General mapping of multi-qu$d$it entanglement conditions to non-separability indicators for quantum optical fields

We show that any multi-qudit entanglement witness leads to a non-separability indicator for quantum optical fields, which involves intensity correlations. We get, e.g., necessary and sufficient conditions for intensity or intensity-rate correlations to reveal polarization entanglement. We also derive separability conditions for experiments involving multiport interferometers, now feasible with integrated optics. We show advantages of using intensity rates rather than intensities, e.g., a mapping of Bell inequalities to ones for optical fields. The results have implication for studies of non-classicality of"macroscopic"systems of undefined or uncontrollable number of"particles".

Non-classicality due to entanglement initially was studied using quantum optical multiphoton interferometry, see e.g., [1]. The experiments were constrained to defined photon number states, e.g., the two-photon polarization singlet [2]. This includes Greenberger-Horne-Zeilinger (GHZ) [3] inspired multiphoton interference, with an interpretation that each detection event signals one photon. Spurious events of higher photon number counts contributed to a lower interferometric contrast. Still, states of undefined photon numbers, e.g., the squeezed vacuum, can be entangled [4][5][6].
This form of entanglement of quantum optical fields served e.g., to show that a strongly pumped two-mode (bright) squeezed state allows one to directly refute the ideas of EPR [7], as it approximates their state, and a form of Bell's Theorem can be shown for it [4]. The trick was to use displaced parity observables. Recently it has been shown that this is also possible for four-mode bright squeezed vacuum [8], which can be produced via type II parametric down-conversion, see e.g [5,6]. In this case the state approximates a tensor product of two EPR states, and interestingly can also be thought of as a polarization "super-singlet" of undefined photon numbers [9]. The approach of Ref. [8] used (effectively) intensity observables, which are less experimentally cumbersome.
With the birth of quantum information science and technology, entanglement became a resource. We have an extended literature on detection of entanglement for systems of finite dimensions, essentially "particles", see e.g., [10]. It is well known that not all entangled states violate Bell inequalities. Still there is theory of entanglement indicators, called usually witnesses, which allow to detect entanglement, even if a given state for finite dimensional systems (essentially, qudits) does not violate any known Bell inequalities. The case of two-mode entanglement for optical fields was studied in trailblazing papers of [11,12], which discussed "two-party continuous variable systems", and with a direct quantum optical formalism in [13]. The entanglement conditions reached in the papers did not involve intensity correlations.
An entanglement condition for four-mode fields, which was borrowing ideas from two spin-1/2 (two-qubit) correlations, involved correlations Stokes operators and was first discussed in [5]. The resulting indicator was used to measure efficiency of an "entanglement laser". The output of the "laser" was bright four-mode vacuum. We shall present here the most extensive generalization of such an approach, i.e., entanglement indicators for optical fields which are derivatives of multi-qudit entanglement witnesses involving intensity correlations. In Supplementary Material [14] we give examples of entanglement conditions based on such an approach. Some of them are more tight versions of the entanglement conditions mentioned above.
As a growing part of the experimental effort is now directed at non-classical features of bright (intensive, "macroscopic") beams of light, e.g., [15][16][17][18][19][20][21] so the time is ripe for a comprehensive study of such entanglement conditions. All that may lead to some new schemes in quantum communication and quantum cryptography, perhaps on the lines of Ref. [9]. The emergence of integrated optics allows now to construct stable multiport interferometers [22][23][24][25][26][27][28][29], and is our motivation of going beyond two times mode case.
We present a theory of mapping multi-qudit entanglement witnesses [10] into entanglement indicators for quantum optical fields, which employ intensity correlations or correlations of intensity rates. By intensity rates we mean the ratio of intensity at a given local detector and the sum of intensities at all local detectors (in some case the second approach leads to better entanglement detection). The method may find applications also in studies of non-classicality of correlations in "macroscopic" many-body quantum systems of undefined or uncontrollable number of constituents, e.g., Bose-Einstein condensates [30], other specific states of cold atoms [31,32].
