## Abstract

It is a fundamental consequence of the superposition principle for quantum states that there must exist nonorthogonal states, that is, states that, although different, have a nonzero overlap. This finite overlap means that there is no way of determining with certainty in which of two such states a given physical system has been prepared. We review the various strategies that have been devised to discriminate optimally between nonorthogonal states and some of the optical experiments that have been performed to realize these.

© 2009 Optical Society of America

## 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:

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, $\mid {\psi}_{i}\u27e9$ (or more generally mixed states with density operators ${\widehat{\rho}}_{i}$) 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 $\mid {\psi}_{i}\u27e9$, 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 $\widehat{O}$ (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 $\mid {o}_{m}\u27e9$, then we can write the operator $\widehat{O}$ in the diagonal form

*O*will give the result ${o}_{m}$ isConsider, 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 beThe probability that a measurement of this property on a photon prepared in the circularly polarized state $\mid L\u27e9$ 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 ${\widehat{P}}_{H}=\mid H\u27e9\u27e8H\mid $, the projector onto the state $\mid H\u27e9$, is the required operator. More generally, for our operator $\widehat{O}$, the probability that a measurement gives the value ${o}_{m}$ isThe projectors have four important mathematical properties, and it is helpful to list these:

- The projectors are Hermitian operators, ${\widehat{P}}_{m}^{\u2020}={\widehat{P}}_{m}$. This property is associated with the fact that probabilities are, themselves, observable quantities.
- They are positive operators, which means that $\u27e8\psi \mid {\widehat{P}}_{m}\mid \psi \u27e9\ge 0$ for all possible states $\mid \psi \u27e9$. 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 ${\sum}_{m}{\widehat{P}}_{m}=\widehat{\mathbb{1}}$, so that the sum of the probabilities for all possible measurement outcomes is unity.
- They are orthonormal in that ${\widehat{P}}_{m}{\widehat{P}}_{n}=0$ 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 $\left\{{\widehat{\pi}}_{m}\right\}$, each of which we wish to associate with a measurement outcome such
that the probability that our measurement gives the result labeled *m*
is

- The probability operators are Hermitian: ${\widehat{\pi}}_{m}^{\u2020}={\widehat{\pi}}_{m}$.
- They are positive operators: $\u27e8\psi \mid {\widehat{\pi}}_{m}\mid \psi \u27e9\ge 0$ for all possible states $\mid \psi \u27e9$.
- They are complete: ${\sum}_{m}{\widehat{\pi}}_{m}=\widehat{\mathbb{1}}$.

Note that Hermiticity follows from positivity, since if all expectation values are nonnegative, $\u27e8\psi \mid {\widehat{\pi}}_{m}\mid \psi \u27e9\ge 0$ for all possible states $\mid \psi \u27e9$, then in particular they are all real numbers and ${\widehat{\pi}}_{m}$ 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, $\mid {\psi}_{0}\u27e9$, $\mid {\psi}_{1}\u27e9$, with associated probabilities ${p}_{0}$, ${p}_{1}=1-{p}_{0}$. If outcome 0 (associated with the probability operator ${\widehat{\pi}}_{0}$) is taken to indicate that the state was $\mid {\psi}_{0}\u27e9$, and outcome 1 (associated with ${\widehat{\pi}}_{1}$) is taken to indicate that the state was $\mid {\psi}_{1}\u27e9$, the probability of making an error in determining the state is given by

*Helstrom bound*[1],

### 3.1a. Minimum Error Conditions

The above analysis is easily extended to two mixed states ${\widehat{\rho}}_{0}$, ${\widehat{\rho}}_{1}$, in which case the optimal measurement becomes a projective
measurement onto the subspaces corresponding to positive and negative eigenvalues
of ${p}_{0}{\widehat{\rho}}_{0}-{p}_{1}{\widehat{\rho}}_{1}$. In the general case of *N* possible states $\left\{{\widehat{\rho}}_{i}\right\}$ with associated *a priori* probabilities $\left\{{p}_{i}\right\}$, the aim is to minimize the expression

