1. Introduction
The state of a quantum system is a mysterious object and has been the subject of much attention since the earliest days of quantum theory. We know that it provides a way of calculating the observed statistical properties of any desired observable but that it is not, itself, observable. This means that we cannot determine by observation the state of any single physical system. If we have some prior information, however, then we may be able to use this to determine, at least to some extent, the state. Consider, for example, a single photon that we know has been prepared with either horizontal or vertical polarization. A suitably oriented polarizing beam splitter can be used to transmit the photon if it is vertically polarized and reflect it if its polarization is horizontal. Determining the path of the photon by absorbing it with a suitable detector then determines the state to have been one of horizontal or vertical polarization.
Suppose, however, that we are told that our photon was prepared with either horizontal or with left-circular polarization. These quantum states of polarization are not orthogonal in that states of circular polarization are superpositions of those of both vertical and horizontal polarization:
If we subject our photon to the polarization measurement outlined in the preceding paragraph, then a left-circularly polarized photon will appear to be horizontally polarized with probability
1
/
2
and vertically polarized with the same probability.
The problem of discriminating between such states is fundamental to the quantum theory of communications [1, 2, 3, 4] and underlies the secrecy of the now well-reviewed science of quantum cryptography [5, 6, 7, 8, 9]. Indeed, we can use the connection between quantum state discrimination and quantum communications to motivate the problem of state discrimination. We suppose that two parties, conventionally named Alice and Bob, wish to communicate by using a quantum channel. To do this Alice selects from a given set of states,
∣
ψ
i
⟩
(or more generally mixed states with density operators
ρ
̂
i
) with a given set of probabilities
p
i
. The selected state is encoded in the preparation of a given physical system, such as photon polarization, and this is sent to Bob. Bob will know both the set of possible states and the associated preparation probabilities. His task is to determine, by means of a suitable measurement, the state selected by Alice and hence the intended message. This, then is the quantum state discrimination problem: how can we best discriminate among a known set of possible states
∣
ψ
i
⟩
, each having been prepared with a known probability
p
i
.
The quantum state discrimination problem, as posed here, has been the subject of active theoretical investigation for a long time [1, 2, 3, 10, 11, 12, 13, 14], but it is only comparatively recently that experiments have been performed, and most of these have been based on optics. There exist in the literature a number of reviews of and introductions to quantum state discrimination [4, 15, 16, 17, 18, 19, 20, 21]. Our purpose in preparing this review is twofold: first, to bring the rapidly developing field up to date and, second, to introduce the idea of state discrimination to a wider audience in optics. It seems especially appropriate to do this as it is in simple optical experiments that the ideas are most transparent and where most of the important practical developments have been made.
2. Generalized Measurements
Most of us are introduced to the idea of measurements in quantum theory in a manner that is, essentially, that formulated by von Neumann [22]. Each observable property
O
is associated with a Hermitian operator
O
̂
(or more precisely a self-adjoint one), the eigenvalues of which are the possible results of a measurement of
O
. If the eigenvalues and eigenvectors are
o
m
and
∣
o
m
⟩
, then we can write the operator
O
̂
in the diagonal form
If the system to be measured has been prepared in the state
∣
ψ
⟩
, then the probability that a measurement of
O
will give the result
o
m
is
Consider, for example, a measurement to determine whether the polarization of a single photon is horizontal or vertical. A suitable operator, corresponding to this measurement, would be
The probability that a measurement of this property on a photon prepared in the circularly polarized state
∣
L
⟩
will give the result
H
, corresponding to horizontal polarization, is
It is helpful, in what follows, to rewrite the above probabilities as the expectation value of an operator. In this way the probability that a measurement of optical polarization shows the photon to be horizontally polarized is
where
P
̂
H
=
∣
H
⟩
⟨
H
∣
, the projector onto the state
∣
H
⟩
, is the required operator. More generally, for our operator
O
̂
, the probability that a measurement gives the value
o
m
is
We note that the measured value, itself, makes no explicit appearance in this probability; it is not the eigenvalue but only the corresponding eigenvector that determines the form of the projector and hence the probability for the associated measurement outcome.
The projectors have four important mathematical properties, and it is helpful to list these:
-
The projectors are Hermitian operators,
P
̂
m
†
=
P
̂
m
. This property is associated with the fact that probabilities are, themselves, observable quantities.
-
They are positive operators, which means that
⟨
ψ
∣
P
̂
m
∣
ψ
⟩
≥
0
for all possible states
∣
ψ
⟩
. This reflects the fact that the expectation value of the projector is a probability and must, therefore, be positive or zero.
-
They are complete in that
∑
m
P
̂
m
=
1
̂
, so that the sum of the probabilities for all possible measurement outcomes is unity.
-
They are orthonormal in that
P
̂
m
P
̂
n
=
0
unless
m
=
n
. This property is sometimes associated with the fact that measurement outcomes must be distinct (you can only get one of them). This view is, as we shall see, not correct. You can indeed only get one outcome, but this does not require the orthonormality property.
The theory of generalized measurements can be formulated simply by dropping the final requirement. To see how this works, we introduce a set of probability operators
{
π
̂
m
}
, each of which we wish to associate with a measurement outcome such that the probability that our measurement gives the result labeled
m
is
We insist on the first three of the properties of the projectors, as these are required if we are to maintain the probability interpretation (
8), but we drop the final requirement so that our probability operators have the following properties:
-
The probability operators are Hermitian:
π
̂
m
†
=
π
̂
m
.
-
They are positive operators:
⟨
ψ
∣
π
̂
m
∣
ψ
⟩
≥
0
for all possible states
∣
ψ
⟩
.
-
They are complete:
∑
m
π
̂
m
=
1
̂
.
