DDiscriminating distinguishability
Stasja Stanisic
1, 2 and Peter S. Turner Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory andDepartment of Electrical & Electronic Engineering, University of Bristol, UK Quantum Engineering Centre for Doctoral Training, University of Bristol, UK (Dated: June 5, 2018)Particle distinguishability is a significant challenge for quantum technologies, in particular pho-tonics where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantuminterference. We take a representation theoretic approach in first quantisation, separating particles’Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantuminformation inspired bipartite model where distinguishability can arise as correlation with an en-vironment carried by the particles themselves. This makes clear that the HOM experiment is aninstance of a (mixed) state discrimination protocol, which can be generalised to interferometers thatdiscriminate unambiguously between ideal indistinguishable states and interesting distinguishablestates, leading to bounds on the success probability of an arbitrary HOM generalisation for multipleparticles and modes. After setting out the first quantised formalism in detail, we consider severalscenarios and provide a combination of analytical and numerical results for up to nine photonsin nine modes. Although the Quantum Fourier Transform features prominently, we see that it issuboptimal for discriminating completely distinguishable states.
I. INTRODUCTION
Interference lies at the heart of quantum mechanics,and thus its promise of fundamental advantages over non-quantum technologies, with far-reaching ramifications incommunication, metrology, simulation and computation.The nemesis of quantum interference is distinguishabil-ity, with the Hong-Ou-Mandel (HOM) effect [1] being aprototypical example. Recent advances in scaling linearoptics for universal quantum computation [2–4], and therace to demonstrate quantum computational ‘supremacy’via analog computations that sample the scattering am-plitudes of multipartite states [5–13], highlight the needfor a thorough understanding of distinguishability in mul-timode quantum interference [14–23].Rather than the usual second quantized approach, wecan gain insight by bringing quantum information con-cepts to bear in first quantization [24–27]. Distinguisha-bility can then be modelled, for example, as entangle-ment between controlled and uncontrolled degrees of free-dom of individual particles, with loss of interference beingcaused by the decoherence that results when the uncon-trolled Hilbert space is marginalized. This can be for-malized by observing that bosonic (and fermionic) Fockstates of two (sets of) degrees of freedom can have nat-ural Schmidt decompositions, corresponding to so calledunitary-unitary duality in many-body physics [28].An example of a pertinent idea from quantum informa-tion is state discrimination [29–31]; we start by showinghow this reproduces the well known HOM distinguisha-bility test for two particles. In principle the formal-ism accommodates any number of particles and modes,and we show how this generalises for multimode quan-tum interference, taking a representation theoretic ap-proach (Sections II and III); this complements a num-ber of generalizations in the literature [32–40]. We setup the state discrimination problem in the linear optical framework, assuming we have access to passive trans-formations (networks of phaseshifters and beamsplitters)and projective measurements via photon number count-ing detectors (Sec. IV). This restriction on the allowedmeasurements yields a highly nontrivial constraint on themixed state discrimination scenario – this new problemis what we study here. In particular, the optimisationproblem that results is nonlinear, as is usually the casein multiphoton interferometry [41], necessitating numer-ical techniques described in Sec. IV C.The results are as follows: in Sec. V A we presenttwo general upper bounds valid for any photon number N when discriminating (i) a state with a single distin-guishable photon from the completely indistinguishablestate, and (ii) the completely distinguishable from thecompletely indistinguishable state; in Sec. V B we showwhy the HOM test is the only test of distinguishabilityfor arbitrary states of two photons, and demonstrate thegenerality of the formalism by considering three photonsin two modes; in Sec. V C we use a mix of analyticaland numerical techniques to argue the optimality of abalanced three mode network (tritter) as a discrimina-tor for both completely distinguishable and singly distin-guishable states; in Sec. V D 1 we look at discriminationof singly distinguishable states with higher photon num-bers up to N = 9 and show that the quantum Fouriertransform (QFT) saturates the bound, suggesting it isthe optimal interferometer for all N ; finally in Sec. V D 2we look at the discrimination of completely distinguish-able states with higher photon numbers and give exam-ples of the best known interferometers up to N = 8,found by observing a pattern emerging from the optimi-sations. Most of these results are summarised in Table I.Although not surprising that the QFT features heavily,the results show that it is not optimal for discriminatingcompletely distinguishable states, motivating the searchfor optimal discriminating networks for other states of a r X i v : . [ qu a n t - ph ] J un interest. II. MOTIVATIONA. Hong-Ou-Mandel interference
We will use the HOM scenario as an example that setsout the main features of our distinguishability model, andits relationship to state discrimination. Each HOM pho-ton has two pertinent degrees of freedom: one is spatial,namely the interferometer arms, and the other is tempo-ral, namely the time of arrival. We are usually interestedin the case where it is the spatial degree of freedom overwhich we have control (via interferometry), and so wecall this the ‘System’ degree of freedom. We interpretthe temporal degree of freedom as a ‘Label’ – in generalthis would include all the particles’ degrees of freedomwhich we do not control. Since complete control of theSystem includes the possibility of putting photons in thesame spatial mode, we view the Label as determining theparticles’ distinguishability, via correlations between theSystem and Label degrees of freedom. In a real HOM ex-periment we are interested in preparing situations withvarying distinguishability, so we do in fact manipulatethe temporal Label degree of freedom as well, but forapplications we usually think of the System-Label corre-lations as having been determined by means beyond ourcontrol.The HOM scenario has two spatial System modeswhich we will call “top” and “bottom” ( s = ↑ , ↓ ), andtwo photons, requiring two temporal Label modes thatwe will call “early” and “late” ( l = ← , → ). (Note thatthese symbols will need to be ordered – we have avoidedthe obvious choice of s and l = 1 , a † sl [42], giving rise to Fock states whichwe can write as arrays where rows correspond to Systemmodes and columns to Label modes. An example of acompletely distinguishable two photon state is | ψ d (cid:105) = ˆ a †↑← ˆ a †↓→ | vac (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) , (1)with an early photon in the top arm and a late one inthe bottom, while | ψ i (cid:105) = ˆ a †↑← ˆ a †↓← | vac (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (2)corresponds to an indistinguishable state where bothphotons are early.Ideally an interferometer acts only upon the System,corresponding to a unitary transformation on the twospatial modes ˆ a † sl (cid:55)→ (cid:88) t ˆ a † tl U ts . (3) Here U is a 2 × × × ↑← , ↑→ , ↓← , ↓→ ) is given by U ⊗ . (4)It is tempting to interpret the tensor product in Eq. (4) asthat between the System and the Label. A quantum in-formation theoretic approach to distinguishability wouldthen ignore (trace out) the Label, arriving at reducedstates on the System where all the nontrivial transfor-mations and measurements occur. However, this is notthe tensor product structure of the four harmonic oscil-lators in the second quantized model, and so one cannotmarginalise, for example, the columns in Eqs. (1,2). Inorder to trace out the Label we will use a first quantizeddescription.Second quantized Fock states can be related to firstquantized single particle states as follows. Viewing eachexcitation of our four mode aggregate as a particle withfour available states ( ↑← , ↑→ , ↓← , ↓→ ), and recognizingthat as bosons the total state must be symmetric un-der particle exchange, we have a one-to-one relationshipbetween the Fock states of two bosons in four