Necessary and sufficient conditions that must be satisfied by the POM achieving minimum error in distinguishing between the states $\left\{{\widehat{\rho}}_{i}\right\}$, occurring with probabilities $\left\{{p}_{i}\right\}$, are given by

Note that these conditions are not independent; the second may be derived from the first, as shown in the text.If $\left\{{\widehat{\pi}}_{i}\right\}$ corresponds to an optimal measurement, then for all other POMs $\left\{{\widehat{\pi}}_{i}^{\prime}\right\}$ we require

For any POM satisfying this condition, it follows that the operator $\widehat{\Gamma}={\sum}_{i}{p}_{i}{\widehat{\rho}}_{i}{\widehat{\pi}}_{i}$ is positive, and therefore Hermitian. Thus we have

*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 $\widehat{\Gamma}$ we obtain

### 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:

*N*symmetric pure states occurring with equal

*a priori*probabilities ${p}_{i}=1\u2215N$, considered by Ban

*et al.*[34], and given by

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

*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 $\{{\widehat{\pi}}_{i}=\widehat{\mathbb{1}},{\widehat{\pi}}_{j}=0,\forall j\ne i\}$, where *i* is such that ${p}_{i}\ge {p}_{j},\forall j$. Condition (17)
holds trivially for this POM. Thus the no-measurement solution is optimal when
condition (16) holds, which then
reads as

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

Consider therefore the operators

*a priori*probabilities, ${p}_{0}={p}_{1}={\scriptstyle \frac{1}{2}}$, the minimum value or Ivanovic–Dieks–Peres (IDP) limit [53, 54, 55] is given by $P(?)=\mathrm{cos}\phantom{\rule{0.2em}{0ex}}2\theta =\mid \u27e8{\psi}_{0}\mid {\psi}_{1}\u27e9\mid $ and is achieved by the measurement

*a priori*more probable. Clearly, when $\sqrt{{p}_{0}\u2215{p}_{1}}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}2\theta >1$, this no longer defines a physical measurement; the optimal measurement then is simply the von Neumann measurement given by Eq. (36). In this case $\mid {\psi}_{1}\u27e9$ always gives the inconclusive result, and the probability of failure is $P(?)={p}_{0}{\mid \u27e8{\psi}_{0}\mid {\psi}_{1}\u27e9\mid}^{2}+{p}_{1}$. 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] $\mid \alpha \u27e9$ or $\mid -\alpha \u27e9$. If we interfere the pulse with a second pulse prepared in the state $\mid i\alpha \u27e9$ by using a 50:50 symmetric beam splitter, then one of the output modes will be left in its vacuum state $|0\u3009$:

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### 3.2a. $N>2$ Pure States

In the general case of discriminating unambiguously between *N*
pure states $\left\{\mid {\psi}_{i}\u27e9\right\}$, $i=0,\dots ,N-1$, we wish to find probability operators $\left\{{\widehat{\pi}}_{i}\right\}$ such that

*j*is obtained only if the state is $\mid {\psi}_{j}\u27e9$, in which case it occurs with probability ${P}_{j}$. We first note that this is only possible if the states $\left\{\mid {\psi}_{i}\u27e9\right\}$ are linearly independent, as was shown by Chefles [59]. When this is the case, we can construct states $\mid {\psi}_{j}^{\perp}\u27e9$ such that

*N*linearly independent states. The occurrence of outcome

*j*indicates unambiguously that the state was $\mid {\psi}_{j}\u27e9$. 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 $\mid {\psi}_{j}\u27e9$ is the same for all

*j*$({P}_{j}=P,\forall j)$, 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 ${\widehat{\rho}}_{0}$, ${\widehat{\rho}}_{1}$, which may be written in terms of their eigenvalues and eigenvectors as follows:

*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 $2r$-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 ${\widehat{\rho}}_{i}$, 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

*i*that were due to state ${\widehat{\rho}}_{i}$. In a single-shot measurement, this therefore corresponds to the probability that it was state ${\widehat{\rho}}_{i}$ that gave rise to outcome

*i*. Thus, we can think of this quantity as our confidence in taking outcome

*i*to indicate state ${\widehat{\rho}}_{i}$. In terms of the probability operator ${\widehat{\pi}}_{i}$ associated with outcome

