Implementing Non-Projective Measurements via Linear Optics: an Approach Based on Optimal Quantum State Discrimination

Transcription

1 Ipleenting Non-Projective Measureents via Linear Optics: an Approach Based on Optial Quantu State Discriination Peter van Loock 1, Kae Neoto 1, Willia J. Munro, Philippe Raynal 2, Norbert Lütkenhaus 2 Trusted Systes Laboratory HP Laboratories Bristol HPL Noveber 28, 2005* generalised easureents, linear optics We discuss the proble of ipleenting generalized easureents (POVMs) with linear optics, either based upon a static linear array or including conditional dynaics. In our approach, a given POVM shall be identified as a solution to an optiization proble for a chosen cost function. We forulate a general principle: the ipleentation is only possible if a linear-optics circuit exists for which the quantu echanical optiu (iniu) is still attainable after dephasing the corresponding quantu states. The general principle enables us, for instance, to derive a set of necessary conditions for the linear-optics ipleentation of the POVM that realizes the quantu echanically optial unabiguous discriination of two pure nonorthogonal states. This extends our previous results on projection easureents and the exact discriination of orthogonal states. * Internal Accession Date Only 1 National Institute of Inforatics, Hitotsubashi, Chiyoda-ku, Tokyo , Japan 2 Quantu Inforation Theory Group, Institute of Theoretical Physics I, Institute of Optics, Inforation and Photonics (Max-Planck Forschungsgruppe), Universität Erlangen-Nürnberg, Staudtstr.7, D Erlangen, Gerany Approved for External Publication Copyright 2005 Hewlett-Packard Developent Copany, L.P.

2 Ipleenting Non-Projective Measureents via Linear Optics: an Approach Based on Optial Quantu State Discriination Peter van Loock 1, Kae Neoto 1, Willia J. Munro 1,2, Philippe Raynal 3, and Norbert Lütkenhaus 3 1 National Institute of Inforatics, Hitotsubashi, Chiyoda-ku, Tokyo , Japan 2 Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdo 3 Quantu Inforation Theory Group, Institute of Theoretical Physics I, Institute of Optics, Inforation and Photonics (Max-Planck Forschungsgruppe), Universität Erlangen-Nürnberg, Staudtstr.7, D Erlangen, Gerany We discuss the proble of ipleenting generalized easureents (POVMs) with linear optics, either based upon a static linear array or including conditional dynaics. In our approach, a given POVM shall be identified as a solution to an optiization proble for a chosen cost function. We forulate a general principle: the ipleentation is only possible if a linear-optics circuit exists for which the quantu echanical optiu (iniu) is still attainable after dephasing the corresponding quantu states. The general principle enables us, for instance, to derive a set of necessary conditions for the linear-optics ipleentation of the POVM that realizes the quantu echanically optial unabiguous discriination of two pure nonorthogonal states. This extends our previous results on projection easureents and the exact discriination of orthogonal states. PACS nubers: Hk, Hz, Dv I. INTRODUCTION The ipleentation of positive operator-valued easures (POVMs) for photonic quantu state signals is an essential task in any quantu inforation protocols. In general, in order to ipleent such easureents, a nonlinear interaction of the signal states, described by a Hailtonian at least cubic in the optical ode operators [1], is needed. With current technologies, however, these nonlinear effects are hard to obtain on the level of single photons. Apart fro hybrid schees based on weak nonlinearities and strong coherent probe pulses [2], an alternative approach for inducing a nonlinear eleent is to exploit the effective nonlinearity associated with a easureent. In particular, for photonic-qubit states, universal quantu gates and hence any POVM can be realized deterinistically or asyptotically (neardeterinistically), using linear optics, photon counting, entangled auxiliary photon states, and conditional dynaics (feedforward) [3 6]. Moreover, cheaper resources ay suffice for the ipleentation of non-deterinistic gates and POVMs, using feedforward [3, 7] or a static array of linear optics [3, 8 13]. Here we will focus on the ipleentation of POVMs using either static linear optics or feedforward and, in particular, photon counting. Although there are soe specific results on this issue [14], a general and practical solution to the proble as to whether a given POVM can be ipleented by linear optics is not known. Only for the special class of projective easureents, a set of siple criteria has been derived [15]. In the special case of a projection easureent, the signal states to be distinguished (i.e., the basis that spans the space to be projected on) are orthogonal. In this case, quantu echanically, an exact discriination with unit probability for a conclusive result is possible. However, if the ipleentation of the projection easureent is restricted to a liited class of transforations such as passive linear optics or Gaussian transforations, unit probability ight be unattainable [15, 16]. The prie exaple to which such a no-go stateent applies is the Bell easureent for polarization-encoded photonic qubit states [15 18]. Of course, such a no-go stateent for exact state discriination does not rule out the possibility for near-deterinistic or non-deterinistic ipleentations. For exaple, the siplest approxiation to the single-photon qubit Bell easureent only requires a syetric bea splitter and photon counting. This schee achieves a success probability of one half, thus attaining the upper bound when using linear optics and photon counting, but neither auxiliary photons nor feedforward [19]. A hierarchy of siple criteria for the exact discriination of orthogonal states can be derived via a dephasing approach [15]. The idea of this approach is to siulate the actual detection, for instance, in the photon nuber basis through a dephasing of the linearly transfored states, turning the into ixtures diagonal in the Fock basis. Any ter in these ixtures represents a possible detection pattern for a given input state and a given linear-optics circuit. The requireent for an exact discriination of the signal states is then that the overlap of the dephased density operators vanishes, corresponding to the non-existence of any coinciding patterns. Expressing the overlap in ters of the fidelity, this eans that the fidelity of the orthogonal states ust reain zero after the linear transforation and the dephasing operation have been applied to the states. In order to extend the analysis of projection easureents [15] to generalized easureents, the first obvious approach is to consider von Neuann easureents in a larger Hilbert space. Suitably chosen, these are then equivalent to the POVM in the saller signal space. In fact, any POVM can be expressed in such a way via the