Note that Hermiticity follows from positivity, since if all expectation values are nonnegative,
⟨
ψ
∣
π
̂
m
∣
ψ
⟩
≥
0
for all possible states
∣
ψ
⟩
, then in particular they are all real numbers and
π
̂
m
must be Hermitian. The set of probability operators characterizing the possible outcomes of any generalized measurement is called a probability operator measure, usually abbreviated to POM [1, 4]. You will often find this set referred to as a positive operator-valued measure or POVM [23, 24]. If the latter name is used then the probability operators become elements of a positive operator-valued measure.
The differences between the projectors and more general probability operators are best appreciated by reference to some simple examples, and these will be given in the following sections. There are, however, some important and perhaps even surprising points, and it is sensible to emphasize these here. First, the three properties described above have a remarkable generality in that (i) any measurement can be described by the appropriate set of probability operators and (ii) any set of operators that satisfy the three properties correspond to a possible measurement [4, 23]. This means that we can seek the optimum measurement in any given situation mathematically, by searching among all sets of operators that satisfy the required properties. Having found this optimum measurement, we know that a physical realization of it will exist, and we can seek a way to implement it in the laboratory. The second point to emphasize is that the number of (orthogonal) projectors can only be less than or equal to the dimension of the state space. For optical polarization, for example, there are only two orthogonal polarizations, and the state space is therefore two dimensional. It follows that any von Neumann measurement of polarization can have only two outcomes. By dropping the requirement for orthogonality, we allow a generalized measurement to have any number of outcomes, so a generalized measurement of polarization can have three, four, or more different outcomes. Finally, a generalized measurement allows us to describe the simultaneous observation of incompatible observables, such as position and momentum or, in the context of quantum optics, orthogonal field quadratures [25, 26]. Perhaps the first reported generalized optical measurement was of precisely this form [27, 28].
3. State Discrimination—Theory
3.1. Minimum Error Discrimination
In quantum state discrimination we wish to design a measurement to distinguish optimally between a given set of states. As we have seen in Section 2, any physically realizable measurement can be described by a POM. Thus, by mathematically formulating a figure of merit describing the performance of a measurement, we can search for the set of probability operators describing the optimal measurement. There are several possible figures of merit, each one corresponding to a different strategy. Possibly the simplest criteria that may be applied is to minimize the probability of making an error in identifying the state. We begin with the special case where the state is known to be one of two possible pure states,
∣
ψ
0
⟩
,
∣
ψ
1
⟩
, with associated probabilities
p
0
,
p
1
=
1
−
p
0
. If outcome 0 (associated with the probability operator
π
̂
0
) is taken to indicate that the state was
∣
ψ
0
⟩
, and outcome 1 (associated with
π
̂
1
) is taken to indicate that the state was
∣
ψ
1
⟩
, the probability of making an error in determining the state is given by
where in the last line we have used the completeness condition
π
̂
0
+
π
̂
1
=
1
̂
. This expression takes its minimum value when the second term reaches a maximum, which in turn is achieved if
π
̂
0
is a projector onto the positive eigenket of the operator
p
0
∣
ψ
0
⟩
⟨
ψ
0
∣
−
p
1
∣
ψ
1
⟩
⟨
ψ
1
∣
. Note that two pure states define a two-dimensional space, and without loss of generality we can choose an orthogonal basis
{
|
0
〉
,
|
1
〉
}
of this space such that the components of each state in this basis are real. Thus we can express
∣
ψ
0
⟩
,
∣
ψ
1
⟩
as follows:
and the eigenvalues of
p
0
∣
ψ
0
⟩
⟨
ψ
0
∣
−
p
1
∣
ψ
1
⟩
⟨
ψ
1
∣
can be calculated directly as
The minimum probability of making an error is then given by the so-called Helstrom bound[
1],
and the optimal measurement is simply a von Neumann measurement. In particular, for
p
0
=
p
1
=
1
∕
2
, the optimal measurement is a projective measurement onto the states
These are symmetrically located about the signal states, as may be expected from the symmetry of the problem. As
p
0
is increased,
∣
φ
0
⟩
moves closer to
∣
ψ
0
⟩
, and the optimal measurement becomes biased toward making fewer errors when the more probable state is sent (see Fig.
1). Finally, as may be expected intuitively, if
p
0
is much bigger than
p
1
, the optimal measurement is very close to simply asking, “Is the state
∣
ψ
0
⟩
or not?”
3.1a. Minimum Error Conditions
The above analysis is easily extended to two mixed states
ρ
̂
0
,
ρ
̂
1
, in which case the optimal measurement becomes a projective measurement onto the subspaces corresponding to positive and negative eigenvalues of
p
0
ρ
̂
0
−
p
1
ρ
̂
1
. In the general case of
N
possible states
{
ρ
̂
i
}
with associated a priori probabilities
{
p
i
}
, the aim is to minimize the expression
or, equivalently, to maximize
The optimal measurement is known only in certain special cases; however, necessary and sufficient conditions that must be satisfied by the optimal POM for the general case are known [
1,
12,
13] and are given in Eqs. (
16,
17). For simplicity, we prove only sufficiency of the conditions here, but we note that there is also a straightforward proof of their necessity [
29].
If
{
π
̂
i
}
corresponds to an optimal measurement, then for all other POMs
{
π
̂
i
′
}
we require
Inserting the identity
∑
j
π
̂
j
′
=
1
̂
gives
Note that
π
̂
j
′
≥
0
; thus the above holds if Eq. (
16) holds, which is therefore a sufficient condition.
For any POM satisfying this condition, it follows that the operator
Γ
̂
=
∑
i
p
i
ρ
̂
i
π
̂
i
is positive, and therefore Hermitian. Thus we have
where we have used the fact that the probability operators form a resolution of the identity
∑
i
π
̂
i
=
1
̂
. As both
∑
i
p
i
π
̂
i
ρ
̂
i
−
p
j
ρ
̂
j
and
π
̂
j
are positive operators, each term in the sum over
j
must be identically zero. Using similar reasoning we can argue that each term in the sum over
i
must be identically zero. Thus, in terms of
Γ
̂
we obtain
Eliminating
Γ
̂
gives Eq. (
17), which is therefore also a sufficient condition.