The essential ideas are presented for polarization measurements by two observers and the most simple model of intensity observable: photon-number in the observed The experiments (two parties). Two multi-mode beams propagate to two spatially separated measurement stations. Each station consists of a d input d output tunable multi-port beamsplitter-interferometer (MPBS) and detectors at its outputs. For polarization measurements put dA = dB = 2, and treat the paths as polarization modes.
mode. Next, we present further generalization of our approach, and examples employing specific indicators involving intensity correlations for unbiased multiport interferometers. We discuss generalizations to multi-party entanglement indicators. We show that the use of rates leads to a modification of quantum optical Glauber correlation functions, which gives a new tool for studying non-classicality, and that it also gives a general method of mapping standard Bell inequalities into ones for optical fields.
We discuss spatially separated stations, X = A, B, ... with (passive) interferometers of d X input and output ports, FIG. 1. In each output there is a detector which measures intensity. One can assume either a pulsed source, sources acting synchronously [33,34] or that the measurement is performed within a short time gate. Each time gate, or pulsed emission, is treated as a repetition of the experiment building up averages of observables.
Stokes parameters.-For the description of polarization of light, the standard approach uses Stokes parameters. Using the photon numbers they read Θ j = â † jâ j −â † j ⊥â j ⊥ , where j, j ⊥ denote a pair of orthogonal polarizations of one of three mutually unbiased polarization bases j = 1, 2, 3, e.g., The zeroth parameter Θ 0 is the total intensity: N = â † jâ j +â † j ⊥â j ⊥ . Alternative normalized Stokes observables were studied by some of us [35][36][37]. They were first intorduced in [38], however a different technical approach was used. Following [35] one can put Ŝ j = NΠ , and Ŝ 0 = Π , whereΠ = 1−|Ω Ω| and |Ω is the vacuum state for the considered modes, a j |Ω =â j ⊥ |Ω = 0. Operationally, in the r-th run of an experiment, we register photon numbers in the two exits of a polarization analyzer, n r j and n r j ⊥ , and divide their difference by their sum. If n r j + n r j ⊥ = 0 , the value is put as zero. This does not require any additional measurements, only the data are differently processed than in the standard approach. In [35][36][37] examples of two-party entanglement conditions and Bell inequalities using normalized Stokes operators were given. Here we present a general approach. Map from two-qubit entanglement witnesses to entanglement indicators for fields involving Stokes parameters.-Pauli operators σ = (σ 1 ,σ 2 ,σ 3 ) andσ 0 = 1 form a basis in the real space of one-qubit observables. Thus, any two-qubit entanglement witness,Ŵ , has the following expansion: First, we show that (1) holds for any pure state ψ AB . Let us denote the polarization basis H and V aŝ x H =x 1 andx V =x 2 . Normalized Stokes operators in arbitrary direction can be put as m · S X , where m is an arbitrary unit real vector, or in the matrix form We introduce a set of states where k, m ∈ {1, 2}. This allows us to put where the matrix elements ofR AB ψ are Ψ AB km Ψ AB ln . As a Gramian matrix,R AB ψ is positive. Except for |ψ AB describing vacuum at one or both sides, we have 0 < TrR AB ψ = Π AΠB ≤ 1. Thus,R AB ψ =R AB ψ / Π AΠB is an admissible density matrix of two qubits.
For mixed states , i.e., convex combinations of ψ AB λ 's with weights p λ , one getsR AB = λ p λR AB λ which is positive definite, and its trace is λ p λ TrR AB λ ≤ 1. Thus after the re-normalization one gets a proper two-qubit density matrixR AB . As purity of a field state ψ AB λ does not warrant that the corre-spondingR AB λ is a projector,R AB does not have to have the same convex expansion coefficients in terms of pure two-qubit states, as in terms of ψ AB λ 's. For any separable pure state of two optical beams |ψ AB prod , defined as F † A F † B |Ω , where F † X is a polynomial function of creation operators for beam (modes) X, and |Ω is the vacuum state of both beams, the matrixR AB factorizes:R AB =R ARB . Simply, can be shown to be a qubit density matrix and Ŵ sep ≥ 0, therefore for pure separable states of the optical beams Ŵ S prod ≥ 0. Obviously, Ŵ S sep ≥ 0 also for all mixed separable states.