3 2 Naiark extension. For signal states having only one photon, already the Naiark extension approach reveals that any POVM can be ipleented with linear optics. A deonstration of this can be found in App. A. In general, for signal states with arbitrarily any photons, to decide whether a linear-optics ipleentation of a given POVM is possible is a non-trivial proble. Here, in order to address this question, we refer to a fundaental principle, independent of the Naiark extension. In order to apply this principle, first, the POVM shall be identified as a solution of an optiization proble for soe cost function. In ters of this cost function, the principle then states that the ipleentation is only possible if a linear-optics circuit exists for which the quantu echanical optiu (iniu) is still attainable after dephasing the corresponding quantu states. Whether linear optics or ore general linear transforations including ulti-ode squeezing are sufficient to ipleent the corresponding POVM depends on the ability of these tools to obey the above general rule. Applying this rule to the fidelity of two nonorthogonal states will enable us to derive a set of necessary conditions for the ipleentation of the quantu echanically optial unabiguous state disciination (USD), extending our analysis of discriinating orthogonal states [15]. The USD of non-orthogonal states is a siple exaple for a non-projective POVM, where soe easureent results are inconclusive, but the reaining results correctly identify the signal state. The plan of the paper is as follows. First, in Sec. II, we are going to explain how the effect of the detection behind a linear-optics circuit can be described via dephasing. This enables us to present the ain result of the paper, a general principle for the ipleentation of POVMs with linear optics. In Sec. III, we briefly review how the known criteria for linear-optics projection easureents follow fro this general principle as a siple special case. Finally, we turn to the ipleentation of non-projective POVMs in Sec. IV, where our ain focus will be on the unabiguous discriination of two pure non-orthogonal states. II. THE DEPHASING APPROACH TO POVMS Given a general non-projective POVM, via the Naiark extension approach it is pretty hard to decide whether the POVM can be ipleented with linear optics. Here we propose an alternative strategy independent of the Naiark extension, based upon a dephasing approach. The dephasing effect will be used to iic the projection of the individual odes onto the detection basis. Let us first introduce the dephasing foralis. The dephasing basis is deterined by the detection echanis of the ipleentation. This ight be either the discrete photon nuber basis (photon counting) or the continuous quadrature eigenstate basis (hoodyne detection). In the following, we will use the Fock basis as the dephasing basis. This basis can be easily substituted by other appropriate bases [15]. If the signal states ˆρ are linearly transfored into the states ˆρ H, and the photon nuber of the odes will be detected, the corresponding dephasing effect can be described as ˆρ H ˆρ H = 1 (2π) N dφ N e i a D a ˆρ H e i a D a. (1) Here, we used dφ N dφ 1 dφ 2...dφ N, the diagonal N N atrix D, (D) ij = δ ij φ i, and the vectors a = (â 1, â 2,..., â N ) T and a = (â 1, â 2,..., â N ), representing the annihilation and creation operators of all the electroagnetic odes involved. The effect of the dephasing is that it turns the linearly transfored states into a Fock-diagonal density atrix. This ixture contains all possible photon nuber patterns for a given input state and a given linearoptics circuit. The weights of the different ters in this ixture are deterined by the probabilities for obtaining the corresponding pattern via photon detection. Thus, the dephasing foralis is an equivalent description for the effect of the detection after the linearoptics transforation (see Fig. 1). An exaple for a pure signal state ˆρ = χ χ, and hence a pure transfored state ˆρ H χ H χ H, would be χ H = α β 101 +γ 002. In this case, the dephased state becoes ˆρ H = α β γ , corresponding to the possible detection patterns 110, 101, and 002. The advantage of the dephasing foralis is that the effect of the detection is described on the level of the state transforations and the final states becoe as classical as they can get. These close-to-classical states can then be analyzed with respect to a given quantu inforation task. One ay also consider only partially dephased states which are Fock-diagonal only with respect to the dephased odes. Partial dephasing iics those protocols where only a subset of the odes is detected and a subsequent linear-optics transforation is applied to the reaining odes conditioned upon the easureent outcoes (conditional dynaics). Using the dephasing foralis, we now propose the following strategy in order to decide whether a given POVM can be ipleented via linear optics. First, the POVM shall be identified as a unique solution to an optiization proble. For the cost function to be optiized, we then refer to a general principle: the ipleentation is only possible if a linear-optics circuit exists for which the quantu echanical optiu (iniu) is still attainable after dephasing the corresponding quantu states. Thus, optiizing the cost function for the dephased states ust yield the sae iniu as for the original signal states. The linear-optics circuit ust be chosen such that C optial linear optics, dephasing! = C optial quantu echanics, (2)

4 3 FIG. 1: An equivalent description of the detection echanis after a unitary state transforation via dephasing. In the case of photon counting, the dephased density atrix is a ixture of all possible photon nuber patterns for a given input state and a given state transforation. Here, we are ainly concerned about linear-optics transforations. where the sybol C denotes the corresponding cost functions [20]. This general criterion is a necessary and sufficient condition for the possibility of ipleenting the corresponding POVM. The sufficiency here is due to the close-to-classical character of the totally dephased output states which are directly linked to the click patterns of the ipleentation. In the case of only partially dephased states corresponding to a conditional-dynaics protocol, the stateent in Eq. (2) is no longer sufficient but only necessary for the ipleentability of the POVM. Siilarly, if the POVM is not a unique solution to the optiization proble, the condition in Eq. (2) is only necessary. In general, it will be highly non-trivial to find the quantu echanical optiu of the corresponding cost functions. In any cases, neither for pure states, as typically given before the dephasing, nor, in particular, for ixed states, as obtained after dephasing, a closed expression for the optiu exists. However, for instance, for the non-projective POVM that is an optial solution to the unabiguous discriination of two pure non-orthogonal states, the corresponding cost function is the failure probability and its optiu/iniu before dephasing is siply the overlap (fidelity) of the states. For the ixed states after dephasing, in this case, at least a lower bound for the cost function can also be given in ters of the fidelity of the states. It is then possible to derive a relatively siple set of necessary conditions for the ipleentability of the corresponding POVM. Later we will discuss this exaple in detail. However, before applying the general principle in Eq. (2) to non-projective POVMs, let us first review how the known criteria for projection easureents follow fro this principle as a siple special case. III. PROJECTION MEASUREMENTS Following the approach of the preceding section, given a projection easureent, we shall consider this easureent as the optial solution to the discriination of orthogonal states. A suitable cost function for an error-free state discriination is the failure probability, i.e., the probability for obtaining an inconclusive result. Now the optial strategy in order to discriinate states within an orthogonal set is to do a projection easureent on the space spanned by these orthogonal states. This strategy will always lead to a conclusive error-free result. Since this iplies zero cost for discriinating orthogonal states, C optial quantu echanics = 0, a linear-optics ipleentation of exact state discriination eans that! = 0 according to Eq. (2). C optial linear optics, dephasing In order to discriinate any two pure orthogonal states fro the projection easureent basis, the quantu echanically optial/inial failure probability is given by the overlap of the states to be discriinated. Expressing the overlap in ters of the fidelity, F (ˆρ 1, ˆρ 2 ) ( Tr ) 2, ˆρ1 ˆρ 2 ˆρ1 for two pure orthogonal signal states, + and, of course, we have F (ˆρ +, ˆρ ) = 0. Hence after dephasing, the inial failure probability ust not becoe nonzero, in order to satisfy our principle in Eq. (2). Since in any ixed-state discriination schee, the squared failure probability is lower bounded by the fidelity of the ixed states [21], the condition for ipleenting the exact state discriination becoes F (ˆρ +,H, ˆρ,H ) =! 