Symmetry-protection of multiphoton states of light

PAPER • OPEN ACCESS Symmetry-protection of multiphoton states of light To cite this article: Jon Lasa-Alonso et al 2020 New J. Phys. 22 123010 View the article online for updates and enhancements. This content was downloaded from IP address 161.111.10.229 on 23/03/2021 at 12:40 New J. Phys. 22 (2020) 123010 https://doi.org/10.1088/1367-2630/abcb2d OPEN ACCESS RECEIVED 13 July 2020 REVISED 1 November 2020 ACCEPTED FOR PUBLICATION 17 November 2020 PUBLISHED 15 December 2020 Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. PAPER Symmetry-protection of multiphoton states of light Jon Lasa-Alonso1,2,∗,Mart ´ ın Molezuelas-Ferreras1,JJMiguelVarga 1,2, Aitzol Garc´ ıa-Etxarri2,3,G ´ eza Giedke2,3and Gabriel Molina-Terriza1,2,3,∗ 1Centro de F´ ısica de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-San Sebastián, Spain 2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain 3IKERBASQUE, Basque Foundation for Science, Mar ´ ıa D ´ ıaz de Haro 3, 48013 Bilbao, Spain ∗Author to whom any correspondence should be addressed. E-mail: [email protected] and [email protected] Keywords: quantum entanglement, quantum optics, quantum information with photons, quantum nanophotonics, multiphoton states, symmetry-protection Abstract In this manuscript we analyze the emergence of protected multiphoton states in scattering problems with cylindrical symmetry. In order to do that, we first provide a formal definition of the concept of postselected symmetry-protection. We show that the notion of symmetry-protection is not limited to oneor two-photon states, on the contrary, it can be formally extended to the multiphoton case. In addition, we prove for the case of cylindrical symmetry that all possible multiphoton protected states are constructed from a small set of oneand two-photon states. Finally, we point out possible applications that symmetry-protected states may have in quantum communications, concretely, in the construction of decoherence-free subspaces. 1. Introduction The processing of quantum information carried by photons has reached such a level of maturity that photonic quantum computers are becoming competitive in this technological field [1,2]. As was shown in 2001 in a seminal work [3], passive linear optics, i.e. an interferometer, is sufficient for universal photonic quantum computing if combined with single-photon state preparation and feedback based on photon number measurements. More recently it was shown that, even without feedforward, these photonic devices can efficiently perform computational tasks that are supposed to be computationally hard on classical computers (‘boson sampling’) [4–6], something which has been demonstrated in proof-of-principle experiments [7–9]. In fact, the quantum interference of photons is at the heart of the enhancement associated to quantum applications such as the processing and transmission of quantum information, which is essential to establish a quantum network of communications [10]. Quantum information can be encoded in photons within different degrees of freedom, such as transverse momentum, spatial path or time-bin, among others. In particular, the framework based on total angular momentum and helicity [11,12] has gained especial relevance due to applications such as the generation of states in high-dimensional Hilbert spaces [13,14], light–matter interactions [15], data transmission [16], and sensing of chirality in molecules [17–19]. One fascinating feature of this framework is that it allows to describe on the same footing the paraxial and non-paraxial regimes of light [12,20]. This is interesting, because most of the control of light for quantum optics experiments is performed in the paraxial regime, while light–matter interactions typically occur in subwavelength structures, such as atoms, molecules, or nanostructures. Therefore, in order to maximize the interaction in scattering problems, light beams must be strongly focused onto the samples, often leaving the paraxial regime. In fact, the study of the interaction between light and subwavelength structures is receiving a growing interest within the community [21–25]. Although the interaction of light with these structures can be described from the scattering of the electromagnetic modes, at least in the linear regime, the scattering properties of multiphoton states can be rather complex. This is due to quantum © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al Figure 1. In (a) cylindrically symmetric scatterers are depicted. In (b) the splitting of the helicity of a Bessel mode is shown when it interacts