*i*, we can write

*a priori*density operator. We note that ${\widehat{\pi}}_{i}$ 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 thatand a physically realizable measurement may be constructed by adding an inconclusive result. Thus the only constraint we need worry about is that ${\widehat{\pi}}_{i}\ge 0$. Optimization of this figure of merit is greatly facilitated by the use of the ansatzwhere, by construction, ${\widehat{\pi}}_{i}\ge 0$ if ${\widehat{Q}}_{i}\ge 0$. With this, Eq. (52) becomes

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

*a priori*probabilities ${p}_{i}=1\u22153$, $i=0,1,2$. 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

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:

### 3.3a. Other Similar Strategies

A related strategy may be constructed by applying a worst-case optimality criterion to the conditional probability considered here, $P({\widehat{\rho}}_{i}\mid i)$ [82]. This approach does not allow for inconclusive results, but searches for the measurement for which the smallest value of $P({\widehat{\rho}}_{i}\mid i)$ 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 ${P}_{\mathrm{RC}}$, can never be larger than the largest value of $P{({\widehat{\rho}}_{i}\mid i)}_{\mathrm{max}}$ 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) ${\widehat{\rho}}_{i}$ such that [from Eq. (58)]

### 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 $\mid {\psi}_{i}\u27e9$ or simply in any one of the other states in a given set $\left\{\mid {\psi}_{j}\u27e9\right\}$, $j\ne i$. 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 ${\widehat{\pi}}_{i}$ 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 ${\widehat{\rho}}_{i}$ 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 $P({\widehat{\rho}}_{i}\mid i)$. If this figure of merit is equal to unity for some state ${\widehat{\rho}}_{i}$, the optimal measurement is such that, when outcome
*i* is obtained, we can be absolutely certain that ${\widehat{\rho}}_{i}$ 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

*et al.*[71] that it is possible to apply unambiguous discrimination to the problem of determining whether a system is in a given state $\mid {\psi}_{0}\u27e9$ or in either of two other possible states, $\mid {\psi}_{1}\u27e9$, $\mid {\psi}_{2}\u27e9$, 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]. Letwhere $|0\u3009$, $|1\u3009$ are the eigenkets of ${\rho}_{1}$, $0<q<1$, and without loss of generality we can write $\mid {\psi}_{0}\u27e9=\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta \mid 0\u27e9+\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \mid 1\u27e9$. 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 ${\rho}_{1}$, while the result 0 is interpreted as inconclusive. However, this measurement never tells us if the state was $\mid {\psi}_{0}\u27e9$. 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 ${\widehat{\rho}}_{1}$ with certainty, but also sometimes identifies ${\widehat{\rho}}_{0}$ 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

#### 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 $\left\{{a}_{i}\right\}$ and the reception event, *B*, is one of the set $\left\{{b}_{j}\right\}$, then the mutual information is defined to be

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

**eliminating**one of the states so thatA 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

### 3.6b. Maximum Confidence Measurements

The confidence in identifying a given state $\mid {\psi}_{j}\u27e9$ as a result of a state discrimination measurement on the ensemble $\{\mid {\psi}_{i}\u27e9,{p}_{i}\}$ is simply the probability that it was state $\mid {\psi}_{j}\u27e9$ that gave rise to the measurement outcome observed. Consider now the entangled state

where $\left\{{\mid {\psi}_{i}\u27e9}_{L}\right\}$ are nonorthogonal states of the left system which together span a $D\le N$-dimensional space, and $\left\{{\mid i\u27e9}_{R}\right\}$ 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 ${\mid {\psi}_{j}\u27e9}_{L}$ which gave rise to the observed outcome is equivalent to the probability that the right system is now found in state ${\mid j\u27e9}_{R}$. Thus, if measurement outcome*j*causes the right system to transform to ${\widehat{\rho}}_{R\mid j}$, we can writeIt may be shown (by reference to the Schmidt decomposition of $\mid \Psi \u27e9$ [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 ${\widehat{P}}_{D}$ 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 ${}_{R}\u27e8j\mid {\widehat{\rho}}_{R\mid j}{\mid j\u27e9}_{R}$ is restricted by the requirement that ${\widehat{\rho}}_{R\mid j}$ lies in this subspace, and is clearly bounded by the magnitude of the projection of ${\mid j\u27e9}_{R}$ onto this space:

*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 $|0\u3009$ and $|1\u3009$ states of a qubit:

Other polarizations are superpositions of these states. In particular, as illustrated in Fig. 4, linear polarization at $\pm 45\xb0$ to the horizontal and circular polarizations are the superpositionsIt 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 byWe 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 $+\theta $ and $-\theta $ to the horizontal, so that the angle between them is $2\theta $, for a range of values of *θ* between 0
and $\pi \u22154$. 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 $+\theta $ or $-\theta $ to the horizontal. These were then measured by using a
polarizing beam splitter oriented so as to transmit light polarized at $+45\xb0$ 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

and the tetrad ensemble of four equiprobable statesTo measure more than two orthogonal states of polarization we need to introduce an additional degree of freedom, and a suitable one is provided by the path of the light beam. We shall illustrate this idea only for the trine ensemble, the experimental setup for which is shown in Fig. 9. Details for the tetrad ensemble can be found in [110]. The input polarizing beam splitter separates, coherently, the polarization components by transmitting the horizontal component and reflecting the vertical component. This allows us to manipulate these components independently. A half-wave plate placed in the path of the horizontally polarized beam rotates the polarization so that only the requisite fraction of it is transmitted at the next polarizing beam splitter. The vertically polarized beam is transformed into a horizontally polarized beam so that it can be recombined coherently with what is left of the originally horizontally polarized beam. Thus the polarization of this combined beam is analyzed by using a final polarizing beam splitter. The photon ends up in one of the three photodetectors, and we can think of each of the trine polarization states being transformed into a superposition of exit paths from the interfermometer [110]:

*i*. This measurement device is optimal, as it correctly identifies the initial polarization state with probability $2\u22153$.

#### 4.3. Unambiguous Discrimination

Unambiguous discrimination between nonorthogonal polarization states, like the minimum error measurements described above, requires an extension of the two-dimensional state space, and an interferometer is the ideal device for implementing this. The idea is depicted in Fig. 10. We have two possible linear polarization states, each of which has a larger vertical component of polarization than horizontal. The double-headed arrows are intended to represent the magnitudes of the probability amplitudes at various places. The input polarizing beam splitter reflects the vertical component and transmits the horizontal component. The mirror in the upper arm of the interferometer transmits just enough for the reflected field to have the same amplitude as that in the lower arm. If the photon escapes from the interferometer at this point, then the measurement is inconclusive. If it does not, however, then the amplitudes for the vertical and horizontal fields are equal in magnitude and become orthogonal when recombined at the output polarizing beam splitter. At this stage they can be discriminated with certainty by using a final, suitably oriented, polarizing beam splitter.

The first demonstration of unambiguous discrimination between nonorthogonal polarization states used a specially selected length of polarization maintaining fiber [111]. This has the effect of maintaining, with low losses, the horizontal component of polarization but attenuating the orthogonal vertical component. If the length of the fiber is chosen appropriately, then any light exiting the fiber will be in one of two orthogonal polarizations and so can be discriminated with certainty. An interferometric experiment has the advantage that it allows us to measure the ambiguous results, also. The experimental setup [112] is very similar to that for the minimum error discrimination of the three trine states, but with the three measured outputs now corresponding to the unambiguous identification of the states $\mid {\psi}_{0}\u27e9$, $\mid {\psi}_{1}\u27e9$ and to the ambiguous result. The results of this experiment are shown in Fig. 11.

We have presented here only the simplest experiments, but more complicated problems have also been addressed. In particular, unambiguous discrimination has been demonstrated for three possible states and also between nonorthogonal pure and mixed states [113]. The generalized measurements described here have all been implemented by using light, but the principles are independent of the system used. It should be noted, particularly in the context of quantum information, that nonorthogonal states encoded in the energy levels of atoms or ions can similarly be subjected to generalized measurements with unoccupied levels used to assist in the process [114].