3.1b. Square-Root Measurement
For any given set of states we can construct an associated measurement, the square-root measurement [30, 31, 32, 33], as follows:
where
ρ
̂
=
∑
i
p
i
ρ
̂
i
. It is clear that the probability operators
{
π
̂
i
}
are positive and sum to the identity, and thus form a complete measurement. For many of the cases in which the optimal minimum error measurement is known, it is the square-root measurement [
34,
35,
36,
37,
38,
39]. We will present here the example of
N
symmetric pure states occurring with equal a priori probabilities
p
i
=
1
∕
N
, considered by Ban
et al.[
34], and given by
for some unitary operator
V
̂
satisfying
V
̂
N
=
1
̂
. For this set we have
and it is useful to note that
where in the last line we have used the property
V
̂
N
=
1
̂
. Thus
and
ρ
̂
commutes with
V
̂
. The square-root measurement consists of the operators
and condition (
17) is equivalent to the requirement
Noting that
we see that this requirement is satisfied. We now proceed to evaluate
Γ
̂
:
To satisfy condition (
16) we require
Writing
Γ
̂
=
(
1
∕
N
)
⟨
ψ
i
∣
ρ
̂
−
1
∕
2
∣
ψ
i
⟩
ρ
̂
1
∕
2
,
we can show that
where we have used the Cauchy–Schwarz inequality. Thus condition (
16) holds, and the square-root measurement is optimal. Note that the case of two equiprobable pure states discussed above is an example of a symmetric set. In this case
V
̂
=
σ
̂
z
, and it may easily be verified that
σ
̂
z
∣
ψ
0
⟩
=
∣
ψ
1
⟩
and
σ
̂
z
2
=
1
̂
. Another example of a symmetric set is the so-called trine ensemble [
32,
40], given by
and obtained from one another by rotation through an angle of
2
π
∕
3
. These states form a resolution of the identity, and the square-root measurement consists of equally weighted projectors onto the states themselves,
π
̂
i
=
2
3
∣
ψ
i
⟩
⟨
ψ
i
∣
.
The above solution has been extended to multiply symmetric states [37] and mixed states [38, 39]. The square-root measurement has also been generalized by Mochon [41], who considered measurements of the form
where
σ
̂
=
∑
i
p
i
′
ρ
̂
i
, i.e., the square-root measurements corresponding to the same set of states but constructed using different a priori probabilities. For pure states, each such measurement is optimal for at least one discrimination problem with the same states, occurring with probabilities given analytically in [
41].
3.1c. Other Results
Most of the known results for minimum error discrimination correspond to one of the two cases discussed above: that of just two states, or those for which the square-root measurement is optimal. Another example that is interesting to note is the no-measurement strategy [42]. Sometimes the optimal discrimination strategy is not to measure at all, but just to guess the state which is a priori most likely, a measurement that may be described by the POM
{
π
̂
i
=
1
̂
,
π
̂
j
=
0
,
∀
j
≠
i
}
, where
i
is such that
p
i
≥
p
j
,
∀
j
. Condition (17) holds trivially for this POM. Thus the no-measurement solution is optimal when condition (16) holds, which then reads as
Clearly this is never optimal if
ρ
̂
i
is pure; a necessary (but not sufficient) condition is that the eigenvectors of
ρ
̂
i
span the entire Hilbert space in which the states
{
ρ
̂
j
}
lie. A practical example is discriminating signal states from random noise, described by the density operator
ρ
̂
i
∝
1
̂
. If the signal-to-noise ratio is small enough, the minimum error strategy is to always guess that the state received was random noise [
42]. It is therefore useful to know the noise threshold at which this occurs, which may be deduced from condition (
35).
Other examples for which explicit results are known include three mirror symmetric qubit states, for both pure [43] and mixed states [44], and the case of equiprobable pure states, a weighted sum of which equals the identity operator [13]. The form of the solution for any set of qubit states has also been explored in some detail by Hunter [45, 46], including a complete characterization of the solution for equiprobable pure qubit states. In the general case, for which explicit results are not known, it is possible to deduce both upper [47, 48] and lower [49, 50] bounds on the error probability. Alternatively, numerical algorithms exist that can find the optimal measurement for a specified set of states to within any desired accuracy [51, 52].
3.2. Unambiguous Discrimination
In the minimum error measurement, each possible outcome is taken to indicate some corresponding state. It is perhaps surprising that it is sometimes advantageous to allow for measurement outcomes that do not lead us to identify any state. Suppose again that we wish to discriminate between the two pure states given by Eq. (10), occurring with a priori probabilities
p
0
,
p
1
. Consider the von Neumann measurement
If outcome ?, associated with the probability operator
π
̂
?
, is realized, we cannot say for sure what state was prepared. However, note that
⟨
ψ
1
∣
π
̂
0
∣
ψ
1
⟩
=
0
, and thus when outcome 0, corresponding to POM element
π
̂
0
, is realized, we can say for certain that the state was
∣
ψ
0
⟩
. Thus, by allowing for measurement outcome ?, which does not lead us to identify any state, we can construct a measurement that sometimes allows us to determine unambiguously which state was prepared. This measurement, however, only ever identifies the state
∣
ψ
0
⟩
; ideally we would like to design a measurement that can identify either state unambiguously, at the cost of sometimes giving an inconclusive result. The generalized measurement formalism outlined above allows for exactly such a measurement, a possibility that was first pointed out by Ivanovic [
53], Dieks [
54], and Peres [
55].