0. Thus, we have Tr(ˆρ +,H ˆρ,H ) = 0, since always 0 Tr(ˆρ 1 ˆρ 2 ) F (ˆρ 1, ˆρ 2 ). Using the dephasing integral fro Eq. (1), one can then derive a hierarchy of siple conditions for the exact discriination of two or even ore states [15]. These conditions are necessary and sufficient for the possibility of exactly ipleenting the corresponding projection easureent. For a two-diensional projection easureent, corresponding to the discriination of two orthogonal states χ + and χ, the necessary and sufficient conditions for an exact ipleentation via linear optics and, for instance, photon counting, are given by [15], χ + ĉ jĉj χ = 0, j, (3) χ + ĉ jĉ j ĉ jĉ j χ = 0, j, j, χ + ĉ jĉ j ĉ j ĉ jĉ j ĉ j χ = 0, j, j, j,. = Here, the ode operators ĉ j = Û â j Û = i U jiâ i are those corresponding to the output odes of the linearoptics circuit. In the reainder of this section, we will add soe new and useful observations to the results of Ref. [15] on projection easureents. Assuing signal states with a fixed nuber of photons (say N photons), there is an obvious interpretation for the highest order conditions (i.e., the N th order condi-.

5 4 tions), because for these we have χ + ĉ jĉ j ĉ j ĉ jĉ j ĉ j χ = χ +,H â jâ j â j â jâ j â j χ,h Ψ (j, j, j,... +) Ψ(j, j, j,... ), (4) where Ψ(j, j, j,... ±) is the probability aplitude for detecting a photon in ode j and another photon in ode j, etc., when the input was the + or state. Thus Ψ(j, j, j,... ±) represents the probability aplitude for any possible pattern to be detected at the output. Now it becoes clear why any highest order ust vanish for exact state discriination. Only those patterns that do not occur at all and the successful patterns that can be triggered only by one of the two states lead to Ψ (j, j, j,... +) Ψ(j, j, j,... ) = 0. In contrast, for any failure pattern, the product of the probability aplitudes becoes nonzero, Ψ (j, j, j,... +) Ψ(j, j, j,... ) 0. As a result, the highest order conditions alone are necessary and sufficient for exact state discriination. Fulfilling all the highest order conditions then iplies that all lower order conditions are satisfied as well. However, note that the converse does not hold. The lower order conditions are only necessary, but not sufficient for exact state discriination. Thus, if the lower order conditions are satisfied, the highest order conditions ay well be violated. As the lower orders are easier to calculate than the higher orders, one would norally start by coputing the lowest orders. In order to rule out the possibility of exact state discriination, it is then sufficient to find a violation of any lower order condition ( no-go stateent). However, for a go stateent, the lower orders alone do not suffice. In this case, for verifying that exact state discriination is possible, one has to either calculate the higher orders as well or directly check a possible solution inferred fro the lower orders. All these observations also indicate that for unabiguously discriinating two nonorthogonal signal states of fixed photon nuber, there ust be at least one highest order condition that is violated (corresponding to the existence of at least one failure pattern and hence a nonzero failure probability). Let us now consider non-projective POVMs including the optial unabiguous discriination of nonorthogonal states via linear optics. IV. NON-PROJECTIVE POVMS Our goal is now, siilar to the criteria for projection easureents, to derive relatively siple conditions for the ipleentation of a given non-projective POVM. Our approach shall be based upon the general principle expressed in Eq. (2). We have seen already that there are state estiation probles with trivial optial POVM solutions. For instance, discriinating orthogonal states optially eans to perfor the corresponding projection easureent. A very natural way to optially discriinate quantu states drawn fro a set of linearly independent states is to perfor a POVM that iniizes the probabiltity of identifying the wrong states. This so-called iniu error discriination (MED) can always be described by a projection easureent onto a suitably chosen basis in the signal Hilbert space [22]. Therefore, in order to decide whether for a given set of quantu states MED can be ipleented via linear optics, we can also directly apply the conditions for projection easureents. An exaple for this is the MED of two syetric coherent states ± α which cannot be accoplished via non-asyptotic linear-optics schees [23]. Another trivial exaple is the optial estiation of an unknown qubit state. In this case, the optial ean optial fidelity F quantu echanics = 2/3 [20] can be attained by randoly choosing an arbitrary qubit basis, easuring in this basis, and estiating the state via the basis vector that corresponds to the outcoe of the easureent. Thus, trivially, the optial estiation of a copletely unknown qubit state α 0 + β 1 in photonic dual-rail encoding, 0 10, 1 01, can be ipleented by directly detecting the photons in the two odes. In fact, in order to satisfy our general principle in Eq. (2), we need optial optial to fulfil 1 F linear optics,dephasing = 1 F quantu echanics = 1/3; this can be accoplished by directly dephasing the input state [24]. An exaple that leads to highly non-trivial POVM solutions is the calculation of the accessible inforation in quantu counication, involving an extreely difficult optiization proble. Although our general principle expressed in Eq. (2) applies to this proble as well, here we are not going to attept to treat a linear-optics ipleentation of the accessible inforation gain. By contrast, a relatively siple optiization leads to the optial unabiguous discriination of quantu states, i.e., a schee that either identifies the signal state correctly or it yields an inconclusive result with the sallest probability allowed by quantu theory. In this case, the cost function is the probability for obtaining an inconclusive result. Now it has been shown that in general, this failure probability squared has a lower bound deterined by the fidelity of the signal states, Prob 2 fail F [21]. For two pure non-orthogonal signal states, the inial failure probability squared exactly coincides with the overlap (fidelity) of the two states (assuing equal a priori probabilities [25 27]). Thus, as for ipleenting optial unabiguous state discriination (USD), we can directly apply our general principle to the fidelities of the states before and after dephasing. The corresponding optial POVM solution is a non-trivial non-projective POVM, consisting of two POVM eleents for the correct identification of the states and one that describes the inconclusive result [28 30]. Let us now consider the question whether this optial USD of two pure non-orthogonal states can be ipleented with linear optics.