#### 4.4. Maximum Confidence Measurements

Maximum confidence discrimination between three symmetric states in two dimensions (the simplest possible case) has also been demonstrated experimentally by using the polarization of light as a qubit [115, 116]. In the experimental realization, the states given in Eq. (59) were encoded in the left–right circular polarization basis, and the setup distinguished between the elliptical polarizations

*N*outcome measurement may be, in principle, performed by using a number of such modules in series [116, 117]. Thus, after PBS2, two orthogonal modes in arm A correspond to outcomes 0 and ?, while two orthogonal modes in arm B correspond to results 1 and 2. Finally HWP4, QWP2, and PBS3–4 are used to separate these modes, which are then detected at the photodetectors in the output arms. The results of this experiment demonstrated an improvement over the minimum error measurement in the confidence figure of merit for linearly dependent states and are shown in Fig. 13.

#### 4.5. Mutual Information

The strategies for maximizing the mutual information for two pure states require us
to perform a minimum error measurement [91].
With more states we require, in general, a generalized measurement [90, 92].
For the trine and tetrad states we obtain the accessible information by eliminating,
with certainty, one of the possible states. This can be realized experimentally by
using the same device as that devised for the minimum error measurement, simply by
interchanging everywhere the horizontal and vertical components of polarization. In
other words, the device for maximizing the mutual information for the trine or tetrad
states is the same as that for minimizing the error in discriminating between a set
of states **orthogonal** to the given trine or tetrad. For more than four
states of linear polarization, we can maximize the mutual information by performing a
measurement with just three possible outcomes [92].

The experiment to realize the minimum error discrimination between two nonorthogonal polarization states [109] also provided the maximum mutual information. For the pure states of Eqs. (10) with $\theta =15\xb0$, corresponding to linear polarizations at an angle of 30°, the mutual information derived from the measurements was [18]

## 5. Conclusion

Quantum theory allows us to prepare, at least in principle, even the simplest system in an uncountable infinity of different ways. The polarization for a single photon, for example, can be prepared in a state that corresponds to any point on the surface of the Poincaré sphere. It is a fundamental consequence of the superposition principle, however, that no measurement can discriminate with certainty between two nonorthogonal quantum states. The challenge for quantum state discrimination is to perform this task as well as is possible.

It is evident that selecting the best possible measurement in any given situation usually requires us to perform a generalized measurement. These are general in the sense that they represent, not just projective measurements of the kind envisaged by von Neumann [22], but rather the most general measurements possible within the confines of quantum theory. The POM formalism is, as we have seen, a remarkable tool in the search for optimal measurements. That this is the case is a consequence of the facts that (i) any set of probability operators satisfying the required properties listed in Section 2 correspond to a possible quantum measurement and (ii) all possible measurements can be described by an appropriate set of probability operators. This means that we can separate the mathematical task of finding the theoretically optimum measurement from the practical one of designing a measurement to implement it.

We have seen that optimal measurements have been found to minimize the error in identifying the state, to discriminate between states unambiguously, and to determine the state with the maximum level of confidence. These similar sounding goals are all subtly different and correspond, for the most part, to quite distinct measurements. We have also discussed yet another task relevant to quantum communications, that of maximizing the information transferred. The problem of state discrimination acquired much of its significance from considering the problem of quantum communication and in particular from quantum cryptography [4, 5, 6, 7, 8, 9]. All existing implementations of these are based on optics, and it is perhaps not surprising, therefore, that it is in optics that the experimental advances in quantum state discrimination have been made. We have described, in particular, how quantum-limited measurements have been devised on optical polarization to realize the optimal measurements for detection with minimum error and unambiguous discrimination as well as detection with maximum confidence and maximum mutual information. As quantum information technology develops, the ability to optimize performance by performing the best possible measurements can only become more important.

## Acknowledgments

This work was supported, in part, by the UK Engineering and Physical Sciences Research Council (EPSRC), the Royal Society and the Wolfson Foundation (SMB), by the Synergy fund of the Universities of Glasgow and Strathclyde, and by Perimeter Institute for Theoretical Physics (SC). Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.

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