Consider therefore the operators
chosen such that
⟨
ψ
0
∣
π
̂
1
∣
ψ
0
⟩
=
⟨
ψ
1
∣
π
̂
0
∣
ψ
1
⟩
=
0
, and where
0
≤
a
0
,
a
1
≤
1
. Thus when outcome 0 is realized, we can say for sure that the corresponding state was
∣
ψ
0
⟩
, while when outcome 1 occurs, we know the state was
∣
ψ
1
⟩
with certainty. Note that these cannot form a complete measurement for any choice of
a
0
,
a
1
, unless
∣
ψ
0
⟩
,
∣
ψ
1
⟩
are orthogonal, and thus an inconclusive outcome is needed, associated with the probability operator
The probability of occurrence of the inconclusive result is given by
and the unambiguous discrimination strategy may be further optimized by minimizing this probability, subject to the constraints
a
0
,
a
1
≥
0
,
π
̂
?
≥
0
. For equal a priori probabilities,
p
0
=
p
1
=
1
2
, the minimum value or Ivanovic–Dieks–Peres (IDP) limit [
53,
54,
55] is given by
P
(
?
)
=
cos
2
θ
=
∣
⟨
ψ
0
∣
ψ
1
⟩
∣
and is achieved by the measurement
The optimal
P
(
?
)
for arbitrary prior probabilities was first given by Jaeger and Shimony [
56]. As
p
0
is increased, the optimal measurement is given by Eqs. (
37,
38) with
giving
P
(
?
)
=
2
p
0
p
1
cos
2
θ
. Thus the measurement becomes biased toward unambiguously identifying the state that is a priori more probable. Clearly, when
p
0
∕
p
1
cos
2
θ
>
1
, this no longer defines a physical measurement; the optimal measurement then is simply the von Neumann measurement given by Eq. (
36). In this case
∣
ψ
1
⟩
always gives the inconclusive result, and the probability of failure is
P
(
?
)
=
p
0
∣
⟨
ψ
0
∣
ψ
1
⟩
∣
2
+
p
1
. Thus for
p
0
much bigger than
p
1
, the optimal strategy is the one that rules out the less probable state, in contrast to the minimum error measurement, which in this regime (approximately) identifies or rules out the more probable state.
A simple example from quantum optics might help to illustrate the main idea [57]. Let us suppose that we have an optical pulse known to have been prepared, with equal probability, in one of the two coherent states [58]
∣
α
⟩
or
∣
−
α
⟩
. If we interfere the pulse with a second pulse prepared in the state
∣
i
α
⟩
by using a 50:50 symmetric beam splitter, then one of the output modes will be left in its vacuum state
|
0
〉
:
The state can be identified simply by detecting the light in the associated output mode. The ambiguous outcome is a consequence of the fact that the coherent states have a nonzero overlap with the vacuum state, and the probability for this result is
which is the Ivanovic–Dieks–Peres (IDP) limit.
3.2a.
N
>
2
Pure States
In the general case of discriminating unambiguously between
N
pure states
{
∣
ψ
i
⟩
}
,
i
=
0
,
…
,
N
−
1
, we wish to find probability operators
{
π
̂
i
}
such that
where
0
≤
P
i
≤
1
. Thus outcome
j
is obtained only if the state is
∣
ψ
j
⟩
, in which case it occurs with probability
P
j
. We first note that this is only possible if the states
{
∣
ψ
i
⟩
}
are linearly independent, as was shown by Chefles [
59]. When this is the case, we can construct states
∣
ψ
j
⊥
⟩
such that
i.e.,
∣
ψ
j
⊥
⟩
is orthogonal to all allowed states except
∣
ψ
j
⟩
. The POM elements
thus satisfy Eq. (
44) and unambiguously discriminate between the linearly independent states
{
∣
ψ
i
⟩
}
. As before, an inconclusive outcome is necessary to form a complete measurement
The above defines the unambiguous discrimination strategy for
N
linearly independent states. The occurrence of outcome
j
indicates unambiguously that the state was
∣
ψ
j
⟩
. As in the two-state case, a further condition that may be applied is to minimize the probability of obtaining an inconclusive result. Analytical solutions for the minimum achievable
P
(
?
)
are not known in the general case, but the solution for three states is given by Peres and Terno [
60], who also discuss how the method used can be extended to higher dimensions. For the special case in which the probability of unambiguously identifying a state
∣
ψ
j
⟩
is the same for all
j
(
P
j
=
P
,
∀
j
)
, the minimum probability of obtaining an inconclusive result is known [
59]. Further, the optimal strategy minimizing this probability is given for
N
linearly independent symmetric states in [
61]. For the general case, upper [
62] and lower bounds [
63,
64] have been given for the probability of successful unambiguous discrimination of
N
linearly independent states, and numerical optimization techniques have also been considered [
64,
65].
3.2b. Mixed States
It is only relatively recently that unambiguous discrimination has been extended to mixed states [66], where it may be applied to problems such as quantum state comparison [66, 67], subset discrimination [68], and determining whether a given state is pure or mixed [69]. Consider the problem of discriminating between two mixed states
ρ
̂
0
,
ρ
̂
1
, which may be written in terms of their eigenvalues and eigenvectors as follows:
where
0
<
λ
i
(
j
)
≤
1
. Define the projectors
such that
Λ
̂
ker
(
0
)
ρ
̂
0
=
Λ
̂
ker
(
1
)
ρ
̂
1
=
0
. These are the projectors onto the kernels of
ρ
̂
0
and
ρ
̂
1
, respectively [
70]. If we now define
π
̂
1
to lie in the kernel of
ρ
̂
0
, then
π
̂
1
=
Λ
̂
ker
(
0
)
π
̂
1
Λ
̂
ker
(
0
)
, and clearly
Thus if there exists a positive operator
π
̂
1
in the kernel of
ρ
̂
0
for which
Tr
(
ρ
̂
1
π
̂
1
)
≠
0
, then
ρ
̂
1
may be unambiguously discriminated from
ρ
̂
0
. Similarly,
π
̂
0
should lie in the kernel of
ρ
̂
1
. Thus a necessary and sufficient condition for unambiguous discrimination between two mixed states is that they have nonidentical kernels, and thus nonidentical supports [
66]. Unless the states are orthogonal, an inconclusive outcome will be needed, as before,
π
̂
?