with a non-dual scatterer. Modes of same frequency, ω, and linear momentum, pz, are postselected at the output. interference effects and the fact that one can equally describe multiphoton states with different sets of orthogonal modes [26]. In this work, we analyze the emergence of a very specific set of multiphoton states in generic scattering problems. While it is always possible to find eigenstates of a given scatterer, i.e. states which are left invariant in the interaction with the system, these eigenstates normally depend on the particularities of the chosen sample. However, there are situations in which certain states are left invariant by all the scatterers compatible with certain symmetry operations. These so-called ‘symmetry-protected states’4[27]canbe non-trivial and in some situations hard to find. Here, we consider initial states of a known number of photons in a given set of angular momentum light modes and investigate their scattering on cylindrically symmetric structures. We restrict ourselves to the cases where the final state is postselected to contain all input photons in a certain set of output modes. We observe that the symmetries of the physical problem strongly constrain the possible output states. In particular, if a state is left invariant by all the scattering matrices symmetric under rotations and mirror operations, we say that the input state is symmetry-protected in the scattering process. We also show that states that are protected in postselected scattering at cylindrically symmetric structures (figure 1(a)) can only be constructed in the subspace of input states with total angular momentum equal to zero, agreeing with previous results shown in reference [27]. These symmetry-protected states can be useful for sensing the geometrical asymmetries present in nanostructures. Furthermore, studying these states may also pave the way to efficient transmission channels of entangled multiphoton states and decoherence-free subspaces. Actually, due to the generality of the arguments used in this work, these considerations may apply to macroscopic structures such as optical fibers, but also to nanostructures such as nanofibers [31], nanoholes [32] or nanospheres [33,34]. The rest of the manuscript is organized as follows. After setting the general stage on the notion of symmetry-protection in section 2,wespecializeinsection3on the case of cylindrically symmetric systems and introduce the set of modes that we are going to use in this work. In section 4we present the results found for two-photon states, both for modes with null angular momentum and with arbitrary non-zero integer value. In section 5we generalize the results to an arbitrary number of photons, N.Insection6we discuss the applications that symmetry-protected states may have in quantum communications. Finally, in section 7we summarize the main conclusions of the manuscript. 2. Symmetry-protection: general considerations We consider the scattering of a system of photons with mode space Hon a linear passive sample that is invariant under a set of symmetry operations G.WedenotebyTthe full single-particle scattering matrix (usually unitary, though it may include linear losses such that ρ→ TρT†is a trace-nonincreasing completely positive map) and its Fock space representation by ˆ T.ForasubspaceHs⊂Hof modes we denote by HN sthe space of Nindistinguishable photons in the modes Hsand the isometry from the full Fock space to Hsby PN s= N  n1,...,nM−1=0|n1,...,nM−1,nMn1,...,nM−1,nM|, 4We should note that the terminology we use here (introduced in reference [27]) has little connection with ‘symmetry-protected topological phases/order’ [28–30]. The topological notion of symmetry-protection refers to phases of (many-body) quantum systems, where two distinct phases are said to be symmetry-protected topological phases if they cannot be smoothly transformed into each other without closing the gap if the Hamiltonian respects the protecting global symmetry. In contrast, the definition we use here is the property of an N-photon state that is an eigenstate of all postselected scattering matrices that obey the protecting symmetry. 