6 5 A. Optial unabiguous state discriination The optial unabiguous state discriination (USD) of two pure non-orthogonal states χ + = α 0 + β 1, χ = α 0 β 1, (5) where α > β are assued to be real and { 0, 1 } are two basis states, corresponds to a projection onto the orthogonal set (see also App. A), w µ = u µ + N µ, (6) in an extended Hilbert space. Here, the { u µ } are state vectors in a Hilbert space K such that Ê µ = u µ u µ (7) are the POVM operators of a three-valued POVM, µ = 1, 2, 3, with µ ʵ = 11. The vectors { N µ } are defined in the copleentary space K orthogonal to K, with the total Hilbert space H = K K. For the optial USD, one can show that u 1/2 = 1 ( ) β 2 α 0 ± 1, N 1/2 = 1 1 β2 2 α 2 2, u 3 = 1 β2 α 2 0, N 3 = β α 2, (8) and 2 0 = 2 1 = 0. The first two POVM eleents (µ = 1, 2) here refer to the two signal states, whereas the third POVM eleent (µ = 3) corresponds to the inconclusive result. To ake the discriination unabiguous, we have indeed Tr(Ê1 χ χ ) = Tr(Ê2 χ + χ + ) = 0 with ʵ fro Eq. (7) and Eq. (8). To ake it optial, we have Prob succ = Tr(Ê1 χ + χ + )/2 + Tr(Ê2 χ χ )/2 = 1 Prob fail = 1 Tr(Ê3 χ + χ + )/2 Tr(Ê3 χ χ )/2 = 1 χ + χ = 1 (α 2 β 2 ) = 2 β 2. (9) As for the linear-optical ipleentation, using onephoton signal states and ultiple-rail encoding, 0 100, 1 010, 2 001, one can directly ipleent the corresponding POVM for the optial USD, as described in App. A for general single-photon based POVMs [Eq. (A3) and Eq. (A4)]. In this case, the output states after the linear-optics circuit, 100, 010, and 001, uniquely refer to one of the three orthogonal states w µ, and hence identify the signal states χ + and χ with the best possible probability. However, in general, for arbitrary signal states, it turns out to be very hard to decide whether the optial USD can be ipleented, because of the infinite nuber of possible Naiark extensions. In the following, we will investigate the optial USD of two pure states independent of the Naiark extension, using the general principle introduced in the preceding sections and expressed in Eq. (2). A suitable cost function for the USD of two pure nonorthogonal states is the failure probability. When optiized over all possible POVMs, the inial failure probability corresponds to the overlap of the states. Thus, according to Eq. (2) and since after dephasing the ixedstate USD failure probability is bounded fro below by the fidelity [21], we obtain the condition F (ˆρ +,H, ˆρ,H)! = F (ˆρ +, ˆρ ), (10) where F (ˆρ +, ˆρ ) is the fidelity of the input states and F (ˆρ +,H, ˆρ,H ) is the fidelity after linear optics and dephasing. Note that the fidelity of the dephased density atrices only yields a lower bound on the failure probability and the optial failure probability ay well exceed this bound. Thus, even for a fixed array of linear optics [31], described by totally dephased density atrices, the criterion in Eq. (10) is, in general, only a necessary condition for optial USD. As a result, for the optial USD of two pure states via linear optics and subsequent photon counting (of all odes after static linear optics or only one first ode in a conditional-dynaics schee), we have the following rule: the linear-optics circuit ust be chosen such that the overlap of the two states in ters of the fidelity is the sae before and after dephasing. This stateent, as expressed by Eq. (10), extends the exact discriination of orthogonal states to the ore general scenario for optial discriination of nonorthogonal states. Whether linear optics or, ore generally, linear transforations including ulti-ode squeezing (corresponding to arbitrary quadratic interactions) are sufficient to ipleent optial USD depends on the ability of these tools to obey the above rule. When focusing on the special case of USD, a ore direct derivation of the fidelity criterion in Eq. (10) is possible and given in App. B. Let us now exaine the stateent in Eq. (10) in ore detail for a fixed array of linear optics. B. Optial USD via a fixed linear network For a fixed array of linear optics, all output odes will be detected at once. Therefore, Eq. (10) refers to totally dephased density atrices. In order to check the criterion in Eq. (10), we find that the fidelity before and after linear optics becoes F (ˆρ +, ˆρ ) = F (ˆρ +,H, ˆρ,H ) =,n α α n β β n, (11)

7 6 because after the linear-optics transforation, the output states will always take on the following for, consequence of this result is that the overlap of the input states can be written as χ +,H = k χ,h = l α k {k} + α {} β l {l} + β {}, (12) χ + χ = χ +,H χ,h = α β = α β. (17) where the coefficients depend on the linear-optics circuit chosen in a particular ipleentation. The indices k and l denote photon nuber patterns, i.e., N-ode Fock states, that exclusively occur in the expansion of χ +,H and χ,h, respectively. Hence these patterns unabiguously refer to the + state or to the state. However, because of the finite overlap of the input states, we ust include patterns that occur in the expansion of both states. These abiguous patterns are denoted by the index. In general, the aplitudes of the abiguous N-ode Fock states in the expansions, and hence the probabilities for the corresponding patterns to be detected, ay be different for the + and the state. After dephasing, the output states take on the following for ˆρ +,H = k P + k {k} {k} + P + {} {} Let us now look at the first-order expression fro the conditions in Eq. (3). We obtain χ + ĉ jĉj χ = χ +,H â jâj χ,h (18) = e iφ α β {} â jâj {}, because annihilating a photon in the jth ode of both states only leads to nonzero contributions fro coinciding patterns. Using Eq. (17) and Eq. (18), there are two observations we can ake. First, the odulus of any first