=
1
̂
−
π
̂
0
−
π
̂
1
. The problem of finding the strategy that minimizes the probability of occurrence of the inconclusive result is again a difficult one, and one that has received much attention in the past few years. The solutions for some special cases are known; some examples are when both states have one-dimensional kernels [
66], unambiguous discrimination between a pure and a mixed state, first in two dimensions [
71] and later extended to
N
dimensions [
72]; other examples may be found in [
73,
74,
75]. Reduction theorems given in [
76] show that it is always possible to reduce the general problem to one of discriminating two states each of rank
r
, which together span a
2
r
-dimensional space. Thus the simplest case that is not reducible to pure state discrimination is the problem of two rank-2 density operators in a four-dimensional space, which was recently analyzed in detail by Kleinmann
et al.[
77]. Upper and lower bounds for the general case are given in [
66,
78,
79], a further reduction theorem in [
74], and numerical algorithms are discussed in [
80].
3.3. Maximum Confidence Measurements
As pointed out in the previous section, unambiguous discrimination is possible only when the allowed states are all linearly independent. If this is not the case, there will always be errors associated with identifying some states, even if an inconclusive outcome is allowed. Nevertheless, for more general sets of states we can construct an analogous measurement, one that allows us to be as confident as possible that when the outcome of measurement leads us to identify a given state
ρ
̂
i
, that was indeed the state prepared [81]. Just as with unambiguous discrimination, this measurement is concerned with optimizing the information given about the state by particular measurement outcomes, specifically the posterior probabilities
Physically, in many runs of an experiment, this probability refers to the proportion of occurrences of outcome
i
that were due to state
ρ
̂
i
. In a single-shot measurement, this therefore corresponds to the probability that it was state
ρ
̂
i
that gave rise to outcome
i
. Thus, we can think of this quantity as our confidence in taking outcome
i
to indicate state
ρ
̂
i
. In terms of the probability operator
π
̂
i
associated with outcome
i
, we can write
where
ρ
̂
=
∑
j
p
j
ρ
̂
j
is the a priori density operator. We note that
π
̂
i
appears in both the numerator and the denominator of this expression and thus can be determined only up to a multiplicative constant. It is always possible, therefore, to choose the overall normalization such that
and a physically realizable measurement may be constructed by adding an inconclusive result. Thus the only constraint we need worry about is that
π
̂
i
≥
0
. Optimization of this figure of merit is greatly facilitated by the use of the ansatz
where, by construction,
π
̂
i
≥
0
if
Q
̂
i
≥
0
. With this, Eq. (
52) becomes
where we have used the cyclical property of the trace. Note that
Q
̂
i
∕
Tr
(
Q
̂
i
)
is a positive, trace one operator, and so has the mathematical properties of a density operator. The density operator that has largest overlap with any operator
A
̂
is simply a projector onto the largest eigenvector of
A
̂
(or any density operator in the eigensubspace corresponding to the largest eigenvalue if this is degenerate). For pure states the optimal probability operators are therefore given by
while for mixed states they may be written as
where
σ
̂
i
is any density operator lying in the eigensubspace of
ρ
̂
−
1
∕
2
p
i
ρ
̂
i
ρ
̂
−
1
∕
2
corresponding to its largest eigenvalue. Finally, the limit is given by
where
γ
max
(
A
̂
)
denotes the largest eigenvalue of
A
̂
.
The simplest nontrivial example of a set of linearly dependent states is that of three states in two dimensions. To illustrate this strategy we consider the problem of discriminating between the states
where
0
≤
θ
≤
π
∕
4
, occurring with equal a priori probabilities
p
i
=
1
∕
3
,
i
=
0
,
1
,
2
. These states are symmetrically located at the same latitude of the Bloch sphere, as shown in Fig.
2. The a priori density operator for this set is
and the maximum confidence POM elements may be readily calculated by using Eq. (
56). These have the form
π
̂
i
=
α
i
∣
φ
i
⟩
⟨
φ
i
∣
, where we have some freedom in choosing the constants
α
i
,
i
=
0
,
1
,
2
, and
These states correspond to reflections of the input states in the equatorial plane of the Bloch sphere, and are also shown in Fig.
2. It is not possible in general to choose
α
0
,
α
1
,
α
2
such that
{
π
̂
i
}
form a complete measurement, and thus an additional operator,
π
̂
?
=
1
̂
−
∑
i
π
̂
i
, associated with an inconclusive result, is needed. We may choose to complete the measurement by minimizing the probability of an inconclusive result:
As
P
(
?
)
is a monotonically decreasing function of
α
i
, the optimal values of these parameters lie on the boundary of the allowed domain, defined by the constraint
π
̂
?
≥
0
. It is straightforward to show that this leads us to choose
α
0
=
α
1
=
α
2
=
(
3
cos
2
θ
)
−
1
, giving
It is useful to compare this measurement with the minimum error (ME) measurement, which for this set is given by the square-root measurement discussed earlier:
where
In the Bloch sphere representation, these states correspond to the projection of the input states onto the equatorial plane, as can be seen in Fig.
2. The minimum error and maximum confidence figures of merit are shown for each measurement in Fig.
3. For the minimum error measurement, each outcome leads us to identify some state, and the average probability of making an error is minimized. However, the confidence in identifying a state may be increased by allowing for an inconclusive result, as may be seen from the plots. When a noninconclusive result is obtained in the maximum confidence measurement, the probability that the state prepared really was the one identified is
2
3
, compared with
1
3
(
1
+
sin
2
θ
)
for the minimum error measurement.