modesand symmetric states of two four-dimensional particles,(qu d its, here with d = 4). Applying this procedure tothe indistinguishable state of Eq. (2), we have | ψ i (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (5)= Sym ( |↑←(cid:105) |↓←(cid:105) ) (6)= 1 √ |↑←(cid:105) |↓←(cid:105) + |↓←(cid:105) |↑←(cid:105) ) (7)= 1 √ |↑↓(cid:105) S + |↓↑(cid:105) S ) |←←(cid:105) L , (8)where the subscripts 1 and 2 have been used as (fictitious)particle labels that get permuted, and we have rearrangedthe tensor product structure in the last line to arrive at astate in the S(ystem) ⊗ L(abel) basis. Similarly, one finds | ψ d (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (9)= Sym ( |↑←(cid:105) |↓→(cid:105) ) (10)= 1 √ |↑↓(cid:105) S |←→(cid:105) L + 1 √ |↓↑(cid:105) S |→←(cid:105) L . (11)We see that Eq. (8) is in a product state (Schmidt rank1) of System and Label [43], so the Label states are un-correlated to the System states; learning the Label doesnot allow one to learn anything about the System, asexpected for indistinguishable particles. Equation (11)is entangled (Schmidt rank 2), with the System statesperfectly correlated to the Labels ( ↑ to ← and ↓ to → ),making the photons completely distinguishable.It will be useful to rewrite states of both the Sys-tem and Label according to their permutation symmetry.Schur-Weyl duality [28, 44] ensures that this basis alsohas good quantum numbers for the unitary group actionof the interferometer, in this case U(2) [45]. The irre-ducible representations (irreps) of U(2) are well known,and for only two particles Young diagrams provide a com-pact notation for the basis states that carry these irreps;they are (for arbitrary, ordered single particle quantumnumbers x, y ) the symmetric triplet | x x (cid:105) = | xx (cid:105) , (12) √ | x y (cid:105) = | xy (cid:105) + | yx (cid:105) , (13) | y y (cid:105) = | yy (cid:105) , (14)and the antisymmetric singlet √ (cid:12)(cid:12)(cid:12) xy (cid:69) = | xy (cid:105) − | yx (cid:105) . (15)We can now rewrite Eqs. (8, 11) as | ψ i (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) = | ↑ ↓ (cid:105) S | ←← (cid:105) L , (16) | ψ d (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) = 1 √ | ↑ ↓ (cid:105) S | ←→ (cid:105) L + 1 √ (cid:12)(cid:12)(cid:12) ↑↓ (cid:69) S (cid:12)(cid:12)(cid:12) ←→ (cid:69) L . (17)Note that total exchange symmetry is preserved be-cause the System and Label states in the second termof Eq. (17) are both antisymmetric. We can now seeclearly that in this case the Schur-Weyl bases provide aSchmidt decomposition of the Fock arrays, and that thecompletely distinguishable state has nonzero amplitudeoutside the totally symmetric irrep; we will discuss thegeneralisation of these features in Sec. III B.Tracing out the Label degree of freedom, we arrive atthe reduced density matrices that describe the state ofthe System. Another feature of the Schur-Weyl basisis that these states will be block diagonal, each blockcorresponding to an irrep. Thus, ordering our triplet-singlet basis as (cid:110) | ↑ ↑ (cid:105) , | ↑ ↓ (cid:105) , | ↓ ↓ (cid:105) , (cid:12)(cid:12)(cid:12) ↑↓ (cid:69)(cid:111) , we have ρ i = Tr L [ | ψ i (cid:105) (cid:104) ψ i | ] = , (18) ρ d = Tr L [ | ψ d (cid:105) (cid:104) ψ d | ] = 12 . (19)A coincidence count occurs when both the top andbottom modes are occupied, defining the coincidence sub-space spanned by (cid:110) | ↑ ↓ (cid:105) , (cid:12)(cid:12)(cid:12) ↑↓ (cid:69)(cid:111) . The projector onto this subspace has matrix representation M (1 , = , (20)where we have used an occupation (one excitation in eachof the two System modes) in the subscript.The unitary evolution of these input states due to theinterferometer is given by the two-photon representationof the transfer matrix. Again, in the Schur-Weyl basisthis is block diagonal, specifically a direct sum of thetriplet and singlet matrix representations of U(2). Thematrix elements in the coincidence subspace for an ar-bitrary two mode interferometer with transfer matrix U are U ⊗ ∼ = U ⊕ U = ∗ ∗ ∗∗ per U ∗∗ ∗ ∗ det U , (21)where per and det are the matrix permanent and deter-minant functions, ∗ are matrix elements for events out-side the coincidence subspace, and we use ∼ = to denotethe fact that U ⊗ U only equals U ⊕ U after the basischange. This can be confirmed by direct calculation fromEq. (3), or equivalently by using the Schur-Weyl trans-formation, which for U(2) is the familiar Clebsch-Gordantransformation of angular momentum theory.The probability of a coincidence count is given by theBorn rule, which from Eqs. (18–21) is given by P (1 , = Tr (cid:20)(cid:16) U ⊕ U (cid:17) ρ (cid:16) U ⊕ U (cid:17) † M (11) (cid:21) (22)= Tr (cid:20)(cid:18) U ρU † + U ρU † (cid:19) M (11) (cid:21) (23)= (cid:40) | per U | if ρ = ρ i12 | per U | + | det U | = per | U | if ρ = ρ d (24)where we have written | U | for the elementwise absolutevalue squared of a matrix U .It follows that in order to see no coincidences for anindistinguishable state, which has only a triplet com-ponent, we need an interferometer whose transfer ma-trix permanent vanishes. By parametrising an arbitrary U ∈ U(2) one can confirm that only a balanced beamsplitter has this property (see Sec. V B). We also see thatthe distinguishable state has a singlet component thatscatters through any U according to the determinant,and since any element of U(2) has | det U | = 1, this com-ponent will always give rise to coincidences. Thus, ina HOM experiment one uses a balanced beam splitterto see a “dip” in coincidence counts in the System asone manipulates the Label degree of freedom from dis-tinguishable to indistinguishable and back again. B. State discrimination
By choosing to measure a coincidence count as wellas U to be a balanced beamsplitter, the HOM situationdescribed above ensures that P (1 , = 0 when the input is ρ i , while P (1 , happens to be maximised when the inputis ρ d (see Sec. V B). This is reminiscent of what is knownas unambiguous mixed state discrimination [46].A general state discrimination protocol [30, 31] consistsof two parties, a source (Alice) and a detector (Bob), whoagree on an ensemble of states { p k , ρ k } to be discrim-inated. The source draws a random sample from thisensemble according to the distribution { p k } and sends itto the detector, whose task is to identify which state wassent as best as possible. This is accomplished by find-ing a measurement, given by a set of POVM elements { E k } that maximise the expected probability of success: (cid:80) k p k Tr[ ρ k E k ]. For unambiguous discrimination (UD),we have the further constraint that no mistakes are al-lowed to be made, that is, Tr[ ρ k E j ] = 0 for all k (cid:54) = j , atthe price of having to add an outcome E ? to the POVMthat corresponds to failing to identify the state.Rearranging Eq. (22) and defining M (1 , ( U ) = (cid:16) U ⊕ U (cid:17) † M (1 , (cid:16) U ⊕ U (cid:17) , (25)the HOM measurement scenario described above can nowbe summarised byfind U maximising Tr (cid:2) ρ d M (1 , ( U ) (cid:3) (26)subject to Tr (cid:2) ρ i M (1 , ( U ) (cid:3) = 0 . (27)That is, find an interferometer that maximises the prob-ability of seeing a coincidence for a distinguishable inputstate, subject to the constraint that it never gives coin-cidences for an indistinguishable input state. It is nowclear this is an instance of an UD problem, with the so-lution being a balanced beamsplitter in the HOM case.This gives a direction in which to generalise the HOMscenario to any number of particles in any number ofmodes as a UD problem. A key distinction from generalUD is the restricted form of the available POVM ele-ments, which must be projective measurements definedby the interferometer U and the N -photon occupation n being detected. In particular, we expect that knownoptimal measurements for two-state discrimination willnot be available in linear optics. When speaking gener-ally about measurements we will use the notation E forPOVM elements, while, as above, M n ( U ) is reserved forphoton counts. Because M n ( U ) is degree N in the vari-ables U and U † , this measurement restriction makes theUD optimisation problem nonlinear. III. BACKGROUND:MANY PARTICLES AND MODES
From the HOM example (e.g. Eq. (17)), we see thatsymmetry of the states in the full System-Label space and the correlations within it play a key role in the dis-tinguishability of the particles. Therefore we proceedwith an analysis for any arbitrary number of particlesand modes using Schur-Weyl duality, and then furthergeneralise for particles with two degrees of freedom usingunitary-unitary duality [28].