order expression χ + ĉ jĉj χ is bounded fro above such that χ + ĉ jĉj χ N χ + χ, j, (19) where N is the axiu photon nuber in the states. In addition, we have ˆρ,H = l P l {l} {l} + P {} {}, (13) χ + ĉ jĉj χ χ + ĉ = χ + ĉ j ĉ j χ jĉj χ χ + ĉ j ĉ j χ, j, j, (20) corresponding to a dephasing of the states in Eq. (12) with the probabilities given by P + k = α k 2, P l = β l 2, P + = α 2, and P = β 2. The fidelity after linear optics and dephasing is now given by ( ) 2 F (ˆρ +,H, ˆρ,H) = Tr ˆρ +,H ˆρ,H ( 2 = P + P). (14) Thus, the fidelity criterion fro Eq. (10) can be expressed by,n P + P + n P P n! =,n α α n β β n, (15) using Eq. (14) and Eq. (11). This, however, iplies that α α n β β n,n! = α α n β β n,n e i(φ φ n +φ+ n φ+ ), (16) where α = α e iφ+ and β = β e iφ, etc. The only possible way to satisfy Eq. (16) is for e i(φ φ n +φ+ n φ+ ) = 1,, n. Thus, we have φ φ + = φ,. A direct provided that χ + ĉ jĉj χ and χ + ĉ j ĉ j χ are both nonzero. Siilarly, for the second-order expressions, we obtain χ + ĉ jĉ j ĉ jĉ j χ = χ +,H â jâ j â jâ j χ,h (21) = e iφ α β {} â jâ j â jâ j {}. This leads to χ + ĉ jĉ j ĉ jĉ j χ N(N 1) χ + χ, j, j, and χ + ĉ jĉ i ĉjĉ i χ χ + ĉ jĉ i ĉjĉ = χ + ĉ j ĉ i ĉ j ĉ i χ i χ χ + ĉ j ĉ i ĉ j ĉ i χ, j, i, j, i, (22) (23) provided that χ + ĉ jĉ i ĉjĉ i χ and χ + ĉ j ĉ i ĉ j ĉ i χ are both nonzero. Moreover, for all non-vanishing expressions, also the phases of different orders ust coincide. As a result, we have proven the following theore for the ipleentability of optial USD of two pure nonorthogonal states with linear optics.

8 7 Theore: it is a necessary (but, in general, not sufficient) criterion for the possibility of ipleenting the optial USD of two pure nonorthogonal states χ ± via static linear optics and photon counting that the hierarchies of conditions, χ + χ χ + χ = χ + ĉ jĉj χ χ + ĉ = χ + ĉ j ĉ j χ jĉj χ χ + ĉ j ĉ j χ = χ + ĉ jĉ i ĉjĉ i χ χ + ĉ jĉ i ĉjĉ = χ + ĉ j ĉ i ĉ j ĉ i χ i χ χ + ĉ j ĉ i ĉ j ĉ i χ, etc., j, i, j, i, etc., (24) for any non-vanishing orders, and χ + ĉ jĉj χ N χ + χ, j, χ + ĉ jĉ j ĉ jĉ j χ N(N 1) χ + χ, j, j,.. (25) χ + ĉ jĉ j ĉ j ĉ jĉ j ĉ j χ N! χ + χ, j, j, j,..., are satisfied, where N is the axiu photon nuber in the states. For the existence of a linear-optics solution to the POVM that realizes the optial USD, output ode operators ĉ j ust be found such that a unitary atrix U can be constructed with ĉ j = i U jiâ i and the hierarchies of conditions are satisfied for these operators. Note that for fixed photon nuber, the first set of conditions of the theore [that for the phases in Eq. (24)] iplies the second one [that for the absolute values in Eq. (25)], however, the converse does not hold. The reason is that in any linear-optics schee, when the input states have a fixed nuber of photons N, the total su of a given order satisfies a relation siilar to χ + j ĉ jĉj χ = N χ + χ. (26) Here, the su of the first-order expressions leads to the total photon nuber operator for the output odes, and the relation in Eq. (26) follows fro the photon nuber conservation property of linear optics. Analogous conditions can be found for the total su of the higher order expressions. Now for the su of the first orders, for instance, according to Eq. (26), we obtain χ + ĉ jĉj χ = N χ + χ, (27) j and, provided the phases of all non-vanishing first orders coincide as required to obey Eq. (24), χ + ĉ jĉj χ = j j χ + ĉ jĉj χ. (28) The two relations of Eq. (27) and Eq. (28) together iply that the first-order conditions in Eq. (25) are autoatically satisfied by any linear-optics transforation that fulfils the first-order conditions in Eq. (24). The hierarchies of conditions in Eq. (24) and Eq. (25) are necessary for optial USD of two nonorthogonal states. In words, these criteria ean that for optial USD, the phases of all non-vanishing orders ust coincide. For exaple, any real orders ust have the sae sign in optial USD. In the special case of orthogonal states, χ + χ = 0, one can easily see in Eq. (25) that the hierarchy for exact state discriination fro Eq. (3) can be retrieved. An alternative derivation of the conditions in Eq. (24) and Eq. (25), independent of the fidelity criterion in Eq. (10), is given in App. C. Let us now look at an exaple of two nonorthogonal states with only two photons (which is the siplest nontrivial extension to the trivial case of one-photon states). We are going to consider the two-photon toy odel state α 20 ± β 11, where, without loss of generality, α and β are assued to be real. For the special case of an orthogonal pair α = β, it is know that there is no linear-optics solution for optially and hence exactly discriinating these two states, including feedforward and arbitrary auxiliary states [15]. Here we consider the nonorthogonal case without auxiliary photons, but arbitrarily any additional vacuu odes. Defining U j1 ν 1 and U j2 ν 2 for the eleents of the jth row of the unitary atrix in ĉ j = i U jiâ i, the first-order expression for ode j then becoes χ + ĉ jĉj χ = 2α 2 ν 1 2 β 2 ( ν ν 2 2 ) + 2αβ(ν 1 ν 2 c.c.). (29) Now assuing a fixed array of linear optics, the conditions in Eq. (24) are necessary for optial USD. In order to satisfy the first-order conditions there for any odes j, j, etc., the expression in Eq. (29) ust becoe real for any j, j, etc., because the 0th order χ + χ = α 2 β 2 is real. The sae arguent applies to the second-order expressions, χ + (ĉ j )2 ĉ 2 j χ = 2 ν 1 2[ α 2 ν 1 2 2β 2 ν αβ(ν 1 ν 2 c.c.) ]. (30) Knowing that all these expressions ust be real, let us evaluate the first-order and second-order conditions for positive and negative 0th order χ + χ, α 2 > β 2 or α 2 < β 2, respectively. According to Eq. (24), we obtain ν a 0, and ν b 0, (31)