2 New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al with nM=N−n1−n2−···−nM−1.The(N,Hs)-postselected scattering matrix is defined as: ˆ S≡PN sˆ T(PN s)†, which describes the quantum operation acting on HN sobtained after scattering, conditioned on finding all Nphotons again in the modes in Hs. We call an N-photon state, |ψ∈H N s,(Hs)-symmetry-protected (by G)ifitisaneigenstateofall (N,Hs)-postselected scattering matrices that are compatible with G, i.e. matrices that commute with the set of operators in G. The vacuum state |0and all states with dim(Hs)=1 are trivially symmetry-protected since postselection projects on the one-dimensional spacespannedbythestateitself.Thenotionbecomes interesting, however, for N⩾1anddim(Hs)⩾2, which ensures that postselection projects on a subspace of dimension greater than 1. In that case, most states are not protected. There are two reasons why a state |ψmay fail to be protected. First, photons may be scattered between the modes in Hs,performingan( ˆ S-dependent) quantum operation. Postselection (to Nphotons in the modes Hs) is insensitive to these changes and the postselected state is different from the input, hence not protected. This can be resolved by using a different subspace H sin which at least one basis mode is uniquely characterized by quantum numbers preserved by all ˆ Scompatible with all the elements in G. Then, it is straightforward to write down N-photon Fock states that are protected. Since all scattering matrices commute with the symmetry operators in G, the corresponding quantum numbers cannot be changed by ˆ S. Therefore, if a vector |ψ=ˆ a† ψ|0inH sis uniquely defined by preserved quantum numbers, then any state (ˆ a† ψ)N|0is(H s)-symmetry-protected. Note that here postselection projects on a high-dimensional Hilbert space (Nphotons in dim(H s)modes)andthatifˆ Swere not compatible with G(and if ψwere not the unique mode in H swith the given preserved quantum numbers), then this state would in general not be an eigenstate of the postselected scattering matrix. These protected states are all Fock states and are all eigenstates of some symmetry operators. In all the previous cases, one might just as well postselect on the one-dimensional initially populated subspace spanned by the protected state, since none of the other states in HN swill be populated through scattering (by construction). However, as we will see, this type of protection can be extended to superposition states and whole subspaces in which postselection on Nphotons in Hsbrings a genuine advantage. In this case, a second source of decoherence has to be taken into account: the probability that photons are scattered out of the modes in Hsis, in general, different for different modes, which would change an initial superposition state in ˆ S-dependent (and, thus, unknown) ways. Similarly, different states may acquire different phase shifts. And since both mechanisms depend on unknown details of ˆ S,theywill lead to decoherence. In the following, we construct states that are protected against both sources of decoherence in scattering problems with cylindrical symmetry, where Gcomprises the rotations around a symmetry axis and mirror reflections at a plane containing it. We construct different classes of entangled protected states and discuss some uses of the states found. 3. Properties of the eigenmodes of angular momentum and helicity Let us consider photonic eigenstates of one component of the total angular momentum, Jz=Lz+Sz,and helicity (Λ=J·p/p), where Lzand Szare, respectively, the zcomponents of the orbital (OAM) and spin (SAM) angular momenta ([35], chapter XIII), pis the linear momentum operator and pits modulus. Regarding the physical significance of these operators, the zcomponent of total angular momentum is associated with the ability of light to make objects rotate around the OZ axis. Helicity, on the other hand, is a physical magnitude associated with the vectorial character of the electromagnetic field. Now, we label the eigenstates with the eigenvalue of Jz,m={−∞,...,−1, 0, 1, ...,∞},andthesignof theeigenvalueofΛ,λ={−1, +1}. Therefore, our set of electromagnetic modes can be labeled as  Em,λ( x,t), where  Eis the electric field associated with this particular mode, and we will drop the spatio-temporal dependence of the mode from now on. As we are concerned only with the symmetries of our system, we are leaving out other degrees of freedom which would uniquely define the electromagnetic mode. In principle, one could also use the optical frequency, ωand the zcomponent of the linear momentum, pz,andthis would define the set of Bessel modes  Eω,pz,m,λ(see figure 1(b)) [12], or the optical frequency and j,the quantum number of the square of the total angular momentum, J2, forming the set of multipolar modes  Eω,j(j+1),m,λ[36]. For our purposes it is sometimes convenient to use, instead of the helicity eigenstates, the eigenstates of the mirror transformation My, describing a reflection at the xz plane, a symmetry of the scatterers we consider; we label them with their eigenvalue τ={1, −1}, which is also conserved in the interaction with cylindrical samples. 