Standard Stokes operators case.-Any standard Stokes operator can be put as m · Θ X = klx † k ( m · σ) klxl . We introduce state vectors Φ AB jk =â jbk ψ AB . One has where the matrixP AB has entries Φ AB km Φ AB ln , it is positive definite, and its trace is N ANB . Thus,P AB = P AB / N AN B is an admissible two-qubit density matrix, and one has Ŵ Θ / N AN B = TrŴP AB . All that leads to Ŵ Θ sep ≥ 0. Note that, for a general stateR AB does not have to be equal toP AB . Still, R AB =P AB for states of defined photon numbers in both beams.
Reverse map.-Any linear separability condition expressible in terms of correlation functions of normalized Stokes Parameters reads: µν ω µν Ŝ A µŜ B ν sep ≥ 0. As two-photon states, with one at A and the other at B, are possible field states, thus for any separable such state we must have µν ω µν Ŝ A µŜ B ν sep−2−ph ≥ 0. This is algebraically equivalent to µν ω µν σ µ ⊗σ ν sep ≥ 0, for any two-qubit state. We get an entanglement witness. Therefore, we have an isomorphism. Similar proof applies to standard Stokes observables. Examples where |F η is the state |F after the above described losses in both channels, and where m X is the total number of photons in channel X, before the losses. Expanding |F in terms of Fock states with respect to different polarizations than i, i ⊥ and j, j ⊥ , does not change the values of m X , and thus the formula stays put for any indices. Again we have a strong resistance of the entanglement indicators with respect to losses. Especially for states with high photon numbers, the entanglement conditions based on normalized Stokes parameters, may be more resistant to losses, because 0 < η < 1 one has Multi-party case.-Consider three parties, and the case of indicators of genuine three-beam entanglement. Any genuine three-qubit entanglement witnessŴ (3) has the property that it is positive for pure product threequbit states |ξ AB,C = |ψ AB |φ C , for similar ones with qubits permuted, and for all convex combinations of such states. With any pure partial product state of the optical beams, e.g. |Ξ AB,C = F † AB F † C |Ω , where F † AB is an operator built of creation operators for beams A and B, etc., one can associate, in a similar way as above, a partially factorizable three-qubit density matrixR AB ψR C φ . Thus, the homomorphism works. Generalizations are obvious.
General Theory.-Consider a beam of d A quantum optical modes propagating toward a measuring station A, and a beam of d B modes toward station B. We associate with the situation a d A × d B dimensional Hilbert Space, X , be an orthonormal, i.e. TrV X iV X j = δ ij , Hermitian basis of the space of Hermitian operators acting on C d X . Therefore, productsV A i ⊗V B j form an orthonormal basis of the space of Hermitian operators acting on C d A ⊗ C d B . Thus, any entanglement witness for the pair of qudits,Ŵ , can be expanded intô with real w jk . The optimal expansion (with the minimal number of terms) is to use a Schmidt basis forŴ . EachV X j can be decomposed to a linear combination of its spectral projections linked with their respective eigenbases, |x , where x = a or b consistently with X and l = 1, ..., d X . If one fixes a certain pair of bases in C d A and C d B as "computational ones", i.e., starting ones, denoted as |l x , one can always find local unitary matrices . The construction of Reck et al. [39] fixes (phases in) a local multiport interferometer, which performs such a transformation. We shall call such interferometers U X (j) ones. In the case of field modes a passive interferometer performs the following mode transformation: is the photon creation operator in the l-th exit mode of interferometer U X (j).