A. Schur-Weyl duality in first quantisation
In the first quantized picture of the HOM exampleabove, each photon was considered as a d -dimensionalquantum system, with d the total number of System andLabel modes available. Schur-Weyl duality states thatthe Hilbert space of N qudits can be decomposed as( C d ) ⊗ N ∼ = (cid:77) λ C { λ } ⊗ C ( λ ) , (28)where C { λ } carries irrep λ of the group of unitary trans-formations on a qudit, U( d ), C ( λ ) carries irrep λ of thegroup of permutations of qudits, S N , and ∼ = signifies thatthe left and right hand sides are related by a change ofbasis (a Schur-Weyl transform). Following [44], a Schur-Weyl basis which realises this decomposition is denoted | λqp (cid:105) where λ labels the irrep of both the unitary and thesymmetric groups simultaneously [47], q = 1 , , . . . , d { λ } indexes a basis of the unitary irrep, and p = 1 , , . . . , d ( λ ) indexes a basis of the symmetric irrep. These dimensionscan be computed, for example, by the Weyl character andhook length formulas respectively [48]. There is an im-plied dependence of q and p on λ , the set of which in turndepends on the number of particles N and the numberof modes d . The irrep λ = ( λ , λ , . . . , λ d ) can be spec-ified using Young diagrams, where λ j is the number ofboxes in row j , λ ≥ λ · · · ≥ λ d , and (cid:80) j λ j = N . Theindices q and p correspond to the different ways of fill-ing boxes with the numbers { , . . . , d } and { , . . . , N } tomake semistandard (with repetition) and standard (with-out repetition) Young tableaux respectively, where num-bers cannot decrease as you move right in a tableaux andmust increase as you go down.We can further refine this notation by observing thatthe basis can be chosen such that the representation the-oretic weight of a state corresponds to the occupation n ,which has also been called a type in this context [44].Subspaces of states with the same occupation are theninvariant under the Schur-Weyl transform in this basis,and the unitary index q can be uniquely specified by anoccupation n and an ‘inner’ multiplicity r (the number ofwhich is also known as a Kostka number), which accountsfor the fact that there can be more than one orthogonalstate with the same weight in a unitary irrep λ . As we arefocusing on the action of the unitary group, p will be re-ferred to as an ‘outer’ multiplicity accounting for the factthat the same unitary irrep λ can occur more than once.We can therefore write Schur-Weyl basis states in theform | λpnr (cid:105) , where the irrep dependence of p , n and r hasagain been suppressed to prevent clutter. We will oftenshorten the notation such that | λpn (cid:105) := | λ, p, n, r = 1 (cid:105) , | λnr (cid:105) := | λ, p = 1 , n, r (cid:105) , | λn (cid:105) := | λ, p = 1 , n, r = 1 (cid:105) , re-ducing clutter when the multiplicity is trivial; since λ and n are vectors while p and r are scalars there shouldbe no ambiguity. Coincident input or output will be de-noted with occupation number 1 = (1 , , . . . , N and d , writing states in terms of Youngtableaux can be more compact, as in the HOM discussionof the previous section. The shape of a tableau is speci-fied by λ , which is filled with mode indices specified by n following the rules for semistandard tableaux. The innermultiplicity r corresponds to different semistandard fill-ings of the same λ and n , while the outer multiplicity p will be labelled with a subscript. For example, for threephotons ( N = 3) in three modes ( d = 3), the coincident n = 1 = (1 , ,
1) subspace for irrep λ = (2 ,
1) = isspanned by four states given by p, r ∈ { , } . If we indexthe modes 1, 2 and 3, the two notations are related as | λ = (2 , , p = 1 , n = (1 , , , r = 1 (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) (29) | λ = (2 , , p = 1 , n = (1 , , , r = 2 (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) (30) | λ = (2 , , p = 2 , n = (1 , , , r = 1 (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) (31) | λ = (2 , , p = 2 , n = (1 , , , r = 2 (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) , (32)while, e.g., the n = (2 , ,
0) subspace for irrep λ = (2 , | λ = (2 , , p = 1 , n = (2 , , , r = 1 (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) (33) | λ = (2 , , p = 2 , n = (2 , , , r = 1 (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) , (34)because the Young tableau is not semistandard andtherefore such states do not exist.
1. Implementation of the Schur-Weyl transform
An example of a Schur-Weyl transformation is thetriplet-singlet basis change given in Eqs. (12 - 15), where(when d = 2) it is the same as the well known Clebsch-Gordan transformation. There are several ways to im-plement this basis change more generally [49, 50]; we usethe method described in Ref. [28], which we will brieflyoutline here.Every irrep { λ } of U( d ) can be assigned a highestweight state, which is annihilated by an appropriate set ofraising operators that are realised in terms of the bosoniccreators and annihilators. Given as a Young tableau,this state can be expressed in terms of single particle(qudit) states using Slater determinants; the single par-ticle basis is indexed by the d modes. In much the sameway as is done for U(2) in angular momentum theory,we then use corresponding lowering operators to find a set of states that span the irrep. The size of this setis known, namely d { λ } . A Gram-Schmidt procedure isthen used to orthonormalise the set, (note that there isfreedom in choosing how to do so when there are mul-tiplicities, see e.g. Sec. III C 2). Outer multiplicities arehandled by utilising the dual S N action to permute ahighest weight state in order to find corresponding high-est weights for the multiple copies of irrep { λ } . Again,the number of these states is known, namely d ( λ ) , and or-thonormalisation is required. The lowering procedure isthen repeated until a complete set of λ states are found.Iterating through all λ then gives a complete set of states {| λqp (cid:105)} , from which we can determine the required basistransformation. Transformations for different N and d can be computed once and stored for later use. B. Unitary-unitary duality
In the HOM example we saw that each photon had twodegrees of freedom, the System and the Label, and that,as bosons, first quantised multiphoton states had to betotally symmetric under particle permutations. Indepen-dently decomposing both the System and Label Hilbertspaces according to Schur-Weyl, one is then led to askwhat states of the form (cid:88) λqpλ (cid:48) q (cid:48) p (cid:48) ψ λqpλ (cid:48) q (cid:48) p (cid:48) | λqp (cid:105) S | λ (cid:48) q (cid:48) p (cid:48) (cid:105) L (35)are totally symmetric? This can be viewed as a couplingproblem for irreps of the symmetric group – we wish toconstruct composite states of ‘permutational momentumzero’. The answer turns out much like it does in angularmomentum theory: that λ , p must equal λ (cid:48) , p (cid:48) , respec-tively, and that the coupling coefficients are all equal andindependent of p [25, 51]. Thus totally symmetric pureSystem-Label states are of the form (cid:88) λqq (cid:48) ψ λqq (cid:48) | λqq (cid:48) (cid:105) SL , (36)where we have defined | λqq (cid:48) (cid:105) SL := 1 (cid:112) d ( λ ) d ( λ ) (cid:88) p =1 | λqp (cid:105) S | λq (cid:48) p (cid:105) L . (37)These states carry the symmetric irrep of the ‘global’unitary group, U( d S d L ), acting on the d S d L modes of thecombined System and Label. As discussed above, we canreplace q with pairs n, r in all of these expressions.Equations (36, 37) imply a decomposition of the to-tally symmetric irrep of U( d S d L ) into irreps of its unitarysubgroups U( d S ) and U( d L ) that act on the System andLabel independently. These irreps are labelled simulta-neously by λ , hence “unitary-unitary duality”:Sym (cid:0) ( C d S ⊗ C d L ) ⊗ N (cid:1) ∼ = (cid:77) λ C { λ } S ⊗ C { λ } L , (38)where we include subscripts on the right hand side toremind us which unitary subgroups the irreps belongto [52]. An interferometer U is given by an element ofthe System unitary subgroup U( d S ), and thus it acts onstates in irrep λ according to the irreducible matrix rep-resentation U λ U : | λqq (cid:48) (cid:105) SL (cid:55)→ (cid:88) q (cid:48)(cid:48) | λq (cid:48)(cid:48) q (cid:48) (cid:105) SL U λq (cid:48)(cid:48) q . (39)Just as with a single degree of freedom, the space ofsecond quantized d S × d L Fock arrays can be put into one-to-one correspondence with first quantized totally sym-metric states by the procedure exemplified in Eqs. (5 -8). Thus we can write an arbitrary partially distinguish-able state, which is an element of the totally symmetricsubspace of ( C d S ⊗ C d L ) ⊗ N , in a basis of first quantizedstates given by Eq. (37). We may now trace out theLabel to arrive at mixed states describing any partiallydistinguishable state of N photons in d S modes. We canorder the basis so that the reduced System state and theaction of any System interferometer will both be blockdiagonal according to irreps λ , a potentially significantsimplification. C. States of interest
We will focus our attention on three types of N -photonstates: completely indistinguishable, singly distinguish-able, and completely distinguishable, described below(the general case will be discussed in Sec. VI). We arenot considering loss (where entire qudits would be tracedout), so N will be fixed throughout. Situations withmixed System-Label states and partial distinguishabilitycan be written in terms of the basis [53]; we give exam-ples of this generality with partial distinguishability fortwo photons in two modes in Sec. V B 1, and of mixedSystem-Label states for three photons in three modes inSec. V C 2. Otherwise we will restrict ourselves to thecase where the total System-Label state is pure, corre-sponding to a source that produces states that are always(in)distinguishable in exactly the same way; generaliza-tion is, in principle, straightforward.The most distinguishable N photons can be is for eachLabel to be in an orthogonal state, and so d L ≤ N . Inpractice the Label space could be much larger, but in or-der to describe the particles’ distinguishability we needonly consider the subspace spanned by the Label states,which can be at most N dimensional. In order to set d S , consider first two photons who share the same stateof either degree of freedom; obviously that state is sym-metric, and so in order to maintain total symmetry – orby unitary-unitary duality – the state of the other de-gree of freedom must also be symmetric, cf. Eq. (8).This restricts the combined state to a subspace of thoseallowed in Eq. (37), and so is not completely general.This argument extends to any number of photons, thus to consider arbitrary distinguishability we must have in-put states that have a single photon in each System mode,thus d S ≥ N . Unless indicated otherwise, we will con-sider the case with d S = d L = N . The reader may wishto refer ahead to Sec. V C for concrete examples of thefollowing.