9 8 for α 2 > β 2, and ν a 0, and ν b 0, (32) for α 2 < β 2, where ( ) ( ) ( ) ν1 ν 2 2α ν 2 2, a 2 β 2 β 2, α 2 b 2β 2. (33) In Eq. (31), ν b 0 iplies that α 2 2β 2, because otherwise, for α 2 < 2β 2, the only way to prevent ν b fro becoing negative is to have ν 1 2 > ν 2 2 for any odes j, j, etc., according to Eq. (24). However, no unitary atrix can be constructed, where all the eleents j, j, etc., in the first two coluns satisfy ν 1 2 > ν 2 2. Siilarly, in Eq. (32), ν a 0 leads to α 2 β 2 (which is also siply given by the 0th order). Thus, there is a regie, β 2 α 2 < 2β 2 (including the orthogonal case α 2 = β 2 ), where optial USD is ipossible for a fixed array of linear optics and without auxiliary photons. For α 2 = 2β 2, the optial solution is a siple 50/50 bea splitter, ν 1 2 = ν 2 2 = 1/2 for odes j = 1, 2. In this case, in agreeent with Eq. (31), we obtain ν a = β 2 > 0 and ν b = 0. The orthogonal set of the corresponding von Neuann easureent becoes w 1/2 = 1 [ ] 1 2 ( ) ± 11, 2 w 3 = 1 2 ( ), (34) choosing for the Naiark extension 2 02 [see Eqs. (5)-(8)]. A syetric bea splitter turns these three states into 20, 02, and 11, respectively, which via photon counting uniquely refer to the three different POVM eleents. Thus, optial USD for α = 2β can be achieved with a siple bea splitter. The two signal states α 20 ± β 11 are transfored by the syetric 2 bea splitter into and , respectively. Indeed, we have Prob succ = 2/3 = 2β 2. Let us now apply the general fidelity criterion to a ore sophisticated linear-optics ipleentation of state discriination, naely one that includes conditional dynaics (feedforward): instead of detecting all output odes after the linear-optics circuit, one ay select only one ode for detection. After this first easureent, one can then send the conditional state of the reaining odes through another linear-optics circuit which depends on the easureent outcoe. In the ost general approach, one can include as any feedforward steps as odes are in the signal states, or even ore by adding auxiliary states. C. Optial USD via conditional dynaics Let s assue, without loss of generality, that ode 1 is detected first, corresponding to a partial dephasing of the states only with respect to that ode. Now instead of writing the states after linear optics as in Eq. (12), we use the following expressions, χ +,H = k χ,h = l α k k 1 γ + k + β l l 1 γ l + α 1 γ + β 1 γ, (35) where this tie, the states k 1 and l 1 represent those nuber states of ode 1 which only occur in the expansion of the + and the state, respectively. The one-ode states 1 lead to the abiguous detection events in ode 1. Finally, the states γ + k, etc., refer to the corresponding conditional states of the reaining odes (after noralization). Siilarly, for the partially dephased density operators, we obtain ˆρ +,H ˆρ,H = k + = l + P + k k 1 k γ + k γ+ k P + 1 γ + γ + P l l 1 l γ l γ l P 1 γ γ. (36) Note that the partially dephased states are no longer diagonal in the Fock basis, i.e., the conditional density atrices ay contain off-diagonal ters. The corresponding fidelities are now F (ˆρ +, ˆρ ) = F (ˆρ +,H, ˆρ,H ) (37) =,n α α n β β n γ + γ γ + n γ n, and ( F (ˆρ +,H, ˆρ,H) = P + P γ + γ 2 ). (38) Finally, we end up having the following condition due to the fidelity criterion in Eq. (10), P + P + n P P n γ + γ γ + n γ n =!,n αα n β βn γ γ + γ n + γ n, (39),n or, in ters of the unnoralized conditional states, γ + γ γ + n γn =! γ,n γ + γ n + γn. (40),n Now the only way to satisfy this condition is through γ + γ γ + γ! = γ+ n γn γ + n γ n, (41)

10 9 for any nonzero overlaps labeled by and n. In other words, for any two inconclusive one-ode detection events, any non-vanishing overlaps of the conditional states coing fro the + signal and the signal ust have equal phases. Finally, we can now again exaine the first-order condition of our criteria, however, here only for the detected ode 1, χ + ĉ 1ĉ1 χ = χ +,H â 1â1 χ,h (42) = γ + γ 1 â 1â1 1. Siilar expressions hold for the higher orders in ode 1. Note that the different orders here are evaluated only for the first ode to be detected, corresponding to the first step in a conditional-dynaics schee. Of course, in addition, one could calculate further expressions using the conditional states of odes 2 through N in order to derive ore criteria for a conditional-dynaics protocol. Here, however, we only focus on the first step in any conditional-dynaics schee, naely the detection of a first ode. Using Eq. (42) and Eq. (41) (for any nonzero overlaps), it becoes clear now that the hierarchies of conditions necessary for optial USD when detecting a first ode j are siply the subset of conditions in Eq. (24) and Eq. (25) referring to this one ode. Thus, we obtain χ + χ χ + χ = χ + ĉ jĉj χ χ + ĉ = χ + (ĉ j )2 ĉ 2 j χ jĉj χ χ + (ĉ j )2 ĉ 2 j χ = = χ + (ĉ j )n ĉ n j χ χ + (ĉ j )n ĉ n j χ, etc., (43) for any non-vanishing orders, and χ + ĉ jĉj χ N χ + χ, χ + (ĉ j )2 ĉ 2 j χ N(N 1) χ + χ, etc. (44) In the next section, we discuss whether our new set of conditions enables us to ake general stateents about the use of auxiliary photons for the optial USD of two nonorthogonal states. D. Auxiliary photons for optial USD Let us investigate whether adding an auxiliary state can ake optial USD via linear optics possible when it is ipossible without an ancilla state. The auxiliary state contains optical odes in addition to the signal odes, and these extra odes ay be occupied by additional photons. Let us use the notation χ ± = s ± ψ aux, where now the states s ± represent the signal states and the state ψ aux is the auxiliary state. Splitting the input odes into a set of signal and a set of auxiliary odes allows us to decopose the output ode operator ĉ j = i U jiâ i into two corresponding parts as (we drop the