Beam splitter as quantum coherence-maker Laura Ares∗ and Alfredo Luis† Departamento de Óptica, Facultad de Ciencias F́ısicas, Universidad Complutense, 28040 Madrid, Spain (Dated: May 20, 2022) The aim of this work is to answer the question of how much quantum coherence a beam splitter is able to produce. To this end we consider as the variables under study both the amount of coherence of the input states as well as the beam splitter characteristics. We conclude that there is an optimal combination of these factors making the gain of coherence maximum. In addition, the two mode squeezed vacuum arises as the studied state most capable of gaining coherence when passing through a beam splitter. These results are qualitatively equivalent for the the l1-norm of coherence and for the relative entropy of coherence. I. INTRODUCTION Quantum coherence is the archetype of quantum cor- relations. It potentially concentrates all the superposi- tion principle consequences, meaning all the fascinating fundamental implications [1–3] along with the prospering practical applications [4–6]. This general role allows co- herence to be transformed into other types of correlations such as entanglement or steering [7–9]. Quantum resources theories [10–12] have proved to be a useful support to take advantage of all these quantum features, for example in metrology [13] or quantum in- formation processing [14]. The general scheme of these theories is common, broadly, to quantify the resource by defining the states lacking the resource, the operations that do not generate it, and finally the quantifier that accounts for the amount of resource present [15, 16]. In this work we focus on characterize one operation that is not free with regard to quantum coherence, the beam splitter, which is able to enhance the coherence previ- ously present in the system [17, 18]. This behaviour of beam splitter as coherence-maker is already present in classical optics. Moreover, beam splitters are the key el- ement involved in generating other quantum correlations from quantum coherence [8, 19]. These two properties make us wonder how the beam splitter parameters and the input states influence the increase in coherence. Accordingly, we investigate which choice of the reflectance-transmittance ratio and phase changes pro- duce maximum coherence for a given input state. As the input states we consider incoherent as well as coherent states, so we can evaluate the final amount of coherence regarding the initial one. We also divide the problem into subspaces of fixed incoming energy and compare the possible distributions of photons between the two input modes. Since coherence is a basis dependent quantity, throughout this analysis we are working in the photon number basis so we are always talking about photon- number-coherence. The quantifiers of coherence utilized are the l1-norm and the relative entropy of coherence [16]. ∗Electronic address: laurares@ucm.es †Electronic address: alluis@ucm.es II. BEAM SPLITTER The action of the beam splitter can be expressed in matrix form as a relation between the input a1,2 and output b1,2 complex-amplitude operators( b1 b2 ) = ( τ1 ρ2 ρ1 τ2 )( a1 a2 ) , (1) where τ1,2 ρ1,2 are the corresponding complex transmis- sion and reflection coefficients as defined in Fig. 1. For a lossless beam splitter the transformation matrix is uni- tary, which means [20] |τ1| = |τ2| = cos θ, |ρ1| = |ρ2| = sin θ, (2) where we have introduced the parameter θ to express the balance between transmission and reflection, with θ ∈ [0, π/2], and δρ1 − δτ2 + δρ2 − δτ1 = π, (3) where δ are the phases of the corresponding coefficients. FIG. 1: Scheme of a beam splitter specifying the modes involved and the definition of the transmission and reflection coefficients τ1,2 ρ1,2. ar X iv :2 20 5. 