3 New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al Figure 2. Interaction of light with a cylindrically symmetric scatterer. In (a), a classical beam with OAM m−λand helicity λis focused on the scatterer. There are two output beams: one with the same components of OAM and helicity, and another one, with a difference of two units of OAM and opposite helicity. In (b), a single-photon state with angular momentum mand helicity λinteracts with the scatterer. At the output, a superposition of states with the same and opposite helicities is found, with probability amplitudes αm,λand βm,λ, respectively. In the following we consider cylindrically symmetric scatterers, that is, G={My,Rz(θ)=eiθJz:θ∈ [0, 2π)}is formed by the rotations around the OZ axis and reflections at the xz plane as mentioned before. Note that, in this case, Gis the point group C∞v.ForHswe take the space spanned by all Bessel modes with fixed frequency ωand linear momentum, pz. The interest in considering cylindrical symmetry lies on the fact that it is present in many elements of common optical setups. Moreover, it is experimentally more feasible to construct photons with well-defined zcomponent of angular momentum (Bessel modes) than, for instance, with well-defined total angular momentum (multipolar modes). To construct the protected states, we look at subspaces of Hswhich map to themselves under the action of rotations around the OZ axis and the mirror transformation. More specifically, in this work we consider the spaces spanned by the bases H0=span  E0,+; E0,−(1) and Hm=span  Em,+; Em,−; E−m,+; E−m,−.(2) Let us briefly remind of the form that relevant single-particle operators take in these subspaces. In the case of the Hilbert space H0,thezcomponent of angular momentum operator is Jz=diag(0, 0) and the mirror operator is My=01 10 ,(3) while the postselected scattering operator (or input–output relations, see figure 2) for a cylindrical target is given for this space by: S=αβ βα ,(4) with α,β∈C.Forthespacegiveninequation(2)Jz=diag(m,m,−m,−m), the mirror operator can be written as My=⎛ ⎜ ⎜ ⎝ 0001 0010 0100 1000 ⎞ ⎟ ⎟ ⎠ (5) and the scattering operator is S=⎛ ⎜ ⎜ ⎝ ηζ00 γ00 00γ 00ζη ⎞ ⎟ ⎟ ⎠ ,(6) with η,ζ,,γ∈C. Note that any operator, S, defined in this way, fixes the whole dynamics of the scattering problem by defining the linear response of the considered input modes. This implies that the evolution of any input state (even in the multiphotonic case) is grounded in the single-photon nature of the interaction. The main goal of this study is to find states of light which are symmetry-protected, i.e. states that are left invariant by all scattering operators which commute with Jzand My(here and in the following ‘left invariant’ always refers to the state after postselection). One can check at once that the single-photon eigenstates of Myin the space given by equation (1), fulfill this condition, i.e. 4 New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al S E0,++τ E0,−=sτ E0,++τ E0,−(τ=±1), (7) where sτ=α+τβ. 4. Interaction of two-photon states with cylindrical samples We proceed by motivating the general case with the simple case of two-photon states. It was experimentally proved in reference [27] that there is one two-photon state which, when interacting with a circular nanoaperture, remains unaffected. This state is a simultaneous eigenstate of the angular momentum operator and mirror operator. For the particular case of modes with m=0, measured in reference [27], the mirror operator and the angular momentum commute. However, this is not true in the general case of modes with arbitrary angular momentum m. Therefore, we divide the section in two subsections: the study of modes in H0and Hm. When dealing with photon states we will use Fock state notation. In the case of H0 we will use |n1,n2,wheren1(n2) is the occupation number of the mode with positive (negative) helicity, except when noted. On the other hand, when considering space Hm, the notation will be |n1,n2,n3,n4. Each of the nioccupation numbers refers to the modes in Hmfollowing the order expressed in equation (2). 4.1. Two photons in H0 For two indistinguishable photons in the modes in H0, one can construct a three-dimensional space given by: HN=2 0=span {|1, 1,|2, 0,|0, 2}.