A two-party entanglement witnessŴ R for optical fields, which uses correlations of intensity rates behind pairs of U X (j) interferometers can be constructed as follows. For the output l x of an interferometer, one defines rate observables asr lx =Π Xnlx The witnessŴ expanded in terms of the computational basis: allows us to form an entanglement witness for fields: For any pure state of the quantum beams |Ψ where the matrixR has elements r klmn (9) Using a generalization of the earlier derivations one can show thatR is a two-qudit density matrix, and so on.
The actual measurements, to be correlations of local ones, should be performed using the sequence of pairs of U X (j) interferometers, which enter the expansion of the two-qudit entanglement witness (5). In the entanglement indicator the rates at output x l (j) of the given local interferometer U X (j) are multiplied by the respective eigenvalue ofV X j related with the eigenstate |x (j) l . To get an entanglement witness for intensitiesŴ I we takeŴ and replace the computational basis kets and bras by suitable creation and annihilation operators: For any pure state of the quantum beams |Ψ one has lâ mbn |Ψ , and has all properties of a two-qudit density matrix.
Example showing further extension to unitary operator bases.-Let d be a power of a prime number. Consider d A = d B = d beams experiment (see Fig. 1), with families of U X (m) interferometers which link the computational basis of a qudit with an unbiased basis m, belonging to the full set of d + 1 mutually unbiased ones [40,41]. We introduce a set of unitary observables for a qudit: ). Thus, we can expand any qudit density matrix as where As the basis observables are unitary the expansion coefficients of an entanglement witness operator in terms of such tensor products of such bases are in general complex. This is no problem for theory, but renders useless a direct application in experiments, as one cannot expect the experimental averages to be real, and thus one has to introduce modifications. Below we present one.
The condition Tr 2 ≤ 1 can be put as Thus, applying Cauchy-Schwartz estimate, we get immediately a separability condition for two qudits: Our general method defines a Cauchy-Schwartz-like separability condition homomorphic with (13) as Heren X j (m) =x † j (m)x j (m) is a photon number operator for output mode j of a multiport m, at station X. For generalized observables based on intensity, one can introduceχ k (m) = d j=1 ω jkn j (m) to get the following separability condition:

(16) Supplemental Material presents other examples [14].
Implications for optical coherence theory.-The approach can be generalized further. Let us take as an example Glauber's correlation functions for optical fields, say G (4) , with normal ordering requiring that op-eratorF X ( x, t) is built out of local annihilation operators. The idea of normalized Stokes operators suggests the following alternative correlation function Γ 4 ( x, t; x , t ) given by where a(X) denotes the overall aperture of the detectors in location X. Obviously one has , and for fixed t and t one can define which behaves like a proper two-particle density matrix, provided one constrains the range of x, y, x , y to appropriate sets of apertures. As our earlier considerations use simplified forms of (17), it is evident that such correlation functions may help us to unveil non-classicality in situations in which the standard ones fail, see e.g. [8].
Bell inequalities.-The above ideas allow one to introduce a general mapping of qudit Bell inequalities to the ones for optical fields. A two-qudit Bell inequality for a final number of local measurement settings α and β has the following form: where P ij (α, β) denotes the probability of the qudits ending up respectively at detectors i and j, when the local setting are as indicated, and j P ij (α, β) = P i (α) and P j (β) = i P ij (α, β). The coefficient matrices K, N, M are real, and L R is the maximum value allowed by local realism. The bound is calculated by putting P ij (α, β) = D i (α)D j (β) and P i (α) = D i (α), P j (β) = D j (β), with constraints 0 ≤ D i (α/β) ≤ 1, and i=1 D i (α/β) = 1. As for a given run of a quantum optical experiment local measured photon intensity rates r i (α) and r j (β) satisfy exactly the same constraints. We can replace P ij (α, β) → r i (α)r j (β) LR , and P i (α) → r i (α) LR , etc., where . LR is an average in the case of local realism. The bound L R stays put. To get a Bell operator we further replace the above by rate observablesr i (α)r j (β), etc. Thus any (multiparty) Bell inequality, see e.g. [42], can be useful in quantum optical intensity (rates) correlation experiments. The presented methods for entanglement indicators and Bell inequalities allow also to get steering inequalities for quantum optics.