index j) ĉ = b s ĉ s + b aux ĉ aux, with real coefficients b s and b aux, where ĉ s acts only upon the signal odes and ĉ aux only on the auxiliary odes. Now we find for the first-order expression χ + ĉ ĉ χ = b 2 s s + ĉ sĉ s s + b s b aux s + ĉ s s ψ aux ĉ aux ψ aux +b s b aux s + ĉ s s ψ aux ĉ aux ψ aux +b 2 aux s + s ψ aux ĉ auxĉ aux ψ aux. (45) For either signal or auxiliary states with a fixed nuber of photons, this becoes χ + ĉ ĉ χ = b 2 s s + ĉ sĉ s s (46) + b 2 aux s + s ψ aux ĉ auxĉ aux ψ aux. Only when the signal states are orthogonal, the firstorder expression without ancilla is proportional to that with ancilla, and hence the first-order condition does not change [15]. For the general case of nonorthogonal signal states, s + s 0, the first-order expression without ancilla is effectively displaced in phase space by adding an ancilla state. Let us look at the second-order for states with a fixed photon nuber, χ + (ĉ ) 2 ĉ 2 χ = b 2 s s + (ĉ s) 2 ĉ 2 s s + b 2 aux s + s ψ aux (ĉ aux) 2 ĉ 2 aux ψ aux + 4b 2 s b 2 aux s + ĉ sĉ s s ψ aux ĉ auxĉ aux ψ aux. (47) Here, in the general case of nonorthogonal states, we obtain again a phase-space displaceent by adding the ancilla. It sees that, in general, we cannot rule out the possibility that adding an ancilla helps to satisfy the conditions for optial USD when they cannot be fulfilled without ancilla. However, if the auxiliary state ψ aux contains no photons and is just the (ulti-ode) vacuu state, we obtain χ + (ĉ ) 2 ĉ 2 χ = b 2 s s + (ĉ s) 2 ĉ 2 s s. As a result, adding auxiliary vacuu odes cannot change the sign or phase of the first order expression, since b 2 s is real and positive. However, adding auxiliary vacuu odes ight be useful and necessary in order to build up a unitary atrix for the ode operators ĉ j = i U jiâ i. In fact, in the one-photon exaple discussed after Eq. (9), adding an extra vacuu ode is essential in order to extend the two-diensional signal Hilbert space to an at least three-diensional space required for the POVM and to construct the corresponding unitary atrix. There are also known exaples, where adding extra photons akes the optial USD of the two signal states via linear optics possible. One such exaple for the case of infinite-diensional signal and auxiliary states, both

11 10 USD of ore than two coherent states, syetrically distributed in phase space, the optial USD [29] cannot be achieved as easily as for the binary case. However, there are asyptotic linear-optics solutions including the use of feedforward [32]. V. CONCLUSIONS FIG. 2: Ipleenting the optial unabiguous discriination of two syetric coherent states via a siple 50/50 bea splitter and an auxiliary coherent state of the sae aplitude. with unfixed and unbounded photon nuber, is the optial USD of so-called binary coherent states. In this case, the optial USD can be easily achieved using a 50/50 bea splitter and an ancilla coherent state (see Fig. 2). In our notation, one has s ± ± α (α assued to be real), ψ aux α, and χ ± = s ± ψ aux. This two-ode state is now transfored by the 50/50 bea splitter into χ +,H = 2α 0 χ,h = 0 2α. (48) For these states, a detector click in ode 1 can only be triggered by the + state, whereas a click in ode 2 unabiguously refers to the state. However, there are inconclusive events corresponding to the two-ode vacuu state, ψ inconcl = e α2 00, using Eq. (C2) fro App. C with φ = 0. Since the failure probability is then given by Prob lin.opt. fail = (e 2α2 + e 2α2 )/2 = e 2α2 = +α α, this schee turns out to be optial. Thus, we expect that the corresponding solution satisfies our criteria for optial USD. For a particular ode j, again using U j1 ν 1 and U j2 ν 2 for the eleents of the jth row of the unitary atrix in ĉ j = i U jiâ i, we obtain the nth-order condition, ( χ + (ĉ j )n ĉ n j χ = +α α ν 1 2 α 2 (49) ) n. ν ψ aux â 2â2 ψ aux 2 Apparently, for any ode j = 1, 2, any order n 1 can be set to zero by choosing a 50/50 bea splitter, ν 1 2 = ν 2 2 = 1/2, and the appropriate ancilla state, ψ aux α. This solution is indeed in agreeent with the conditions that we derived for optial USD. The obvious reason, why all nonzero orders vanish in this exaple, is that the only failure pattern here is 00 which always vanishes upon applying annihilation operators [see, e.g., Eq. (18)]. Fro this observation follows that also any cross orders for odes 1 and 2 will vanish with the above solution. Let us finally note that for the optial We considered the proble of ipleenting generalized easureents (POVMs) with linear optics. Such an ipleentation ay either be based upon a static array of linear optics or it ay include conditional dynaics (feedforward). Extending our previous results on projective easureents, we focused, in particular, on non-projective easureents. Our approach to this proble can be forulated as a general principle in the following way. We start by identifying a given POVM as a solution to an optiization proble for a chosen cost function. The ipleentation is then only possible if a linear-optics circuit exists for which the quantu echanical optiu is still attainable after dephasing the corresponding quantu states. As an exaple for applying this principle to the proble of ipleenting a non-projective POVM, we discussed in detail the optial unabiguous state discriination (USD) of two pure nonorthogonal states. In order to ipleent the POVM that realizes the quantu echanically optial USD with linear optics, according to the general principle, the linear-optics circuit ust be chosen such that the overlap of the states, in ters of the fidelity, is the sae before and after dephasing. This stateent extends the exact discriination of orthogonal states to the ore general scenario for optial discriination of nonorthogonal