3.3a. Other Similar Strategies
A related strategy may be constructed by applying a worst-case optimality criterion to the conditional probability considered here,
P
(
ρ
̂
i
∣
i
)
[82]. This approach does not allow for inconclusive results, but searches for the measurement for which the smallest value of
P
(
ρ
̂
i
∣
i
)
is maximized. This more complicated problem is difficult to solve analytically, but may be cast as a quasi-convex optimization, for which efficient numerical techniques are available. An alternative strategy allows inconclusive results to occur with a certain fixed probability,
P
I
, and maximizes the probability of correctly identifying the state when a noninconclusive outcome is obtained. For linearly independent pure states this approach interpolates between minimum error and unambiguous discrimination [83, 84]. The performance of projective (von Neumann) measurements versus generalized measurements for strategies with both errors and inconclusive results is analyzed in [85]. Here the rate of inconclusive results is minimized for a bounded-error rate, and it is shown that as small, but experimentally realistic errors are allowed, the advantage of generalized measurements over von Neumann measurements is reduced for some sets of states. For arbitrary mixed states an approach that allows for both errors and inconclusive results is also possible [86] and may be interpreted as interpolating between a minimum error measurement and a maximum confidence strategy. It is clear that the probability of obtaining a correct result, renormalized over only the results that are not inconclusive, denoted
P
RC
, can never be larger than the largest value of
P
(
ρ
̂
i
∣
i
)
max
for a given set, regardless of how much we increase
P
I
. This upper bound is achieved by a maximum confidence strategy which only ever identifies the state(s)
ρ
̂
i
such that [from Eq. (58)]
while all other results are interpreted as inconclusive. Although it is difficult to find the optimal measurement for general
P
I
, it is indeed found that
P
RC
is saturated at some value of
P
I
, and the maximum
P
RC
achievable corresponds to the strategy outlined here [
86].
3.3b. Related Problems—Quantum State Filtration
Quantum state filtration refers to the problem of whether the state of a system is a given state
∣
ψ
i
⟩
or simply in any one of the other states in a given set
{
∣
ψ
j
⟩
}
,
j
≠
i
. This problem is less demanding than complete discrimination among all possible states, and in the minimum error approach the probability of error may be smaller in the state filtration case [87]. For the maximum confidence measurement, however, the optimality of the probability operator
π
̂
i
in Eq. (57) is independent of the number and interpretation of other possible outcomes. Thus the confidence in identifying a given state from a set cannot be increased by considering this simpler problem. This figure of merit is dependent only on the geometry of the set, and in this sense can be thought of as a measure of how distinguishable
ρ
̂
i
is in the given set.
3.4. Comments on the Relationships between Strategies
The maximum confidence strategy was introduced as an analogy to unambiguous discrimination for linearly dependent states [81]. In fact, unambiguous discrimination is a special case of maximum confidence discrimination. The maximum confidence measurement maximizes the conditional probability
P
(
ρ
̂
i
∣
i
)
. If this figure of merit is equal to unity for some state
ρ
̂
i
, the optimal measurement is such that, when outcome
i
is obtained, we can be absolutely certain that
ρ
̂
i
was in fact the state received, corresponding to unambiguous discrimination. We can use the maximum confidence formalism to investigate when unambiguous discrimination is possible. Equation (52) may be written as
Clearly the limit of unity may be achieved if there exists any projector
Λ
̂
i
for which
Λ
̂
i
∑
j
≠
i
p
j
ρ
̂
j
Λ
̂
i
=
0
while
Λ
̂
i
ρ
̂
i
Λ
̂
i
is nonzero.
π
̂
i
is then any operator lying in the subspace with projector
Λ
̂
i
. This reproduces the known results that unambiguous discrimination is possible between pure states if the states are linearly independent [
59] and between mixed states if they have distinct supports [
66]. More precisely, a measurement is possible that will sometimes allow us to identify
ρ
̂
i
unambiguously if
ρ
̂
i
has support in the kernel of
∑
j
≠
i
p
j
ρ
̂
j
. This condition is less restrictive than the previous, which does not hold in the case where it is possible to unambiguously discriminate some but not all states in a set. Unambiguous discrimination is still possible in this case, but some states are never identified. For example, it was pointed out by Sun
et al.[
71] that it is possible to apply unambiguous discrimination to the problem of determining whether a system is in a given state
∣
ψ
0
⟩
or in either of two other possible states,
∣
ψ
1
⟩
,
∣
ψ
2
⟩
, even if the states span only two dimensions and therefore are linearly dependent. This may be more easily understood as unambiguous discrimination between a mixed state and a pure state in two dimensions [
66]. Let
where
|
0
〉
,
|
1
〉
are the eigenkets of
ρ
1
,
0
<
q
<
1
, and without loss of generality we can write
∣
ψ
0
⟩
=
cos
θ
∣
0
⟩
+
sin
θ
∣
1
⟩
. It is clear that the von Neumann measurement
can unambiguously discriminate the two possibilities—if outcome 1 is obtained, we can say for sure that the state was
ρ
1
, while the result 0 is interpreted as inconclusive. However, this measurement never tells us if the state was
∣
ψ
0
⟩
. In this case it may be useful to consider unambiguous discrimination within the framework of maximum confidence measurements. It is then possible to construct a measurement that sometimes identifies
ρ
̂
1
with certainty, but also sometimes identifies
ρ
̂
0
as confidently as possible. In general an inconclusive result will also be necessary.