Conclusions.-We present tools for a construction of entanglement indicators for optical fields, inspired by the vast literature [10] on entanglement witnesses for finite dimensional quantum systems. The indicators would be handy for more intense light beams in states of undefined photon numbers, especially in the emerging field of integrated optics multi-spatial mode interferometry (see Supplemental Material [14] for examples). One may expect applications in the case of many-body systems, e.g. for an analysis of non-classicality of correlations in Bose-Einstein condensates, like in the ones reported in [43].

SUPPLEMENTAL MATERIAL
We give here several examples, and more details concerning some derivations. All separability conditions are generalizations or tighter versions of conditions presented in [5,17,35,44,45], which were derived using various less general approaches.  [46] in a form of an inequality, which reads whereσ X j = n X j · σ X for X = A, B, and the unit vectors n X j form a right-handed Cartesian basis triad. If a two-qubit state is entangled, then there exists at least one pair of such triads for which the inequality is violated. The conditions can be put in a form of a family of entanglement witnesses: 3 ). (20) Our homomorphisms can be used to get the following [36]: for normalized Stokes operators and for standard ones The homomorphisms warrant that the violations of conditions (21) and (22) are necessary and sufficient to detect entanglement via measurements of correlations of the Stokes observables. That is, any other condition is sub-optimal, including the ones presented in [5], [15] and [17] for standard Stokes observables. From the necessary and sufficient condition (21) one can derive its corollary, which is a necessary condition for separability: The condition can be thought as a more tight refinement of the result in [17]. It can be derived using the fact that for two qubits any of the observables k s k σ A k σ B k +σ A 0 σ B 0 , for arbitrary s k = ±1 is non-negative for separable states. This can be reached via an application of the Cauchy inequality for a product pure states of a pair of qubits. Next we apply the homomorphism. One can also see that (23) is the separability condition (14) in the main text for d = 2.
For the standard Stokes operators the associated separability condition (23) reads This is a tighter version of the condition given in [17]. For states, which locally lead to vanishing averages of local Stokes parameters, here Ŝ A iΠ B = 0, etc., (e.g., for an ideal four-mode bright squeezed vacuum, see below), the conditions (23) and (21) are equivalent. Thus, in such a case the Cauchy inequality based condition is necessary and sufficient for detection of entanglement with normalized Stokes operators. A similar statement can be produced for the analog condition involving traditional Stokes parameters Θ j , given by (24).
Cauchy-like inequality condition vs. EPR inspired approach.-Consider four-mode (bright) squeezed vacuum represented by where Γ describes a gain which is proportional to the pump power, and ψ n − reads where |Ω is the vacuum state. Perfect EPR-type anti-correlations of which are the main trait of the state allow one to formulate the following appealing separability condition (Simon and Bouwmeester, [5]): Note, that for |Ψ − and each |ψ n − the left-hand side (LHS) of the above is vanishing.