1. Completely indistinguishable states
A completely indistinguishable state is one in whichevery photons’ Label state is the same. As mentionedabove, such a state lies in the symmetric Label subspacewith λ = ( N ). Since the symmetric irrep of S N is one di-mensional, d ( N ) = 1 and Schur-Weyl duality tells us thatthe corresponding unitary irrep is always outer multi-plicity free. Moreover, ( N ) is also inner multiplicity free,(there is only one way to symmetrise a product of sin-gle particle states), so we can replace q with the Systemoccupation 1, and q (cid:48) with the Label occupation ( N, N − a † ˆ a † · · · ˆ a † N | vac (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · ·
01 0 · · · · · · (cid:43) (40)= Sym ( | (cid:105) | (cid:105) · · · | N (cid:105) ) (41)= | ( N ) , (cid:105) S | ( N ) , ( N, (cid:105) L , (42)where we have included N − ρ i = Tr L (cid:2) | ( N ) , (cid:105) | ( N ) , ( N, (cid:105) (cid:104) ( N ) , | (cid:104) ( N ) , ( N, | (cid:3) (43)= | ( N ) , (cid:105) (cid:104) ( N ) , | , (44)supported on the one dimensional intersection of the sym-metric System subspace given by ( N ) with the coincidentsubspace defined by the System occupation number 1.
2. Singly distinguishable states
The next state we consider is one where a single pho-ton has become distinguishable from the rest; assumingall efforts are being made to produce the completely in-distinguishable state, this should be the most likely errorto occur. Ordering our modes so that the ‘bad’ photonis in System mode N and Label mode 2, we haveˆ a † ˆ a † · · · ˆ a † N | vac (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · ·
01 0 · · · · · · (cid:43) (45)= Sym ( | (cid:105) | (cid:105) · · · | N (cid:105) ) , (46)where in the last line we have not yet performed theSchur-Weyl transform. Considering this symmetrisation,one observes that although all N ! permutations of the N distinct System indices will occur, since only two distinctLabel modes are involved there are only N single particlestates available to the Label degree of freedom, namelythose with the j th photon in Label mode 2 and the restin Label mode 1; denote these states | j (cid:105) L . Such a La-bel state will be perfectly correlated to all System stateswith the j th photon in mode N ; for each j we can factorthese ( N − | N j (cid:105) S . Thus in the System-Label basis,the singly distinguishable state can be written asSym ( | (cid:105) | (cid:105) · · · | N (cid:105) ) = 1 √ N N (cid:88) j =1 | N j (cid:105) S | j (cid:105) L , (47)e.g. Eqs. (8, 11). These sets of states are orthonormal,and we recognise this as an entangled state with Schmidtcoefficients 1 / √ N .Now consider Schur-Weyl transforming this state intothe form of Eq. (36). Because there are only two distinctLabel modes involved, the only Label irreps that can oc-cur are those whose Young diagrams have two or fewerrows. Moreover, because only a single photon is ‘bad’,the only two rowed diagram allowed is that with a singlebox in the second row. Thus the Label state is supportedonly by irreps λ = ( N ) and ( N − , N ) isalways both inner and outer multiplicity free; for irrep( N − , d (( N − , = N −
1. Itremains only to work out the inner multiplicities for irrep( N − , N − , ,
0) respectively (the marginals of the Fock ar-ray). There is only one Young tableau of shape ( N − , N − , , N − N − , N in the second row box), and so the Systeminner multiplicity is N −
1. Inserting these observationsinto Eq. (36), the Schur-Weyl transformed state is ψ ( N ) , , , ( N − , , , | ( N ) , , , (cid:105) S | ( N ) , , ( N − , , , (cid:105) L + N − (cid:88) r =1 ψ ( N − , , ,r, ( N − , , , √ N − N − (cid:88) p =1 | ( N − , , p, , r (cid:105) S × | ( N − , , p, ( N − , , , (cid:105) L . (48)We can factor the second term and redefine coefficients to yield another Schmidt decomposition: ψ ( N ) | ( N ) , , , (cid:105) S | ( N ) , , ( N − , , , (cid:105) L + ψ ( N − , √ N − N − (cid:88) p =1 (cid:32) N − (cid:88) r =1 φ r | ( N − , , p, , r (cid:105) S (cid:33) × | ( N − , , p, ( N − , , , (cid:105) L . (49)Because the Schur-Weyl transformations yieldingEq. (37) are performed independently, the System-Labelentanglement cannot be changed. From Eq. (47) weknow that the Schmidt coefficients are all 1 / √ N , so wemust have ψ ( N ) = 1 / √ N and ψ ( N − , = (cid:112) ( N − /N .The amplitudes φ r do not affect this entanglement atall – they depend on how one chooses to orthonormalisemultiplicities in the Schur-Weyl transform, and encodethe fact that we chose the ‘bad’ photon to be in Systemmode N . We can always choose r = 1 to correspondto this specific situation, and then use the subgroup ofU( d S ) that permutes System modes to find the statescorresponding to the ‘bad’ photon being in any othermode.Making this choice and tracing out the Label inEq. (49) yields the singly distinguishable reduced state(now suppressing trivial multiplicities) ρ s = 1 N | ( N ) , (cid:105) (cid:104) ( N ) , | + 1 N N − (cid:88) p =1 | ( N − , , p, , (cid:105) (cid:104) ( N − , , p, , | . (50)We see that this is mixed over N dimensions of the coin-cident subspace, overlapping the symmetric and ‘almostsymmetric’ ( N − ,
1) irreps.
3. Completely distinguishable states
A completely distinguishable state has each particlein a distinct Label mode, paired with a unique Systemmode. We can choose to order the modes such that thecorresponding Fock array is diagonal, cf. Eq. (1). Gener-alising the symmetrisation procedure of Eqs. (5 - 8) to N particles, one finds that all N ! possible terms will occur inthe single particle picture, and they will each occur once.The unique pairing of System and Label modes mani-fests as maximal entanglement between the System andLabel single particle states in the coincident subspace. Asabove, because the Schur-Weyl transformations yieldingEq. (37) are performed independently, the System-Labelentanglement is preserved. This means that the trans-formed state must also be maximally entangled with thesame Schmidt rank. Thusˆ a † ˆ a † · · · ˆ a † NN | vac (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · ·
00 1 · · · · · · (cid:43) (51)= Sym ( | (cid:105) | (cid:105) · · · | N N (cid:105) ) (52)= 1 √ N ! (cid:88) λpr | λ, p, , r (cid:105) S | λ, p, , r (cid:105) L , (53)with the sum running over all allowed values of irrep,outer, and inner multiplicities. The completely distin-guishable reduced System state is therefore ρ d = 1 N ! (cid:88) λpr | λ, p, , r (cid:105) (cid:104) λ, p, , r | (54)= 1 N ! | ( N ) , (cid:105) (cid:104) ( N ) , | + 1 N ! (cid:88) λ (cid:54) =( N ) ,p,r | λ, p, , r (cid:105) (cid:104) λ, p, , r | , (55)which is completely mixed over the N ! dimensional coin-cident subspace. D. Unitary parametrisation
FIG. 1. Example of a Reck scheme parametrising anarbitrary unitary transformation on four modes ( d S = 4),grouped into ‘layers’ T k . Each one- (phaseshifter) and two-mode (beamsplitter) subtransformation contributes one realparameter. Only the phaseshifters situated between beam-splitters ( ω , , ω , , ω , ) contribute to our problem. The unitary subgroup U( d S ) corresponds to the setof interferometers that act on the System modes. Wecan parametrise these unitaries with what is known asa Reck scheme in optics [54, 55], decomposing an arbi-trary U into a sequence of single mode unitaries (phase-shifters) and unitaries that act on neighbouring modes(beamsplitters). As shown in Fig. 1, such a scheme canbe viewed as d s − k , each with k phaseshifters and beamsplitters, followed by a final phaseshift on each mode. Because we are only interested innumber state inputs and number counting measurements,only the phaseshifters between beamsplitters play a role.Hereafter when we refer to U we will therefore be re-ferring to this smaller interferometer, without the initialand final sets of phaseshifters. E. Measurements
We will assume that we have access to photon num-ber resolving detectors for the System (see Sec. VI fora discussion of a relaxation). The measurement POVMelements are projections on all states with photon oc-cupation n , M n = (cid:80) λpr | λpnr (cid:105) (cid:104) λpnr | . Note that thisincludes projections onto System states that are not sym-metric; as shown in Eq. (17), distinguishable states cancontain non-symmetric System components that still giverise to clicks. Comparing with Eq. (54), we see that M = N ! ρ d – that is, up to normalisation, a coincidencecount is a projection onto the completely distinguishablestate. As discussed above, we will usually include the in-terferometer in our definition of a measurement, yieldingparametrised POVM elements M n ( U ) = (cid:0) ⊕ λ U λ ⊗ λ (cid:1) † M n (cid:16) ⊕ λ (cid:48) U λ (cid:48) ⊗ λ (cid:48) (cid:17) , (56)where 1l λ corresponds to the irrep of the identity permu-tation in accordance with Eq. (28), (note that we omitthis when it is only one dimensional, e.g. Eq. (21) ). IV. DISCRIMINATION OF DISTINGUISHABLESTATES