09 69 7v 1 [ qu an t- ph ] 1 9 M ay 2 02 2 mailto:laurares@ucm.es mailto:alluis@ucm.es 2 III. INCOHERENT INPUT STATE We start this analysis by considering a beam splitter illuminated by a number state on both input ports. The corresponding incoming number state |n1〉1 ⊗ |n2〉2 = |n1, n2〉 can be expressed in terms of the vac- uum as |n1, n2〉 = 1√ n1!n2! a†n1 1 a†n2 2 |0, 0〉. (4) The output state can be readily obtained in the number basis by inverting the input-output relation (1) to express the input modes a1,2 in terms of the output modes b1,2, and translating the result to Eq. (4) to get |out〉n1,n2 = 1√ n1!n2! ( eiδτ1 cos θb†1 + eiδρ2 sin θb†2 )n1 ( eiδρ2 sin θb†1 + eiδτ2 cos θb†2 )n2 |0, 0〉. (5) Equivalently, the decomposition formulas for the su(2) Lie algebras may be used [21] . After some simple algebra, and using relation (3) we get that the output state (5) becomes |out〉n1,n2 = n1+n2∑ j=0 cj |j, n1 + n2 − j〉, (6) where cj = √ n1!n2!j!(n1 + n2 − j)!eij(δτ1−δρ2 ) n1∑ k=max(0,j−n2) (−1)k cosn2+2k−j θ sinn1−2k+j θ (n1 − k)!k!(n2 + k − j)!(j − k)! . (7) In the particular case of n1 = 0 or n2 = 0 we get that |out〉 are SU(2) coherent states [22]. The computation of the coherence depends just on the modulus of |cj | so that the actual value of the phase δτ1 − δρ2 is irrelevant. As suitable coherence measures we have the l1-norm of coherence, CH , and the relative entropy of coherence, CS , which we will use in their forms adapted to pure states [10, 16] CH = ∑ j |cj | 2 − 1, (8) and CS = − ∑ j |cj |2 ln |cj |2, (9) respectively. If we restrict ourselves to a subspace of fixed energy, n1+n2 = constant, the maximum coherence holds for |cj | = constant [23, 24], this is for the phase-like states |φ〉 = 1√ n1 + n2 + 1 n1+n2∑ j=0 eiφj |j, n1 + n2 − j〉, (10) where φj are arbitrary phases [25, 26]. The corresponding maximum of coherence is Cmax H = n1 + n2, (11) leading to a curious but accidental identification of co- herence with energy. In the case of the relative entropy of coherence the maximum value is Cmax S = −ln [ 1 n1 + n2 + 1 ] . (12) As we are about to see, not all incoming states are able to reach this maximum coherence, irrespective of the beam splitter parameters. This idea leads us to an alter- native expression for the coherence as a the maximum overlap between the system state |ψ〉, assumed pure, and the phase-like states |φ〉 when φ is varied CH = ( √ n1 + n2 + 1 maxφ〈φ|ψ〉)2 − 1. (13) A. One photon This case, n1+n2 = 1, is a rather simple situation since essentially there is just a single configuration, the in- put number state |1, 0〉. The corresponding output state, omitting irrelevant relative phases, is the split photon |out〉1,0 = cos θ|1, 0〉+ sin θ|0, 1〉, (14) that without any calculus shows that maximum coher- ence holds for a 50 % beam splitter θ = π/4, for which |out〉1,0 becomes a phase-like state |φ〉 in Eq. (10). 3 B. Two photons In this situation where n1 + n2 = 2 we have just two meaningful cases, say SU(2) coherent states |2, 0〉 and SU(2) squeezed states |1, 1〉 [22, 27, 28], with output states |out〉2,0 = (15) cos2 θ|2, 0〉+ √ 2 sin θ cos θ|1, 1〉+ sin2 θ|0, 2〉, and |out〉1,1 = √ 2 sin θ cos θ|2, 0〉+ (16)( cos2 θ − sin2 θ ) |1, 1〉 − √ 2 sin θ cos θ|0, 2〉. In Fig. 2 we represent the coherence CH for these states as a function of θ, blue solid line for the |out〉2,0 and red dashed line for |out〉1,1, while the green line marks the maximum coherence in Eq. (11). 