The underlying inequality beyond the condition (27) can be extracted with the use of well-known operator identity (see e.g. [47]): Using this the (27) boils down to which by the way can be generalized to 2 (27), or equivalently (29), cannot be considered as an entanglement indicator for fieldsŴ Θ homomorphic in the way proposed here, with a two-qubit (linear) entanglement witnessŴ . Detection of entanglement with (27) depends on a detector efficiency. The threshold efficiency for entanglement detection, in the case of 2 × 2 mode squeezed vacuum |Ψ − in (25), considered in [5] is given by η crit = 1/3. It does not depend on the gain parameter Γ. Obviously, as N ANB ≤ 1 (29) is not optimal. A more optimal option is to estimate from below the LHS of (29) using a corollary of the Cauchy-like

Simon-Bouwmeester EPR-like condition
≤ N AN B , which is tighter than (29). By combining (24) with (28) we get The new EPR-like necessary condition for separability differs from the one of Simon and Bouwmeester by the second term on the RHS of (30). As the term is always non-negative, this is a stronger condition. For the standard quantum optical model of inefficient detection (see the main text, or Sec. II) the new condition holds for any efficiency. Note, that (28) does not contribute anything to the relation (30), because it is an operator identity. That is, the condition (30) reduces to (24). For normalized Stokes parameters the EPR-like separability condition, which is an analog of (27), reads For a derivation, see [35] (and see also [44,45] for its generalizations to d modes). Entanglement detection with (31) also depends on the detector efficiency, but for the considered bright squeezed vacuum state the threshold efficiency η crit decreases with growing Γ. The η crit is lower than 1/3 for any finite Γ. If one uses the Cauchy-like inequality (23) and the identity 3 i=1Ŝ 2 i =Π +Π 2 NΠ (see [35]), then the following tighter EPR-like separability condition emerges It is equivalent with the much simpler linear condition (23). The condition presented here has much more resistant to losses that the one derived in [35], and generalized in [45], here formula (31).

II. RESISTANCE WITH RESPECT TO LOSSES
Here we derive the dependence on a detector efficiency of average values of entanglement indicators for optical fieldsŴ Θ andŴ S . Our reasoning can be extended to an arbitrary number of quantum optical modes and multiparty cases.
The loss model (an ideal detector and a beamsplitter of transmission amplitude √ η in front of it) is described by a beamsplitter transformation for the creation operators, see e.g. [47], which readŝ whereâ † j refers to the detection channel in j-th mode and c † j refers to the loss channel linked with the mode. First, we shall analyze the problem for standard Stokes operators. Let ψ AB be a pure state of the modes, before the photon losses. The unitary transformationÛ(η) describing losses in all channels leads toÛ(η)|ψ AB = |ψ AB (η) , and we have Notice that as the original state ψ AB does not contain photons in the loss channels, thus in ψ AB n A i (η A )n B j (η B ) ψ AB only the first term of the second line of (35) survives. For the transmission amplitudes η A and η B of beams A and B, we have

36) From this we get the dependence of correlations of Stokes operators on detection efficiency in the form of Θ
j . For normalized Stokes operators, the reasoning is as follows. For Fock states |F = |n Ai , n Ai ⊥ , m Bi , m Bi ⊥ , it is enough to consider only the average value ofŜ A 3 for state |F A = |n A H , m A V , which we shall denote for simplicity as |n, m . Obviously for such a state the intensity rate at the detector measuring output H, with the detection efficiency η for each of the detectors in the station, reads First we notice that k n k = n n−1 k−1 , and rewrite the first summation as from k = 0 to k = n − 1. Next, let us consider a function f (γ, η) of the form which for γ = η gives r 1 (η). Its derivative with respect to γ reads This upon integration with respect to γ, with the initial condition f (γ = 0, η) = 0, gives for γ = η the required result: It is easy to see that this result has a straightforward generalization to the case of more than two local detectors (e.g., see Fig. 1 in the main text). To calculate the dependence on η of the rate at detector i, when we have altogether d detectors at the station, we simply replace in the above formulasn H byn i andn V by j =in j , to get r i (η) = n ntot (1 − (1 − η) ntot ), where n is the number of photons in a Fock state in mode i and n tot is the total number of photons.