We will be interested in two problems: discriminatingthe completely indistinguishable state, ρ i , from the dis-tinguishable states (i) ρ s and (ii) ρ d . From Eqs. (44, 50,55), we observe that each of these states is of the form ρ = αρ i + (1 − α ) ρ ¯i , α (cid:54) = 0 , (57)where ρ i is pure, and ρ ¯i is diagonal in the Schur-Weylbasis with support outside the symmetric subspace λ =( N ). From well known results for the discrimination oftwo mixed states [56], the fact that ρ i lies within thesupport of the mixed state to be discriminated meansthat the optimal measurement is essentially the same foreither Minimum Error or Unambiguous Discrimination;one wishes to project onto the support of ρ ¯i . In partic-ular for UD, the error-free constraint means that we areforced to set E i = 0, and thus the prior probabilities donot affect the optimal choice of measurement operators.This reflects the fact that there is no way to unambigu-ously discriminate the indistinguishable state ρ i – we caneither conclude that the state was distinguishable by ob-serving an output that is completely suppressed by quan-tum interference, or fail to conclude anything at all. Ourtask is therefore to minimise the probability of failure E ? = 1l − E s , d , equivalently maximising the probabilityof unambiguously detecting a singly or completely dis-tinguishable state, respectively.If our measurements are unrestricted, the best choiceof POVM is to project onto the nonsymmetric subspace.This choice is suitable for not only the states ρ s , d , but byextension any state to be discriminated from ρ i . How-ever, as mentioned in Sec. III E, in practice we only haveaccess to number counting measurements – we will there-fore want to approximate this projection as best possible.The approximation will be sensitive to the state we arediscriminating: for example, Eqs. (50, 55) show that ρ s can be optimally discriminated by projecting onto onlythe ( N − ,
1) irrep, while for ρ d one wants to projecton to all of the nonsymmetric irreps. As we will see,this can lead to different interferometers being optimalfor discriminating different distinguishable states in lin-ear optics. A. Restriction to linear optical measurements
In order to discriminate distinguishability in linear op-tics we wish to find the best we can do with the measure-ments we have, namely those in Eq. (56). In the HOMcase, the UD problem described by Eqs. (26, 27) involvesonly a single occupation POVM element, the coincidencecount M n ( U ) with n = (1 , n we find U maximising Tr (cid:2) ρM n ( U ) (cid:3) subject toTr (cid:2) ρ i M n ( U ) (cid:3) = 0. Notice that any n that can be madeto satisfy Tr (cid:2) ρ i M n ( U ) (cid:3) = 0 for a suitable U is an unam-biguous discriminator, but is not necessarily the optimalchoice. In general, it is possible for multiple occupations n to satisfy the UD constraint simultaneously, contribut-ing to the probability of success.We therefore wish to find the subset of all discrimi- nating occupations, call it D , that optimises the successprobability for the same choice of U :find U and D maximising (cid:88) n ∈ D Tr (cid:2) ρM n ( U ) (cid:3) (58)subject to, for all n ∈ D, Tr (cid:2) ρ i M n ( U ) (cid:3) = 0 . (59)Note that the quantity we are maximizing gives us thetotal probability of successful discrimination, which is thesum over all the unambiguously discriminating events inthe set of occupations D .While the first optimisation focuses on giving an op-timal interferometer for discrimination given a specificmeasurement pattern, the second optimisation focuses onthe highest probability of discrimination across all mea-surement patterns. In general we find that these twoproblems give different optimal interferometers; here wewill focus on the latter ‘complete’ optimisation over both U and D , see Sec. VI for a discussion of a variation ofthe problem. B. Scattering probabilities
Let us look at what the probability of a specific mea-surement pattern n being detected at the output of anarbitrary interferometer U is for the states of interest,starting with the completely distinguishable state. FromEqs. (55) and (56),Tr (cid:2) ρ d M n ( U ) (cid:3) = Tr N ! (cid:88) λ,p,r | λ, p, , r (cid:105) (cid:104) λ, p, , r | ( ⊕ µ U µ ⊗ µ ) † (cid:88) λ (cid:48) ,p (cid:48) ,r (cid:48) | λ (cid:48) , p (cid:48) , n, r (cid:48) (cid:105) (cid:104) λ (cid:48) , p (cid:48) , n, r (cid:48) | (cid:16) ⊕ µ (cid:48) U µ (cid:48) ⊗ µ (cid:48) (cid:17) = 1 N ! (cid:88) λ,p,r,r (cid:48) Tr (cid:104) | λ, p, n, r (cid:105) (cid:104) λ, p, n, r | (cid:0) U λ ⊗ λ (cid:1) | λ, p, , r (cid:48) (cid:105) (cid:104) λ, p, , r (cid:48) | (cid:0) U λ ⊗ λ (cid:1) † (cid:105) = 1 N ! (cid:88) λ,p,r,r (cid:48) | (cid:104) λ, p, n, r | U λ ⊗ λ | λ, p, , r (cid:48) (cid:105) | = 1 N ! (cid:88) λ,r,r (cid:48) d ( λ ) | (cid:104) λ, n, r | U λ | λ, , r (cid:48) (cid:105) | , (60)where in the last line we have used the fact that outermultiplicities p give rise to identical copies of unitary ir-reps to write the probability in terms of irreducible uni-tary matrix elements. When r = r (cid:48) = 1 these matrixelements are immanants [57] of a matrix U n whose rowsand columns are determined by the input and outputoccupations of the interferometer given by U [58, 59].Moreover, the completely distinguishable case can beinterpreted as independent classical particles evolvingstochastically [60], leading to the remarkable fact that the sum in Eq. (60) can always be written in terms of thepermanent of the matrix given by the elementwise squareamplitudes of U n , cf. Eq. (24) and note that U = U .The calculation for the singly distinguishable and com-pletely indistinguishable state is the same as Eq. (60),only with fewer irreps occurring. Recalling from0Sec. III C 2 that d (( N − , = N −
1, Eq. (50) givesTr (cid:2) ρ s M n ( U ) (cid:3) = 1 N | (cid:104) ( N ) , n | U ( N ) | ( N ) , (cid:105) | + N − N (cid:88) r | (cid:104) ( N − , , n, r | U ( N − , | ( N − , , , (cid:105) | , (61)where the sum is over all r consistent with n , and Eq. (44)gives Tr (cid:2) ρ i M n ( U ) (cid:3) = | (cid:104) ( N ) , n | U ( N ) | ( N ) , (cid:105) | , (62)where as mentioned above these matrix elements are ex-pressible in terms of per U n [61].We observe that not all occupations are useful for un-ambiguous discrimination. Measurements where all thephotons are bunched into a single mode only occur in thesymmetric irrep, that is, if n = (0 , .., , N, , ..., M n = | ( N ) , , n, (cid:105) (cid:104) ( N ) , , n, | . In this case Eqs. (60)and (61) are proportional to Eq. (62), and since Eq. (59)has to be satisfied, they will always give zero. Completelybunched events can therefore never help discriminate theindistinguishable state, and we will exclude such eventsfrom our searches. C. Numerical optimisation approach
In the Results section there is a mixture of analyticaland numerical results. To construct the cost function forour numerical work we took into consideration the follow-ing criteria: the measurement operator M n can only beincluded in the optimisation if Eq. (59) is satisfied; whenthis is the case it is added to a sum being optimised asper Eq. (58). The cost function chosen was C ( U ) = − (cid:88) n exp (cid:0) − ξ Tr (cid:2) ρ i M n ( U ) (cid:3)(cid:1) Tr (cid:2) ρM n ( U ) (cid:3) , (63)where ξ is adjusted (usually depending on the choice of N , and ranging from 2 to 60) to penalise results where M n might be added to Eq. (58) and optimised withoutsatisfying Eq. (59). A high penalty ξ guarantees that thevalue of Tr (cid:2) ρ i M n ( U ) (cid:3) is close to zero before Tr (cid:2) ρM n ( U ) (cid:3) is optimised and added to the sum. Combining this withthe Eqs. (60) and (61) we have C d ( U ) = − N ! (cid:88) λ (cid:54) =( N ) d ( λ ) (cid:88) n e − ξ |(cid:104) ( N ) ,n | U ( N ) | ( N ) , (cid:105)| × (cid:88) r,r (cid:48) | (cid:104) λ, n, r | U λ | λ, , r (cid:48) (cid:105) | , (64) C s ( U ) = 1 − NN (cid:88) n,r e − ξ |(cid:104) ( N ) ,n | U ( N ) | ( N ) , (cid:105)| × | (cid:104) ( N − , , n, r | U ( N − , | ( N − , , (cid:105) | . (65) Python was used to optimise these functions withthe scipy library function basinhopping using Broy-den–Fletcher–Goldfarb–Shanno (BFGS) as the optimisa-tion algorithm. The seeds were generated using numpyrandom number generation. Though this optimisationfunction will help us explore the space and reach fairlyclose to the global minimum, it can neither guaranteethat minimum is global, nor does it exactly solve theoriginal optimisation problem. This will be problem-atic with minima that are close together, as for exam-ple exp (cid:0) − ξ Tr (cid:2) ρ i M n ( U ) (cid:3)(cid:1) gets closer to 1 for values ofTr (cid:2) ρ i M n ( U ) (cid:3) that are close to 0. In some situations thisvalue can be quite high combined with a high value ofTr (cid:2) ρM n ( U ) (cid:3) , skewing the results towards a possible non-optimal solution for the original problem. We could avoidthis by choosing an appropriately high ξ as a functionof the number of occupations (cid:0) N + d S − N (cid:1) , however, if toohigh, exp (cid:0) − ξ Tr (cid:2) ρ i M n ( U ) (cid:3)(cid:1) will behave like a step func-tion, which does not reward transitional values enough.Therefore, we do not make any strong claims of optimal-ity for the interferometers found numerically when theydo not saturate the general bounds presented in Sec. V A. V. RESULTSA. General bounds