(8) It can be readily seen that this specific basis for HN=2 0is made of eigenstates of helicity, but the states do not have a well-defined mirror eigenvalue, τ. Due to its importance in the scattering of cylindrically symmetric systems, let us study the properties of the mirror operator. Thus, we construct the ˆ Myoperator in HN=2 0 from equation (3). The transformation of Fock space vectors in equation (8)underthemirroroperatoris given in matrix form by: ˆ My=⎛ ⎝ 100 001 010 ⎞ ⎠(9) (note that we have chosen the notation ˆ Oto represent a generic Fock space operator, whereas the hatless form Ois reserved for the mode operators). If we diagonalize this matrix, we obtain the following set of eigenvalues and orthonormal eigenvectors which also constitute a complete basis set for HN=2 0: |Φ1=|1, 1(τ=1) (10) |Φ2=1 √2|2, 0+|0, 2(τ=1) (11) |Φ3=1 √2|2, 0−|0, 2(τ=−1).(12) Two mirror symmetric and one antisymmetric states are found. The mirror antisymmetric state is uniquely characterized by conserved quantum numbers (total angular momentum and mirror eigenvalues) and, thus, it is protected under postselected scattering. On the other hand, the two mirror symmetric states, in principle, could be mixed after undergoing the scattering process (and it is easy to construct a scattering operator that does so) as they both share the τ=1 quantum number. Thus, |Φ3is an example of an entangled two-photon state which is symmetry-protected under the scattering from an arbitrary cylindrical sample. There is another approach which leads to the same result, but that allows us to find two other states which also are two-photon protected states. Instead of starting with eigenmodes of helicity given in equation (1), one can redefine the single-photon Hilbert space basis and use the eigenstates of the mirror operator given in equation (7). With this approach one obtains three symmetry-protected states for the two-photon case we are studying, which are: |S1=1 2|2, 0+√2|1, 1+|0, 2(13) |S2=1 2|2, 0−√2|1, 1+|0, 2(14) and the previously obtained |Φ3state. Interestingly, one finds that all three of them are Fock states in the protected modes given in equation (7)(|2, 0,|0, 2,and|1, 1, respectively, where the sign ‘’isusedto 5 New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al specify that the mirror eigenbasis is being used, see section 5.1). Let us remark here that this is a general consequence of the single-particle nature of the scattering, i.e. that if ˆ a† k|0 are protected then so are Πk(ˆ a† k)nk|0. For brevity, we sometimes refer to the latter state as a ‘product of the states ˆ a† k|0’. In conclusion, |S1,|S2and |Φ3, are symmetry-protected because they can be written as products of protected single-photon states. This is a particularity of the H0spacethatwillbemoredeeplyanalyzedinthenext section. As we will show later, every protected state with Nphotons in the modes which span H0can be writteninthesamefashion. 4.2. Two photons in Hm When the modes under consideration have m=0, the situation is a bit more complex, due to the fact that the mirror operator does not commute with the angular momentum operator on Hm=0. As before, we start with the space given by the modes in equation (2). The necessity of including states of negative angular momentum is now obvious as we want to consider a subspace that the mirror operator leaves invariant. In this case, the accessible part of the Fock space is ten-dimensional: HN=2 m=span {|2, 0, 0, 0,|1, 1, 0, 0,|1, 0, 1, 0,|1, 0, 0, 1, |0, 2, 0, 0,|0, 1, 1, 0,|0, 1, 0, 1,|0, 0, 2, 0, |0, 0, 1, 1,|0, 0, 0, 2}. (15) The elements in equation (15) can be separated in subspaces with different mtot. This can be done because, in a basis of angular momentum eigenmodes, the eigenvalues of the second quantized total angular momentum of the field are mtot =imi, which give the set of values: 0, 2m,−2m. The elements of each of these subspaces are, respectively: S0=span {|1, 0, 0, 1,|0, 1, 1, 0,|1, 0, 1, 0,|0, 1, 0, 1} S+=span {|2, 0, 0, 0,|1, 1, 0, 0,|0, 2, 0, 0} S−=span {|0, 0, 2, 0,|0, 0, 1, 1,|0, 0, 0, 2}. Itcanbenotedthattheonlysubspacewhichisinvariant (whose elements transform to other elements of the subspace) under the action of the mirror operator is S0. Therefore, states belonging to subspace S0are the only ones which can have simultaneously well-defined angular momentum and mirror eigenvalues. Now, transformations under the mirror operator are given by equation (5), which allows us to construct the mirror operator matrix for the S0subspace as: ˆ My=⎛ ⎜ ⎜ ⎝ 1000 0100 0001 0010 ⎞ ⎟ ⎟ ⎠ , (16) whose eigenvectors and eigenvalues are: |Ψ1=|1, 0, 0, 1(τ=1) (17) |Ψ2=|0, 1, 1, 0(τ=1) (18) |Ψ3=1 √2|1, 0, 1, 0+|0, 1, 0, 1(τ=1) (19) |Ψ4=1 √2|1, 0, 1, 0−|0, 1, 0, 1(τ=−1).