Now suppose that, instead of maximizing the conditional probability in Eq. (52) independently for each state in the set, we choose to maximize a weighted average of these probabilities. We would then obtain as our figure of merit
which is precisely the figure of merit maximized by the minimum error measurement. Thus these two strategies can be thought of as applying a different optimality condition to the same quantity. The minimum error measurement also has the additional constraint that the operators
{
π
̂
i
}
must form a complete measurement, as it is never optimal to allow inconclusive results to occur. This constraint makes finding the optimal measurement a difficult problem, although the conditions that the optimal measurement must satisfy are known, as we have shown. By contrast, the maximum confidence strategy allows a closed form solution for an arbitrary set of states. In the special case where the maximum confidence figure of merit is the same for all states
ρ
̂
i
and no inconclusive result is needed, the two strategies coincide. More generally, it is clear by examination of Eq. (
70) that an upper bound for the minimum error figure of merit is given by the largest value of
P
(
ρ
̂
i
∣
i
)
max
for a given set [i.e., the largest value of Eq. (
58)].
3.5. Mutual Information
In communications theory the performance of a communications channel is quantified not by an error probability but rather by the information conveyed. We can give a precise meaning to this by invoking Shannon’s noisy channel coding theorem [88, 89], which states that the maximum communications rate, or channel capacity, is obtained by maximizing the mutual information between the transmitter and receiver. If the transmitted message,
A
, is one of the set
{
a
i
}
and the reception event,
B
, is one of the set
{
b
j
}
, then the mutual information is defined to be
where the logarithm is usually taken to be base 2 so that the information is expressed in bits. For a quantum channel, the state
ρ
̂
i
is selected with probability
p
i
, and the measurement result
b
j
is associated with the probability operator
π
̂
j
. It follows that the mutual information is
where
ρ
̂
=
∑
i
p
i
ρ
̂
i
. The maximum value of the mutual information is found by varying both the preparation probabilities,
p
i
, and the measurement strategies. This is a very difficult optimization problem, and there are very few exact solutions known [
90,
91]. A scarcely simpler problem is to fix the preparation probabilities and then seek the maximum value to give what is referred to as the accessible information [
92].
For two pure states, it is known that the mutual information is maximized if the states are prepared with equal probability and if the minimum error measurement is employed [91]. For three or more states, the accessible information is known if the states are equally likely to be selected and possess a degree of symmetry. In particular, for the so-called trine ensemble of three equally probable states (33), the accessible information is obtained with a generalized measurement with probability operators
Note that the accessible information is obtained not by maximizing the probability for determining the state but rather for eliminating one of the states so that
A similar strategy works well for four equiprobable states arranged so as to form a regular tetrahedron on the Bloch or Poincaré sphere [
90]. For more states, optimal strategies have been demonstrated with fewer measurement outcomes than states [
92].
3.6. No-Signaling Bounds on State Discrimination
Until now we have discussed the limits on quantum state discrimination by mathematically formulating figures of merit that may then be evaluated and compared for any allowed measurement by virtue of the generalized measurement formalism. It is interesting to note, however, that it is possible to place tight bounds on state discrimination without any reference to generalized measurements by appealing to the no-signaling principle, the condition that information may not propagate faster than the speed of light.
Although entanglement appears to allow spacelike separated quantum systems to influence one another instantaneously, it may be shown that quantum mechanical correlations do not allow signaling [93, 94, 95, 96]. Further, owing to the implications of this in reconciling quantum mechanics with special relativity, it has been suggested that the no-signaling principle be given the status of a physical law, which may be used to limit quantum mechanics and possible extensions of it [97, 98]. In practice, bounds on the fidelity of quantum cloning machines [99, 100], the success probability of unambiguous discrimination [101, 102], and the maximum confidence figure of merit [103] have been derived by using no-signaling arguments. In particular, the no-signaling principle may be used to put a tight bound on unambiguous discrimination of two pure states [101] and to derive the maximum confidence strategy [103]. We will discuss these two cases here.
3.6a. Unambiguous Discrimination
Consider the entangled state
where
∣
ψ
0
⟩
L
,
∣
ψ
1
⟩
L
are nonorthogonal states of the left system [given by Eq. (
10)], and
∣
0
⟩
R
,
∣
1
⟩
R
form an orthonormal basis for the right system. The reduced density operator of the right system may be obtained by taking the partial trace over the left system and is given by
According to the no-signaling principle, no operation performed on the left system may be detected by measurement of the right system alone, as this could be used to signal faster than light. Thus, after any physically allowed transformation of the left system, the reduced density operator of the right system must remain the same. Consider now a measurement that discriminates unambiguously between the states
∣
ψ
0
⟩
L
,
∣
ψ
1
⟩
L
of the left system. If outcome 0 is realized, which occurs with some probability
q
0
, the right system is projected into the state
∣
0
⟩
R
, due to the initial entanglement between the systems. Similarly outcome 1 projects the right system into state
∣
1
⟩
R
, with probability
q
1
. There is also the inconclusive result, which transforms the right system to some as yet unknown state
with probability
q
?
. No signaling implies
The task is then to minimize
q
?
subject to the above condition and the conditions
ρ
̂
?
≥
0
,
q
0
,
q
1
,
q
?
≥
0
. This optimization is straightforward [
101] and, remarkably, gives precisely the Jaeger and Shimony result [
56] discussed in Subsection
3.2. Thus the no-signaling condition may be used to place a tight bound on the success probability of unambiguous discrimination, without any reference to generalized measurements.