Recall from Sec. IV the best possible unrestricted dis-crimination measurement is to project onto the non-symmetric subspace, E ( N ) = (cid:80) λ (cid:54) =( N ) ,p,n,r | λpnr (cid:105) (cid:104) λpnr | .Such a POVM element would be equally good for bothsingly and completely distinguishable states, and indeedany distinguishable state of the form in Eq. (57). Thesuccess probability of such a measurement is given byTr (cid:104) ρ (cid:0) ⊕ λ U λ ⊗ λ (cid:1) † E ( N ) (cid:0) ⊕ λ U λ ⊗ λ (cid:1)(cid:105) = Tr (cid:104) ( αρ i + (1 − α ) ρ ¯i ) E ( N ) (cid:105) = 1 − α = (cid:40) − N if ρ = ρ s − N ! if ρ = ρ d , (66)where we have used the fact that any projector ontoirreps is unitarily invariant. These then are universalupper bounds on the success probability for singly andcompletely distinguishable states, respectively. However,since we are restricted to photon number counting mea-surements, we will see that while the first bound is achiev-able, the second is not in general. We will go throughvarious examples in detail in the following sections.1 B. Two modes
1. Two photons in two modes
In the case of two photons in two modes, the states tobe discriminated are, from Eqs. (44), (50) and (55), ρ i = | (cid:105) (cid:104) | , and (67) ρ s = ρ d = 12 | (cid:105) (cid:104) | + 12 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) . (68)Observing that there is only one available state whichis not symmetric, it is easy to write down an arbitrary partially distinguishable System state in this case, sincethere is but one parameter: ρ = α | (cid:105) (cid:104) | + (1 − α ) (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) . (69)As discussed in Sec. IV B, only occupations that do nothave all the photons bunched in the same mode can beused for meaningful discrimination, in this case leavingonly one choice of projector, the coincidence M (1 , = | (cid:105) (cid:104) | + (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) .In our discussion in Sec. II, we claimed that the op-timal discriminator is given by a coincidence count anda balanced beamsplitter; we can now prove this asser-tion. First, note that since there is only one antisymmet-ric state, the antisymmetric irreducible representation ofany U has but one matrix element and so the action ofany interferometer on this state is trivial (in Eq. (21)given by its determinant). Thus the only contributionto non-symmetric part of Eq. (60) is (cid:12)(cid:12)(cid:12)(cid:68) (cid:12)(cid:12)(cid:12) U (cid:12)(cid:12)(cid:12) (cid:69)(cid:12)(cid:12)(cid:12) = 1,and there is nothing to maximise in Eq. (58). All that isleft is to satisfy the constraint, Eq. (59). Parametrising U as (cid:20) e iφ cos θ e iϕ sin θ − e − iϕ sin θ e − iφ cos θ (cid:21) . (70)one finds that the constraint is then per U = cos θ − sin θ = cos 2 θ = 0, with the family of solutions { ( φ, ϕ, π/ | ≤ φ ≤ π, ≤ ϕ ≤ π } . The solutionsdo not depend on the phases φ or ϕ , as we would ex-pect from the discussion in Sec. III D, but only on thechoice of the beamsplitter reflectivity, which is balancedas claimed.We see that not only does unambiguous discrimina-tion return the HOM measurement as was discussed inSec. II A, it is optimal for an arbitrary partially distin-guishable two photon state.
2. Three photons in two modes
As an example of the utility of the formalism, in thissubsection we consider the simplest nontrivial case with N (= 3) > d S (= 2). As mentioned in Sec. III C, this re-stricts the kinds of distinguishable states that can occur; we consider situations with two photons in one Systemmode and the third in the other. The indistinguishablestate is ˆ a † ˆ a † ˆ a † | vac (cid:105) = | (cid:105) , with reduced state ρ i = | (cid:105) (cid:104) | . (71)There are essentially two types of distinguishable statein this situation. The first is ˆ a † ˆ a † ˆ a † | vac (cid:105) = | (cid:105) ,and the second ˆ a † ˆ a † ˆ a † | vac (cid:105) = | (cid:105) . Other statesare equivalent to the above for the reasons discussed inSec. III C 2. Further, the (now incompletely) distinguish-able state ˆ a † ˆ a † ˆ a † | vac (cid:105) = | (cid:105) has a reduced statethat is the same as Eq. (72), and will therefore have thesame discrimination measurement and success probabil-ity. The reduced state for the first case is ρ s = 13 | (cid:105) (cid:104) | + 13 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 13 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) , (72)while that for the second case is ρ s = 46 | (cid:105) (cid:104) | + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) . (73)Note that Eq. (58) does not depend on the amplitude ofthe symmetric part of the state – its contribution has tobe zero by Eq. (59). It only depends on the nonsymmetriccomponents, and since ρ s and ρ s are equally weightedacross the available nonsymmetric states, the optimal dis-criminator will be the same. However ρ s does have halfof the amplitude of ρ s in this subspace, which will halvethe success probability.There are four possible occupations to measure,however as mentioned in Sec. IV B the bunchedones can be disregarded and the optimisation car-ried out on M (2 , and M (1 , . We parametrise U again as in Eq. (70). For M (2 , Eq. (59) reduces to |(cid:104) | U | (cid:105)| = | (cos θ + 3 cos 3 θ ) / | = 0.Since 0 ≤ θ ≤ π , this equation is true for θ ∈ { π/ , arccos ( (cid:112) / , arccos ( − (cid:112) / } .On the other hand, Eq. (59) for M (1 , is |(cid:104) | U | (cid:105)| = | (sin θ − θ ) / | . Thisequation cannot be zero for the above choice of anglesthat ensure |(cid:104) | U | (cid:105)| = 0. Thus, onlyone of the outcomes can be used to discriminate thesestates; without loss of generality, we choose to opti-mise for M (2 , . In this case we want to maximiseTr (cid:2) ρ s M (2 , ( U ) (cid:3) = 2 (cid:12)(cid:12)(cid:12)(cid:68) (cid:12)(cid:12)(cid:12) U (cid:12)(cid:12)(cid:12) (cid:69)(cid:12)(cid:12)(cid:12) = 2 cos θ/ θ = π/
2, we get success probability of 0. When θ = ± arccos ( (cid:112) / /
9. Thus an optimal discriminating interferometer is U = (cid:2) √ − √ (cid:3) / √
3, with success probabilities 4 / ρ s and 2 / ρ s .2 C. Three modes
From now on we will only consider coincident inputwith N = d S . For three photons in three System modes,the completely indistinguishable reduced state, is fromEq. (44), ρ i = | (3) , (cid:105) (cid:104) (3) , | = | (cid:105) (cid:104) | . (74)There are now three different singly distinguishablestates, depending on which System mode the ‘bad’ pho-ton is in. In the Schur-Weyl basis (see Sec. III A 1)their full System-Label states, as per the discussion inSec. III C 2, are √ a † ˆ a † ˆ a † | vac (cid:105) = √ (cid:12)(cid:12)(cid:12) (cid:69) = | (cid:105) | (cid:105) + (cid:12)(cid:12)(cid:12) (cid:69) (cid:12)(cid:12)(cid:12) (cid:69) + (cid:12)(cid:12)(cid:12) (cid:69) (cid:12)(cid:12)(cid:12) (cid:69) , (75) √ a † ˆ a † ˆ a † | vac (cid:105) = √ (cid:12)(cid:12)(cid:12) (cid:69) = | (cid:105) | (cid:105)− (cid:16)(cid:12)(cid:12)(cid:12) (cid:69) + √ (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) (cid:12)(cid:12)(cid:12) (cid:69) − (cid:16)(cid:12)(cid:12)(cid:12) (cid:69) + √ (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) (cid:12)(cid:12)(cid:12) (cid:69) , (76) √ a † ˆ a † ˆ a † | vac (cid:105) = √ (cid:12)(cid:12)(cid:12) (cid:69) = | (cid:105) | (cid:105)− (cid:16)(cid:12)(cid:12)(cid:12) (cid:69) − √ (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) (cid:12)(cid:12)(cid:12) (cid:69) − (cid:16)(cid:12)(cid:12)(cid:12) (cid:69) − √ (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) (cid:12)(cid:12)(cid:12) (cid:69) . (77)While for completely distinguishable states permutingSystem modes has no effect on the reduced state, herethe reduced states will not be invariant. However, be-cause permutations of System modes lie inside the set ofallowed operations, (that