Note that for four-mode bright squeezed vacuum state (25) our entanglement condition for normalized Stokes parameters (23) is fully resilient with respect to losses of the kind described above. This is due to the fact that squeezed vacuum is a superposition entangled states (26), and each of them violates the separability criterion. As the Stokes operators do not change overall photon numbers on each of the sides of the experiments which we consider here, and states ψ n − contain n photons in both beams A and B, an inefficient detection in the case of ψ n − introduces the same reduction factor on both sides of condition (23). The violation of it holds for whatever value of η. The expectation values for the full squeezed state are simply weighted sum of expectation values for its components |ψ n − . The same can be shown for all other squeezed states, and linear separability conditions considered here, including the cases of d > 2.

III. ENTANGLEMENT EXPERIMENTS INVOLVING MULTIPORT BEAMSPLITTERS: HOMOMORPHISM OF SINGLE QUDIT OBSERVABLES AND FIELD OPERATORS
Proof of relation (11) of the main text for qudit states.-We consider a set of unitary qudit observables of the following form in the main text where k = 0, 1, ..., d − 1 and ω = exp(2πi/d), and U (m)|j = |j(m) is a unitary transformation of a computational basis (m = 0) to a vector of a different unbiased basis m. We assume that the bases m = m are all mutually unbiased, and consider only dimensions in which we have d + 1 mutually unbiased bases. We show that the operatorsq k (m)/ √ d with k = 1, ..., d − 1 and m = 0, ..., d, andq 0 (0) = 1 1 form an orthonormal basis in the Hilbert-Schmidt space of (all) d × d matrices.
The orthonormality of the operators can be established as follows. We are to prove that • For k = 0, this is trivial because all k = 0 operators are traceless (as d j=1 ω jk = dδ k0 ). • For m = m , with k = 0 and k = 0, one has 1 d l,j,j ω −jk+j k l(m)|j(m) j(m)|j (m ) j (m )|l(m) where we use the fact that for mutually unbiased bases | j (m )|j(m) | 2 = 1/d.
• For m = m , in the second line of (42) we have j (m )|l(m) = δ lj , and we get in the last line 1 d l ω l(k−k ) = δ kk . As we have (d − 1)(d + 1) + 1 = d 2 such orthonormal operators, the basis is complete. QED.
Remarks on the homomorphism.-We shall now show that for any pure state of a d-mode optical field |ψ , one can always find a d × d one qudit density matrix M for which the following holds whereQ k (m) is defined by (15) in the main text. For the expectation value, which reads we introduce a set of states which for m = 0 gives Then, one can transform (44) into As it was mentioned in the main text, the unitary transformation of the creation operators between input and output beams isâ † l (m) = r U lr (m)â † r , whereâ † r = a † r (m = 0) is a reference operator and U (m = 0) = 1.
Thanks to this the state (45) can be put as Therefore, (47) can be put as Let us introduce a matrix, denoted by M , whose elements are M sr = φ r (0)|φ s (0) . Then d r,s=1 Finally we arrive at where M is a (positive definite) Gramian matrix. Its trace is given by Tr M = Π ≤ 1. We can normalize it to get M = M/ Π , which is an admissible qudit density matrix.
Let us now turn back to qudits, and analyze the structure an expectation of the unitary observable (40). First, consider a pure state |ξ . The expectation value reads where we use |j(m) = r U jr (m) |r and introduce a density matrix M ξ for the state |ξ of elements M ξ rs = r|ξ ξ|s . If we replace |ξ by a density matrix given by = λ p λ |ξ λ ξ λ |, then the expectation (52) becomes where matrix M has elements given by M rs = λ p λ r|ξ λ ξ λ |s . Therefore, (43) holds. Obviously, such reasoning can be generalized to the case of (mixed) states describing correlated beams A and B, in the way it is done in the main text.
For intensity-based observables, we have a similar relation where N is a possible two-qudit density matrix. Note that in general M = N.

IV. NOISE RESISTANCE OF CAUCHY-SCHWARTZ-LIKE SEPARABILITY CONDITION FOR BRIGHT SQUEEZED VACUUM
Observables based on rates can in some cases allow a more noise resistant entanglement detection than the ones based directly on intensities.