states. Using the fidelity criterion, we derived hierarchies of necessary conditions for the possibility of ipleenting the optial USD of two pure nonorthogonal states via linear optics and photon counting. The resulting conditions are a generalization of our previous criteria for projection easureents and the exact discriination of orthogonal states. As for the detection echanis, here we only studied the case of photon counting which leads to dephased states diagonal in the Fock basis. Potential extensions of our results ay include different detection echaniss such as hoodyne detection, as we discussed previously already in the context of projective easureents. Moreover, apart fro passive linear-optics circuits, our criteria can also be applied to arbitrary linear ode transforations, including ulti-ode squeezing. When analyzing those POVMs that realize unabiguous state discriination, one ay also consider the USD of sets of three or ore linearly independent states. Finally, let us ephasize that our approach of choosing suitable cost functions and applying the to the dephased quantu states ight be as well useful for finding bounds on the efficiency of ipleenting POVMs with linear optics.

12 11 Acknowledgents WM, PR, and NL acknowledge the support fro the EU project RAMBOQ. PvL and KN acknowledge funding fro MIC in Japan. PR and NL acknowledge the support of the DFG under the Ey-Noether prograe. APPENDIX A: ONE-PHOTON SIGNAL STATES Let us consider all those POVMs where the signal states contain only one photon. In this typical and iportant case, any unitary operation (gate) can be accoplished with linear optics [33]. This stateent applies to arbitrary qudit states, where each basis vector of the qudit is described by one photon occupying one of d odes, â i 0, i = 1...d ( ultiple-rail encoding ). Siilarly, any POVM can be ipleented solely by eans of linear optics for these one-photon signal states. This can be understood by looking at the corresponding Naiark extension of the POVM. The POVM is then described by a von Neuann easureent onto the orthogonal set w µ = u µ + N µ, (A1) in a Hilbert space larger than the original signal space. Here, the { u µ } are (unnoralized, potentially nonorthogonal) state vectors in a Hilbert space K such that Ê µ = u µ u µ (A2) are the POVM operators of an N-valued POVM, µ = 1...N, with µ ʵ = 11. The vectors { N µ } are defined in the copleentary space K orthogonal to K, with the total Hilbert space H = K K. If the diension of the signal space is n, we have N µ = N i=n+1 b µi v i with soe coplex coefficients b µi and { v i } a basis in K. In the ultiple-rail encoding, this leads to an orthogonal set of vectors w µ = N U µj â j 0, j=1 (A3) with a unitary N N atrix U having eleents U µj. The application of a linear-optics transforation V to this set (in order to project onto it) can be written as w µ w µ = = N j,k=1 U µj V kjâ k 0 N δ µk â k 0 = â µ 0, k=1 (A4) choosing V U. As a result, when detecting the outgoing state, for every one-photon click in ode µ, one can unabiguously identify the input state w µ. This is why it is no surprise that any POVM for one-photon states can be ipleented via linear optics with unit success probability [34]. For states other than one-photon states, it is a priori not clear whether a given POVM can be ipleented with linear optics. A possible approach to deciding this would be to apply the criteria for projective easureents [15] to the orthogonal set in Eq. (6). The ain difficulty then is that one ust consider any possible Naiark extension vectors { N µ } in order to be able to decide whether the POVM can be ipleented or not. In particular, the extension of the signal Hilbert space can be arbitrarily large. Therefore, in an optical ipleentation, arbitrary ancilla states ust be taken into account, including arbitrarily any extra odes and photons. It sees that, in general, ore coplicated approaches are required to deal with the potentially infinite-diensional proble of adding arbitrary auxiliary states [13] (however, see [8]). In this paper, we propose a dephasing approach to the proble of ipleenting POVMs via linear optics, independent of the Naiark extension. APPENDIX B: ALTERNATIVE DERIVATION OF THE OPTIMAL-USD FIDELITY CRITERION Without referring to the general principle in Eq. (2) for arbitrary cost functions and POVMs, here we directly derive the corresponding (necessary) criterion for the special case of optial USD in ters of fidelities. In general, for any state discriination schee based on static linear optics, we have the following fidelity bounds, ( F (ˆρ +, ˆρ ) F (ˆρ +,H, ˆρ,H) Prob lin.opt. fail ) 2. (B1) In words, the fidelity of the linearly transfored and dephased output states is lower bounded by the fidelity of the input states and upper bounded by the squared failure probability in the linear-optics ipleentation of unabiguous state discriination. The lower bound here corresponds to the general rule that the fidelity of two density atrices cannot decrease under CPTP aps [35]. As for the upper bound, we ay note that in any schee, the linearly transfored and dephased output states take on the for of Eq. (13) corresponding to a total dephasing of the states in Eq. (12). Since the two density atrices in Eq. (13) are diagonal in the Fock basis and coute, we have the relation in Eq. (14). Now the failure probability is given by Prob lin.opt. fail = (P+ + P )/2. However, we also have (P + + P )/2 P + P,, thus proving the upper bound in Eq. (B1). According to the fidelity bounds in Eq. (B1), we obtain Eq. (10) as a necessary condition for the optial USD of two states via static linear optics and photon counting, because optial USD requires χ + χ 2 = F (ˆρ +, ˆρ ). ( Prob lin.opt. fail ) 2 =