0.5 1.0 1.5 Θ 0.5 1.0 1.5 2.0 CH FIG. 2: CH for the states |out〉2,0 (blue solid line) and |out〉1,1 (red, dashed line) as functions of the beam splitter’s parameter θ. The green, dotted line marks the maximum coherence (11). We can appreciate that varying the transmittance- reflectance ratio of the beam splitter, the |out〉1,1 state can get larger coherence than the SU(2) coherent state |out〉2,0, which betrays its name a bit. Moreover, it can be clearly seen that the SU(2)- coherent case |out〉2,0 finds its maximum coherence for a balanced beam splitter θ = π/4. On the other hand, the coherence for the |out〉1,1 case finds its maximum for an unbalanced disposition, that can be found analytically to be given by the θ satisfying the equality tan(2θ) = ± √ 2. For these optimum beam splitters we have that the out- put are actually phase-like states (10) |out〉1,1 = 1√ 3 (±|2, 0〉+ |1, 1〉 ∓ |0, 2〉) , (17) while for a balanced beam splitter θ = π/4 we will have the N00N state |out〉1,1 = 1√ 2 (|2, 0〉 − |0, 2〉) . (18) 0.5 1.0 1.5 Θ 0.2 0.4 0.6 0.8 1.0 CS FIG. 3: CS for the states |out〉2,0 (blue solid line) and |out〉1,1 (red, dashed line) as functions of the beam splitter’s parameter θ. The gray, dotted line marks the maximum coherence achieved. This result is reproduced if we consider the relative entropy of coherence. As expected, the absolute values of CH and CS are different, however the tendencies and the optimum configurations of the beam splitter are alike. From Fig. 3 we can highlight that the behaviour around the balanced beam splitter is much smoother in this case. C. Larger photon numbers For photon number larger than two we present nu- merical computations confirming the main results al- ready commented. For the case of four photons, dis- played in Fig. 4a, the larger coherence is obtained by the twin-photon states |out〉2,2 emerging from an unbal- anced beam splitter. Such states |out〉2,2 are no longer phase-like states, Eq. (10), and so the absolute maxi- mum coherence in Eq. (11) is not reached. Nevertheless, it can be shown numerically that for the optimum case, this is |out〉2,2 at optimum θ, there is a symmetric split of the photons between the output modes, this is |〈n1, n2|out〉2,2| = |〈n2, n1|out〉2,2|. (19) This symmetry feature is specially useful in relation to the sensitivity of two-path interferometers [29]. Regarding odd number of photons, that is n1 + n2 = 2k + 1 for integer k, the situation is quite similar, with maximum coherence obtained for input states closer to equal splitting of the photons between input modes, say |k + 1, k〉, and unbalanced beam splitter θ 6= π/4, as shown in Fig. 4b for five photons. The case of three photons is special, as shown in Fig. 4c, since both states |out〉3,0 (blue solid line) and |out〉2,1 (red dashed line) reach the maximum coherence for the balanced beam splitter θ = π/4, although neither do they achieve the maximum coherence. 4 0.5 1.0 1.5 Θ 1 2 3 4 CH (a) CH for the states |out〉4,0 (blue solid line), |out〉3,1 (black, dotted-dashed line), and |out〉2,2 (red,dashed line) 0.5 1.0 1.5 Θ 1 2 3 4 5 CH (b) CH for the states |out〉5,0 (blue solid line), |out〉4,1 (black, dotted-dashed line), and |out〉3,2 (red, dashed line). 0.5 1.0 1.5 Θ 0.5 1.0 1.5 2.0 2.5 3.0 CH (c) CH for the states |out〉3,0 (blue solid line) and |out〉2,1 (red, dashed line). FIG. 4: Plots of the l1-norm of coherence for different incoherent input states as functions of the parameter θ. The green, dotted lines mark the maximum coherence in each subspace (11), and the thicker gray line shows the maximum coherence achieved. The maximum coherence scales linearly with the total number of photons but the overlap between the possible outcomes and the phase-states decreases with the total energy so the maximum coherence Cmax H = n1 + n2 be- comes more and more distant to Eq. (13). We show this evolution in Fig. 5. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à à à à à 5 10 15 20 NT 5 10 15 20 CH FIG. 5: Maximum coherence achieved, CHmax , for incoming number states on each subspace as a function of the total number of photons involved NT (gray squares). In green circles, Cmax H = n1 + n2. The results of this section are also reproduced by the relative entropy of coherence. For the sake of comparison we replicate in Fig. 5 the cases of 3, 4 and 5 photon- subspaces. Likewise, we can see how the maximum CS achieved by this incoming states distances from the maximum avail- able coherence in each subspace, Eq. (12). This compar- ison is presented in Fig. 7. IV. TWO MODE SQUEEZED VACUUM We take advantage of the previously computed coher- ence for the number states to calculate the coherence ob- tained when each port of the beam splitter is illuminated with one of the modes of the two mode squeezed vacuum state (TMSV), |ξ〉 = √ 1− ξ2 ∞∑ n=0 ξn|n, n〉, (20) where ξ is the squeezing parameter, considered real with- out loss of generality. Therefore, we can calculate its coherence as CH =  ∞∑ n=0 √1− ξ2ξn n∑ j=0 |cj | 2 − 1. (21) In Fig. 8 it is represented the final coherence as a function of the parameter θ for different squeezing pa- rameters ξ. The minimum coherence, at θ = 0 coincides with the coherence of the squeezed vacuum without any beam splitter transformation [18] CH = 2ξ 1− ξ . (22) Once more, the results are replicated by the relative en- tropy of coherence (see Fig. 9). Since the coherence of the initial state is not zero, we can define the coherence gained when it goes through the beam splitter as a function of the incoming amount of coherence. To this end we define the gain in coherence as the next percentage G = CHmax CHmin × 100. (23) 5 0.5 1.0 1.5 Θ 0.5 1.0 1.5 CS (a) Relative entropy of coherence for the states |out〉4,0(blue solid line), |out〉3,1 (black, dotted-dashed line), and |out〉2,2 (red, dashed line), as functions of the parameter θ. 0.5 1.0 1.5 Θ 0.5 1.0 1.5 CS (b) Relative entropy of coherence for the states |out〉5,0(blue solid line), |out〉4,1 (black, dotted-dashed line), and |out〉3,2 (red, dashed line), as functions of the parameter θ. 0.5 1.0 1.5 Θ 0.2 0.4 0.6 0.8 1.0 1.2 CS (c) Relative entropy of coherence for the states |out〉3,0(blue solid line), |out〉2,1 (red,dashed line), as functions of the parameter θ. FIG. 6 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à à à à à 5 10 15 20 NT 1.0 1.5 2.0 2.5 3.0 CS FIG. 7: Maximum relative entropy of coherence achieved, CSmax , for incoming number states on each subspace as a function of the total number of photons involved NT (gray squares). In green circles, Cmax S = −ln [ 1 n1+n2+1 ] . Different definitions of this concept have been developed [30]. We can compute G depending on the squeezing of the incoming state, and thus on the mean number of photons. In Fig. 10 it is shown how this gain grows rapidly with the squeezing parameter. V. COHERENT STATE We compute the coherence provided by the beam split- ter when it is illuminated with coherent states on each port. We consider the same beam splitter introduced in Eqs. (1)-(3). The input state in terms of the vacuum state is |α1, α2〉 = eα1a † 1−α ∗ 1a1eα2a † 2−α ∗ 2a2 |0, 0〉, (24) 0.5 1.0 1.5 Θ 1 2 3 4 5 6 CH FIG. 8: CH at the output ports of a beam splitter when the inputs are two-mode squeezed vacuum states with ξ = 0.2 (black, solid line) and ξ = 0.5 (red, dashed line) as a function of the parameter θ. 0.5 1.0 1.5 Θ 0.2 0.4 0.6 0.8 1.0 CS FIG. 9: CS at the output ports of a beam splitter when the inputs are two-mode squeezed vacuum states with ξ = 0.2 (black, solid line) and ξ = 0.5 (red, dashed line) as a function of the parameter