3.6b. Maximum Confidence Measurements
The confidence in identifying a given state
∣
ψ
j
⟩
as a result of a state discrimination measurement on the ensemble
{
∣
ψ
i
⟩
,
p
i
}
is simply the probability that it was state
∣
ψ
j
⟩
that gave rise to the measurement outcome observed. Consider now the entangled state
where
{
∣
ψ
i
⟩
L
}
are nonorthogonal states of the left system which together span a
D
≤
N
-dimensional space, and
{
∣
i
⟩
R
}
forms an orthonormal basis for the right system. Now for any measurement performed on the left system of the entangled pair, the probability that it was state
∣
ψ
j
⟩
L
which gave rise to the observed outcome is equivalent to the probability that the right system is now found in state
∣
j
⟩
R
. Thus, if measurement outcome
j
causes the right system to transform to
ρ
̂
R
∣
j
, we can write
It may be shown (by reference to the Schmidt decomposition of
∣
Ψ
⟩
[
104]) that although the right system lies in an
N
-dimensional Hilbert space, it is confined to a
D
-dimensional subspace (with the projector denoted
P
̂
D
below) because of the entanglement with the left system. The key point, then, is to notice that any operation performed on the left system cannot take the right system out of this subspace, since this could be detected with some probability by a measurement on the right system alone, and thus could be used to signal. Thus
⟨
j
∣
R
ρ
̂
R
∣
j
∣
j
⟩
R
is restricted by the requirement that
ρ
̂
R
∣
j
lies in this subspace, and is clearly bounded by the magnitude of the projection of
∣
j
⟩
R
onto this space:
Further, this bound is achievable and is equivalent to that obtained previously [Eq. (
58)] [
103]. Similar arguments may be applied to the mixed state case, and the maximum confidence strategy is derived in a natural way from no-signaling considerations. Finally, we note that in the case where the states
{
∣
ψ
i
⟩
L
}
are linearly independent,
D
=
N
, and the right system occupies the entire
N
-dimensional Hilbert space. In this case the limit is unity, corresponding to unambiguous discrimination.
4. State Discrimination—Experiments
The theory of generalized measurements has a mathematically appealing generality in that it depends only on the overlaps of the possible states to be discriminated and on the probabilities that each was the state prepared. The nature of the physical states, be they nuclear spins, optical coherent states, or electronic energy levels in an atom, is unimportant. In performing experimental demonstrations, however, the choice of physical system is of primary importance. We require a physical system in which superpositions are relatively stable, easy to prepare and to manipulate, and also, of course, to measure. For all of these reasons, the system of choice has usually been photon polarization and forms the basis of our review.
4.1. Photon Polarization
At least within paraxial optics [105], the electric and magnetic fields are very nearly perpendicular to the direction of propagation of the light. It is conventional to define the polarization by the orientation of the electric field in this transverse plane [106]. Two orthogonal polarizations then correspond to fields in which the electric fields are oriented at 90° to each other. The polarization of a single photon is an excellent two-state quantum system, or qubit [4, 104], as we can identify the states of horizontal and vertical polarization with the logical
|
0
〉
and
|
1
〉
states of a qubit:
Other polarizations are superpositions of these states. In particular, as illustrated in Fig.
4, linear polarization at
±
45
°
to the horizontal and circular polarizations are the superpositions
The set of all possible pure states of polarization can be represented on the surface of a sphere, the Poincaré sphere [
107,
108], which is a representation equivalent to the Bloch sphere used for qubits in quantum information theory [
4,
104]. States of optical polarization can be changed coherently by delaying one polarization compared with the orthogonal polarization, usually by a quarter or half a wavelength, by using birefringent wave plates. A combination of three suitably oriented half- and quarter-wave plates can perform any desired transformation, corresponding to a rotation on the Poincaré sphere through any desired angle about any desired axis. In this way we can realize any desired single-qubit unitary transformation.
It is important, in order to realize generalized measurements, to be able to superpose fields and also to be able to spatially separate different polarizations. These tasks can be performed using beam splitters and polarizing beam splitters. For fully overlapping modes with the same frequency, we can write the output annihilation operators in terms of those for the input modes. For a symmetric polarization-independent beam splitter we find [58]
where the input and output modes are labeled as in Fig.
5.
Enforcing the canonical commutation relations for the output modes constrains the reflection and transmission coefficients:
A polarizing beam splitter is designed to transmit horizontally polarized light and to reflect vertically polarized light. This means that input and output annihilation operators are related by
In correlating photon polarization and direction, a polarizing beam splitter can be used to prepare (filter) light with a desired polarization or, in conjunction with photodetectors placed in each output beam, to measure the polarization. They also allow us to perform different transformations on two orthogonal polarizations, and this is crucial in enabling us to perform generalized measurements.
We should make one important point before describing any of the experiments that have been performed, and this is that they have not been done with single-photon sources. All of them rely on linear optical elements and processes, and for these the single-photon probability amplitudes and the associated probabilities behave in the same way as the amplitudes and intensities of classical optics. Some of the experiments have been performed at light levels in the quantum regime, however, and this suggests strongly that the devices will work in the same way given single-photon sources and detectors.
4.2. Minimum Error Discrimination
4.2a. Two States
The simplest minimum error problem is, as we have seen, that for two pure states, Eqs. (10). For the photon polarizations described above these correspond to two states of linear polarization, oriented at
+
θ
and
−
θ
to the horizontal, so that the angle between them is
2
θ
, for a range of values of
θ
between 0 and
π
∕
4
. If the two states are prepared with equal prior probability, then, as we have seen, the minimum error measurement corresponds to a familiar von Neumann, or projective, measurement with two projectors associated with the orthogonal states, Eqs. (13). For optical polarization, this corresponds to measuring the polarization at 45° to the horizontal. Thus the minimum error strategy in this case is a simple polarization measurement. The experiment to test this [109] was performed by using light pulses with on average 0.1 photons per pulse prepared in the desired polarization state by use of a Glan–Thompson polarizer oriented so as to produce polarized light at the angle
+
θ
or
−
θ
to the horizontal. These were then measured by using a polarizing beam splitter oriented so as to transmit light polarized at
+
45
°
to the horizontal and to reflect the orthogonal polarization. The experimental apparatus is shown in Fig. 6. Results, shown in Fig. 7, were found to be in excellent agreement with the Helstrom value (12) for equal prior probabilities:
4.2b. Three or Four States
Finding a minimum error strategy for discriminating between more than two states is, in general a difficult problem, although very general statements about the solution can be made for qubits [45]. For the trine ensemble of three equiprobable linear polarization states
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