is, S d S ⊂ U( d S )), if we opti-mise for one of these states, the resulting interferometerwill be easily related to the others by including somemode swapping. Therefore we can focus on one of thesestates and the success probabilities that we find will bethe same for the other two; Eq. (75) has the reduced state(cf. Eq. (50)) ρ s = 13 | (cid:105) (cid:104) | + 13 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 13 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) . (78)It is natural to ask about discrimination of mixtures ofthese three states; we will discuss this in Sec. V C 2. The completely distinguishable state corresponding toˆ a † ˆ a † ˆ a † | vac (cid:105) = (cid:12)(cid:12)(cid:12) (cid:69) per Eq. (55) is ρ d = 16 | (cid:105) (cid:104) | + 16 (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) . (79)For the following let us define two sets of measure-ment operators: those with two photons in one mode, M (2 , , = (cid:80) λ (cid:54) = ,p | λ, p, (2 , , (cid:105) (cid:104) λ, p, (2 , , | , M (2 , , , M (1 , , , M (1 , , , M (0 , , , and M (0 , , , which we denote M ; and those with each photon in a different mode,that is M (cid:51) M (1 , , = (cid:80) λ,p,r | λ, p, , r (cid:105) (cid:104) λ, p, , r | . Asdiscussed in Sec. IV B, the measurements M (3 , , = | (cid:105) (cid:104) | , M (0 , , , and M (0 , , will not be helpfulfor discrimination.
1. Discriminating singly distinguishable states
Let ρ λ denote the (unnormalized) part of a statesupported on the subspace of irrep λ . Notice that ρ s has no support in the antisymmetric subspace, so that (cid:80) n Tr (cid:104) ρ s M n ( U ) (cid:105) = 2 / (cid:80) n Tr (cid:20) ρ s M n ( U ) (cid:21) = 0. Inthe best case scenario, we can pick some subset of occu-pations D and U for which Eq. (59) holds and the successprobability will be bounded by 2 /
3. It is well known howto achieve this; use a balanced tritter, U = QF T , andall the occupations from M , where QF T N is defined as QF T N = 1 √ N · · · ω · · · ω N − ... ... ...1 ω N − · · · ω ( N − N − (80)and ω = exp πiN . A parametrisation that realizes a bal-anced tritter is given in Figure 2. FIG. 2. The best known interferometer for discriminat-ing completely indistinguishable from distinguishable statesof three photons in three modes is
QF T , with a success prob-ability of 2 /
3. Up to phases, it consists of two balanced beam-splitters, one 2 : 1 beamsplitter, and one π/
2. Discriminating mixed singly distinguishable states
A short digression regarding mixed System-Labelstates: if we were (uniformly) ignorant about which mode3the ‘bad’ photon was in, we would have an equal mixtureof Eqs. (75, 76, 77). The resulting mixed state is ρ sm := 13 | (cid:105) (cid:104) | + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12) (cid:69) (cid:68) (cid:12)(cid:12)(cid:12) . (81)The overlap (cid:80) n Tr (cid:104) ρ sm M n ( U ) (cid:105) = 2 / M again sat-urates the bound, and a balanced tritter remains the bestchoice of interferometer. This can be seen from the sym-metry of the QF T which treats a ‘bad’ photon in anymode essentially the same way, and so should be true foranalogous singly distinguishable mixed states for all N ,however we will not discuss mixed System-Label statesfurther here.
3. Discriminating completely distinguishable states
Using the cost function from Eq. (64) and a range ofpenalties ξ ∈ { , , , , } we found that the highestsuccess probability in the completely distinguishable caseis 2 /
3. The measurement operators were always the fullset M with a balanced tritter as an example solution,just as in the previous section. However, this does notsaturate the bound in Sec. V A, which is 5 / (cid:80) n Tr (cid:16) ρ d M n ( U ) (cid:17) =2 / (cid:80) n Tr (cid:18) ρ d M n ( U ) (cid:19) = 1 /
6, so that (cid:80) n Tr (cid:0) ρ d M n ( U ) (cid:1) = 5 /
6, which is the discrimi-nation bound. Notice that operators from M donot have support on the anti-symmetric subspace.Therefore, if we only pick operators from M as thediscriminating operators, and assume they can simulta-neously satisfy Eq. (59), then (cid:80) n ∈M Tr (cid:0) ρ d M n ( U ) (cid:1) = (cid:80) n ∈M Tr (cid:16) ρ d M n ( U ) (cid:17) ≤ /
3. This is exactly whathappens for the interferometers from our optimisation.This tells us that if we want the success probability tobe larger than 2 /
3, the only operator left, M (1 , , , wouldhave to be included. Our numerical results show that, onthe contrary, it is unlikely for any D that includes M (1 , , to give a success probability over 1 /
2. We do this witha new cost function, much like Eq. (64) but modified toforce M (1 , , to be included: C d , ( U ) = η Tr( ρ i M (1 , , ( U )) + C d ( U ) , (82)where η is a penalty to ensure Eq. (59) for M (1 , , hasto be satisfied, and C d ( U ) is as defined in Eq. (63). Thispenalty is set to η = 10 making the first term an order of magnitude higher than the second term of Eq. (82),where we took ξ = 6. As we learned in Sec. III D,we can ignore the outside phaseshifters of the standardReck parametrisation, therefore we are only optimizingover 4 parameters, θ , , θ , , θ , , and ω , . The lowestvalue of the cost function found by the optimisation tech-niques in Sec. IV C is − . . / M (1 , , is not included.While this does not give us definitive proof that noscheme that includes a threefold coincidence can give suc-cess probability higher than 2 /
3, it does strongly indicatethat this should be true. Moreover, with the same opti-misation functions we investigated how many of the otheroperators alongside M (1 , , we can pick at the same time,and it seems that the best we can do is to have fourfrom M satisfy Eq. (59) simultaneously. However, inall the situations when this occurs, some of the terms inEq. (58) are zero, thus the success probability remainsat 1 /
2, which can be achieved using just M (1 , , and abalanced beamsplitter.The balanced tritter uses all the measurement oper-ators from M , with each contributing 1 / /
3. To draw attention to thedifference between optimizing a single operator and mul-tiple operators at once, mentioned in Sec. IV A, we no-tice that optimizing for one operator from the set M yields a success probability higher than 1 / U ). Taking this further, we can searchnumerically for the single best outcome, with a costfunction similar to that of Eq. (64), except we now fo-cus only on a single n , that is C ( n, U ) = − (cid:80) λ,r,r (cid:48) exp (cid:0) − ξ | (cid:104) (3) , n | U | (3) , (cid:105) | (cid:1) | (cid:104) λ, n, r (cid:48) | U | λ, , r (cid:105) | . Wefind M (1 , , is a clear winner with a total success prob-ability of 1 /
2, achievable by a balanced beamsplitter asmentioned above. All of the other operators by them-selves only ever give an optimised success probability of1 /
8. Notice that 6 · / / > /
3, showing that thestrategy that gives us the best success chance with a sin-gle operator from M can not be achieved simultaneouslyby all six of them. D. Four and more modes
1. Discriminating singly distinguishable states
Using the numerical optimisation described inSec. IV C, we also examined the discrimination of singlydistinguishable states for N = 4 and 5 photons. Togetherwith the results for N = 2 and 3, we see the optimisationreturn interferometers equivalent to QF T N , each givinga success probability 1 − /N , saturating the bound inSec. V A. We have confirmed this behaviour by directcalculation up to N = 9.4 Singly distinguishable, ρ s Completely distinguishable, ρ d N U
Success probability Success probabilityBound Best Worst Avg Bound2 = 0 . = 0 . = 0 . = 0 . ≈ . ≈ . ≈ . ≈ . = 0 . ≈ . =0 . ≈ . ≈ . ≈ . = 0 . = 0 . = 0 . ≈ . =0 . ≈ . ≈ . ≈ . = 0 . = 0 . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . =0 . ≈ . ≈ . ≈ . ≈ . ≈ . = 0 . ≈ . =0 . ≈ . ≈ . ≈ . = 0 . = 0 . N = 2 to8 photons in N modes. For N = 2 and 3 the quantum Fourier transform ( QF T N ) is optimal for both ρ s and ρ d , but for N ≥ N ; due to the QF T ’s symmetry itdoes not matter which port the ‘bad’ photon (see Sec. III C 2) is in, however this is not true of the ρ d interferometers andso we include best, worst and average success probabilities assuming each port is equally likely to be ‘bad’. The completelydistinguishable state is essentially unique, so there is only one success probability to report; an asterix ∗ indicates extensivenumerical optimisation leads us to believe the N = 3 , , ρ d interferometers have constant optical depth (made up of QF T s followed by QF T s) for each N . Interestingly,the two success probabilities of the QF T are always equal except for N = 6, the only case in the table that is not a power of aprime. The measurement outcomes that lead to these probabilities are specified in Table II.