θ. 6 æ æ æ æ æ æ æ 0.2 0.3 0.4 0.5 0.6 0.7 Ξ 250 300 350 400 450 CHmax CHmin % FIG. 10: Percentage of coherence enhancement for the two-mode squeezed vacuum as a function of the squeezing parameter ξ. and the output becomes |out〉α1α2 = (25) |eiτ1α1 cos θ + eiρ1α2 sin θ, eiτ2α2 cos θ + eiρ2α1 sin θ〉. We compute the coherence of the outcome state when the input state is a coherent coherent of N̄ = 4 on one input port and the vacuum state on the other. In Fig. 11 it can be seen how the optimum beam splitter configu- ration is the one that allows a symmetrical output, with the same mean number of photons on each mode. 0.0 0.5 1.0 1.5 Θ 10 15 20 25 30 35 40 CH FIG. 11: CH for the state |out〉α1=2,α2=0 as a function of the parameter θ. It can be seen that the minimum of CH is the coher- ence of the single-mode coherent state CHmin = CH(|α =√ N̄〉) and it appears when the output state is of the form |α = √ N̄ , 0〉. As in the incoherent case, this result can be reproduced by the relative entropy of coherence as illustrated in Fig. 12. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Θ2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 CS FIG. 12: CS for the states |out〉α1=2,α2=0 as functions of the parameter θ. We may analyze the advantage of the beam splitter in terms of coherence by computing the gain in Eq. (23). In Fig. 13 it can be seen how the gain of coherence increases with the initial coherence of the single-mode state but the growth ratio is smaller for high N̄ . � � � � �� � ��� ��� ��� ��� ����� ����� % FIG. 13: Plot of the percentage of coherence gained as a function of the mean number of photons. A. Comparison with TMSV In the following we compare the performance of the beam splitter when illuminated by the coherent state and by the TMSV when these present identical: a) minimum amount of coherence, CHmin , and b) mean number of photons, N̄ . As introduced, we are considering two states with the same minimum amount of coherence, e. g. CHmin = 3.0. The corresponding coherent state before the beam splitter is |α1 = √ 0.83, 0〉, and the gain in coherence caused by the transformation is Gα ≈ 245%. (26) 7 However, if we consider a TMSV with the same mini- mum coherence available, this is |ξ = 0.6〉, the gain pro- duced by the beam splitter is remarkably higher, Gξ ≈ 364%. (27) Regarding the energy, the mean number of photons of the TMSV is N̄ = ξ2 1− ξ2 , (28) thus, for N̄ = 1 the gain in coherence caused by the beam splitter is Gξ ≈ 470% (29) whereas for a coherent state with |α|2 = 1 it is Gα ≈ 260%. (30) Therefore, the TMSV is able to gain more coherence since the squeezing allows it to better resemble the con- stant statistics of phase-like states. VI. CONCLUSIONS We have performed a detailed study of the role of beam splitters as quantum coherence makers, obtaining the op- timum configuration of the reflectance-transmitance pa- rameters for several incoming states. The optimum con- figuration is such that the outcoming state is as similar as possible to the phase-like states. We have investigated how the amount of energy of the input state concerns the maximum coherence achievable by the system. 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Comput. 16, 1282–1294 (2016). https://doi.org/10.1016/S0079-6638(00)80021-9 https://doi.org/10.1016/S0079-6638(00)80021-9 https://doi.org/10.1016/S0079-6638(00)80021-9 https://doi.org/10.1103/PhysRevA.49.4968 https://doi.org/10.1103/PhysRevA.54.4505 https://doi.org/10.48550/arXiv.1510.06683 I Introduction II Beam splitter III Incoherent input state A One photon B Two photons C Larger photon numbers IV Two mode squeezed vacuum V Coherent state A Comparison with TMSV VI Conclusions References