(20) AsinthecaseofH0,thestate|Ψ4will not mix (under postselected scattering) with other states, either belonging to spaces with a different m=mor the other three mirror symmetric states in the same S0 subspace. Therefore, for every m, the mirror antisymmetric states generated in this way are protected and do not mix with any other by scattering on a cylindrically symmetric sample (figure 3(a)). Also, it is easy to check that none of the mirror symmetric states that diagonalize the scattering matrix are independent of the scattering coefficients, in other words, symmetry arguments alone cannot warrant their protection (figure 3(b)). Finally, notice that in the single-photon Hilbert space Hm, symmetry-protected states cannot be found. Notwithstanding, in the two-photon case such states exist. This is a consequence of quantum interference and, thus, it is a feature of the multiphotonic nature of the states we are considering. 6 New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al Figure 3. Schematic behavior of the states in equations (17)–(20) interacting with a cylindrically symmetric scatterer. In (a), the only mirror antisymmetric state, |Ψ4, is left invariant through scattering. In (b), a mirror symmetric eigenstate, |Ψ1, generates through scattering a superposition of all the mirror symmetric states. 4.3. Summary of two-photon interactions with cylindrically symmetric samples All light modes can be classified according to their angular momentum and helicity. This classification block-diagonalizes the scattering matrix of cylindrically symmetric objects in submatrices given by modes in spaces H0and Hm. We have seen that for pairs of photons we can always find subspaces S0where the total angular momentum of the state is zero. Importantly, each of these subspaces contains a mirror antisymmetric state which is symmetry-protected. In other words, when scattering these photon pairs off a cylindrical scatterer and postselecting for two photons, we always find the same pair: the scatterer cannot redistribute the two photons in the subspace due to conservation laws and the single-particle nature of the scattering we are considering (see appendix A). It may be interesting to point out, that the two-photon protected states we have identified are entangled according to standard criteria for entanglement of indistinguishable particles used in the literature. State |Φ3of equation (12), for instance, has a Slater number 2 and, thus, it can be considered particle-entangled according to [37]. Nevertheless, it can be written as a product state between the mirror-symmetric and anti-symmetric in H0and, therefore, it is not entangled according to most definitions [26,38]. In contrast, state |Ψ4of equation (20) cannot be written as a single product of creation operators applied to the vacuum state in any way (neither with orthogonal nor with non-orthogonal modes) and, thus, it is entangled according to all these definitions [26,37,38]. Finally, while for two-photon states this procedure has been quite direct, there are still a few questions which remain open. The obvious one is: can we generalize this procedure to arbitrary multiphoton states? In the next section we proceed to generalize our study of symmetry-protection to N-photon states. 5. Interaction of multiphoton states with cylindrical samples The search for symmetry-protected states in the multiphoton case is, in general, much more complicated. As the number of particles increases, all the eigenspaces of interest in which to search for protected states increase in dimension, making it harder to find or exclude solutions. In particular, the simple sufficient condition for protection—being a state uniquely characterized (within the postselected space) by ˆ Jzand ˆ My eigenvalues loses its usefulness as all the simultaneous eigenspaces become degenerate for N>2. One can, however, dig into the formal definition of symmetry-protection and try to make it operative. A mathematical procedure to construct or exclude N-photon symmetry-protected states is presented in appendix Bbased on this idea. While the basic reasoning can be used for any type of scattering problem under symmetry constraints, here we exploit the specific relations between the eigenvectors and eigenvalues of cylindrically symmetric scattering matrices. We use it to prove that there are no other symmetry-protected states in HN mapart from products of the state given by equation (20). In what follows, we proceed as before, by studying symmetry-protection separately for HN 0and HN m spaces. 5.1. Nphotons in H0 As explained earlier, to understand symmetry-protection in HN 0, we should begin with the set of single-photon modes which are joint eigenstates of Myand Jzoperators, i.e., |1, 0=1 √2|1, 0+|0, 1(21) and |0, 1=1 √2|1, 0−|0, 1.