2. Discriminating completely distinguishable states
Numerical optimisation for the N = 4 and 5 photoncompletely distinguishable states yields success probabil-ities of 19 /
24 and 31 /
36, respectively. Both of these areless than the general bounds of Sec. V A, (23 /
24 and35 /
36, respectively), and so we cannot conclude they areoptimal. We observe that they do both exceed the singlydistinguishable bound of 1 − /N , consistent with the in-tuition that it is easier to discriminate a completely dis-tinguishable state than one that is less distinguishable.The numerics are sensitive to the penalties used inEq. (64), due to the existence of interferometers withvery similar performance. For N = 4, a penalty ξ =10 returns an interferometer with success probability25 /
32 that minimises the cost function with a valueof − . /
24 exists but gives a higher valueof − . − . − . N = 4, and 10,12, 14, 15, 16, 18, 20, 35, and 60 for N = 5.While the complexity of the calculations precluded anyfurther optimisation for N >
5, we notice that the bestinterferometers for N = 2 , , , QF T followed by QF T s . This suggests a ‘recursive’structure for the best discriminating interferometers; for N = 6, 7, 8 we tried combinations of QF T N , QF T N − and so on, and found that discriminators composed of QF T s followed by QF T s performed the best. This isremarkable as these are of constant optical depth, inde-pendent of N . Indeed, increasing the optical depth be-yond this seems to decrease the success probability, whichallowed us to limit our search to a manageable number ofconfigurations. These are educated guesses however, anddo not rule out the existence of better interferometersthat might be found.Table I contains a summary of these results. We reportthe probabilities for the best interferometers found tosuccessfully discriminate ρ s and ρ d from ρ i up to N = 8.The measurement outcomes that achieve these probabil-ities up to N = 5 are specified in Table II, where inthe interest of saving space we give the occupations thatfail (i.e. correspond to the ambiguous POVM element E ? ) instead of the successful discriminators, because thelatter far exceed the former. For comparison, for eachinterferometer we include success probabilities for bothstates of interest to be discriminated from the completelyindistinguishable state. Note that as discussed above for N = 3, although the QF T N interferometer is optimalfor ρ s no matter which System mode the ‘bad’ photon isin, this will not be true for interferometers that lack thesymmetry of QF T N . Indeed, the best ρ d discriminatordoes not treat each System mode the same way, and so when using such an interferometer to discriminate ρ s wereport best, worst and average success probabilities, as-suming each System mode is equally likely to contain the‘bad’ photon. VI. DISCUSSION AND FURTHER WORK
Although we have focused on single and complete dis-tinguishability, as shown in Sec. III B the formalism ad-mits arbitrary states. Consider for example Fock arrayswith a single excitation in each System mode and an ar-bitrary Label occupation, call it n L . Applying the Schur-Weyl transform and focusing on the symmetric irrep ( N ),where the support is one dimensional, we see that the re-duced system state will be of the form n L ! N ! | ( N ) , (cid:105) (cid:104) ( N ) , | + (cid:18) − n L ! N ! (cid:19) ρ i . (83)This gives a bound of 1 − n L ! /N ! on the probability forsuccessfully discriminating such a state from the com-pletely indistinguishable one, and includes the singly andcompletely distinguishable cases above. The exact formof such states could be found by reasoning as in Sec. III C.We can use the formalism to compute the number ofparameters that describe an arbitrary partially distin-guishable collection of N particles in N (or more gen-erally d ) modes. Because of the maximal entanglementover p in Eq. (37), when we trace out the Label of anarbitrary totally symmetric state in Eq. (36), the re-sulting mixed state has identical blocks for each copyof λ , (the number of identical blocks being equal to theouter multiplicity). Thus the most general mixed stateis described by a single (Hermitian) block for each ir-rep. Recalling that the number of real parameters ina d -dimensional Hermitian matrix is d , we have foran arbitrary partially distinguishable mixed state of N bosons in N modes (subtracting one for normalisation) (cid:80) λ d { λ } − (cid:0) N + N − N (cid:1) − standard Young tableaux, d ( λ ) . This isbecause coincidence implies each single particle state isdifferent, and so semistandard tableau become standard;in this case we have (cid:80) λ d ( λ )2 − N ! − d dimensions has2( d −
1) real parameters, and every pure state added to aset can add at most one parameter beyond those requiredto describe its projection onto the d − (cid:80) Nd =2 (2( d −
2) + 1) = ( N − real parameters inthis case, which agrees with previous analyses [16] but isfar fewer than the general case. Note that all of thesequantities are of course larger than (cid:0) N (cid:1) , the number ofpairwise distinguishabilities classical intuition might leadone to believe are necessary to measure [25].There are many other state discrimination scenarioswe could consider. For example, we could try to un-6 N ρ s ρ d TABLE II. Measurement occupations corresponding to the ambiguous POVM element E ? that do not discriminate the twostates of interest for the numerically optimised interferometers in Table I ( N = 2 , , ,
5) – these are in general far fewer thanthe number of successful discriminating occupations, and so easier to list. Recall that for ρ s , the optimal choice of QF T N does not depend on the mode in which the single distinguishable photon is present, and neither do the occupations. Note thatalthough all of the occupations not listed here satisfy Eq. (59), some might have zero probability of occurring and thereforenot contribute to discrimination. ambiguously discriminate ρ d from ρ s , or two entirelydifferent states, or more than two states. Note thatdue to the ‘nested’ structure of our three states of in-terest (cf. Eq. (57)), attempting to find a UD POVM { E i , E d , E s , E ? } reduces to only being able discriminate ρ d from the rest. Another version of discrimination toconsider is using bucket (yes/no) instead of number re-solving detectors, which are simpler to engineer. Whileour focus has been on optimizing over all the possiblemeasurement patterns to obtain the highest possible suc-cess probability, as mentioned in Sec. III E another typeof optimisation that can be carried out is choosing a fixedset of patterns and optimizing the interferometer U only.The difference would be that in Eqs. (58, 59) D wouldnow be fixed, simplifying the problem. As an example,during the preparation of this manuscript a closely re-lated paper was released [40], where the authors study asingle reference photon input into a QF T N − , followedby QF T HOM tests on the N − N − N photons in 2 N − D is fixed as the set of N -fold coincidences. The approachis different and so it is not surprising that it is subopti- mal for discrimination, however this interferometer’s be-haviour is clear for all N .Finally, we have no doubt that proofs for many of theresults here, such as QF T N optimality for discriminatingsingly distinguishable states, should be possible, but theyare left as further work.The data associated with this paper is available fordownload at the University of Bristol data repository,data.bris [62]. ACKNOWLEDGEMENTS
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