 .
Similar expressions hold for the higher orders in mode
1. Note that the different orders here are evaluated only
for the first mode to be detected, corresponding to the
first step in a conditional-dynamics scheme. Of course,
in addition, one could calculate further expressions using
the conditional states of modes 2 through N in order to
derive more criteria for a conditional-dynamics protocol.
Here, we only focus on the first step in any conditional-
dynamics scheme, namely the detection of a first mode.
Using Eq. (D8) and Eq. (D7) (for any nonzero over-
laps), it becomes clear now that the hierarchies of con-
ditions necessary for optimal USD when detecting a first
mode j are simply the subset of the fixed-array conditions
in Eq. (24) and Eq. (25) referring to this one mode. This
subset of conditions is given in Eq. (36) and Eq. (37).
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