(22) 7 New J. Phys. 22 (2020) 123010 J Lasa-Alonso et al Equation (7) shows that states in equations (21)and(22) are left invariant when impinging on a cylindrical sample. Thus, one can construct multiphoton states which are protected by defining creation and annihilation operators for these states and taking their products as pointed out in section 2.Denotingby ˆ a† 0,s|0, the mirror symmetric state in Equation (21), and ˆ a† 0,a|0, the mirror antisymmetric mode in equation (22), we can identify symmetry-protected states of Nphotons in the following way: |ns,na= t=s,a ˆ a† 0,tnt √nt!|0, (23) where N=na+nsand ns(na) is the occupation number of the mirror symmetric (antisymmetric) photon mode. All these states have well-defined angular momentum and mirror transformations. In particular, their mirror eigenvalue is given by (−1)na. Finally, just for completeness, when Nis odd there are (N+1)/2 mirror symmetric and (N+1)/2 mirror antisymmetric states of this kind. However, in the case of Nbeing even, there are N/2mirror antisymmetric states, and N/2+1 symmetric states. In both cases, the total number of states is N+1. 5.2. Nphotons in Hm Following the reasoning of the previous sections, we know that products of symmetry-protected states are also protected. Therefore, a state of the form |Ψ=ˆ a† m,+ˆ a† −m,+−ˆ a† m,−ˆ a† −m,−N/2|0(24) must be left invariant by any cylindrically symmetric scatterer. Note that this state belongs to the S0 subspace of the N-photon Fock space and its mirror symmetry depends on whether N/2 is even or odd. In general, products of such states constructed from different mand Nvalues are also protected, even the products of these states and the ones obtained in equation (23). Interestingly, the state given in equation (24) is the only symmetry-protected state that can be obtained for a fixed value of mand N. This can be proved from the very general definition of symmetry-protection given in section 2, exploiting the properties of the eigenstates of cylindrically symmetric scattering matrices. The details are given in appendix B. The proof rests on the defining property that a protected state is required to be an eigenstate of all scattering operators S,S,..., etc compatible with the group of symmetry operators G={My,Rz(θ)=eiθJz:θ∈[0, 2π)}. Importantly, this constraint not only determines the possible form of any compatible scattering matrix, as shown in equation (6), but also the transformations between the eigenmodes of two compatible scattering matrices. Finally, we observe that the transformations between two infinitesimally distinct scattering matrices Sand Ssuffice to prove that the symmetry-protected state in equation (24)isunique. 6. Symmetry-protection and decoherence-free subspaces Note that so far we have discussed the protection of one-dimensional subspaces, namely single multiphoton states that are preserved under scattering when postselecting on subspaces of HN 0and HN mN-photon Fock spaces with null total angular momentum. While this provides an interesting characterization of the scatterer and may be useful for certain applications, itisnotsufficienttotransmitqubitsorotherformsof quantum information, since a two-dimensional Hilbert space is needed to encode a qubit. Therefore, to profit from symmetry-protection for the transmission of qubits, at least a two-dimensional protected subspace is required. However, there is no way that cylindrical symmetry alone can guarantee that after postselection a state like (aPm+bPm)|0is unchanged, where Pm|0and Pm|0represent N⩾1-photon protected states as constructed above. Symmetry arguments alone cannot warrant that the scattering transformation of the states is independent of m: while the use of protected states ensures that the transformation is proportional to the identity, both the amplitude and the phase may depend on m, and thus both the relative phase and amplitude of aand bcan change, decohering the qubit. However, as we now show, with one additional assumption on the scatterer, decoherence-free subspaces may be constructed. Moreover, we show that the construction of these subspaces is possible even in the case where losses are present. In what follows we consider that this is, in fact, the case. If the scatterer (and hence the scattering matrix) can be considered to be static, i.e. constant during a time interval [t1,t2], then a protected state Pm|0scattered at time t1or time t2will undergo exactly the same transformation (loss in amplitude and phase change) and, therefore, if we can postselect on no losses, any superposition of the two would be unaffected. Indeed, for the protected states constructed in the 8