Distortion noise.-We take as our working example a d × d mode bright squeezed vacuum in the presence of a specific type of noise, which can be treated as distortion of the state, which lowers the correlations between the beams.
A. 2 × 2 mode bright squeezed vacuum plus noise We build our noise model in following steps. Let us introduce four squeezed vacuum states which are related with the Bell state basis for two qubits. To make our notation concise let us denote by k = 0 the polarization H and by k = 1 polarization V , and let us define that the index values follow modulo 2 algebra. Then one can write down the following (55) and define squeezed vacua related with the Bell states as |Ξ(0, 0) = |Φ + , |Ξ(0, 1) = |Ψ + , |Ξ(1, 0) = |Φ − , and |Ξ(1, 1) = |Ψ − . This notation may look too dense here, but it will help us further on. Our noise model, which is an analog of the "white noise" in the case of two qubits, can be defined as (56) The following properties of the noise are essential. For each i and j, That is the noise itself such that it leads to vanishing correlations between components of the Stokes parameters. This is easy to see when one recalls the local unitary transformations, say on side A, (replaced here by mode transformations) which link the three other twoqubit Bell states with the singlet. Simply they are equivalent to π rotations of Bloch sphere of side A with respect to axes z, x, and y. The second property is For normalized Stokes operators.-Let us start with the analysis of noise in terms of the rate observables. Let v be the visibility, which determines the following noisy state: where 0 ≤ v ≤ 1. We have to find the threshold v above which our separability condition This will be our measure of the resilience with respect to the noise. Applying the technical facts that for |Ψ − one has The threshold visibility v crit is given by The respective terms of (62) are given by that follows from the definition of Π A and where 3 F 2 (1, 1, 3; 2, 2; tanh 2 Γ) is generalized hypergeometric function.
For standard Stokes operators.-Following the same reasoning for observables based rates the threshold visibility v old crit for observables based on intensities is given by We have Comparison of critical visibilities to detect entanglement via the Cauchy-like condition, for four-mode squeezed vacuum Ψ − mixed with "white" noise in (56). The upper curve is for standard Stokes parameters, and the lower for normalized ones. The latter one turns out to lead to a higher noise resistance. Note that the Cauchy-like condition is equivalent in the case of Ψ − , and the mixture of Ψ − with the model noise, with the necessary and sufficient conditions to detect entanglement via measurement of Stokes parameters. Therefore, this graph shows the critical visibilities also for this case. One cannot do better. Obviously the graphs for Ψ + , Φ + and Φ − are identical. The asymptotic limit vcrit = 1/3, for Γ → 0, is concurrent with the white noise threshold for a two-qubit singlet.
The form of (67) was obtained as follows. Let us put x = tanh 2 Γ, and c = cosh 4 Γ. We have Thus, the threshold visibility in function of the amplification gain v old crit (Γ) for the "macroscopic singlet" We compare the critical visibilities obtained with the two approaches (normalized vs. standard Stokes parameters) in Fig. 2.

B. Unitary observables for d-mode
Multimode bright squeezed vacuum.-The bright squeezed vacuum is a state of light of undefined photon number which has, due to entanglement, perfect EPR correlations of numbers of photons between specific modes reaching A and B. Such an entanglement can be observed in multimode parametric down-conversion emission. The interaction Hamiltonian of the process, for a classical pump, is essentiallyĤ = iγ where γ is the coupling constant proportional to a pump power. Thus, d × d mode (bright) squeezed vacuum state is given by where Γ = γt and t is the interaction time, and Noise model.-If we consider the unitary observables, our noise model can look as follows: we build our noise model in following similar steps as for the d = 2 case. Let us now index k stand for local modes k = 0, 1, ..., d − 1 and we shall the modulo d algebra for it. Then one can write down the following

|Ω
(72) with m and l taking values 0, 1, ..., d − 1. Note that these squeezed d-mode vacua are analogs of the following Bell basis for a pair of qudits: 1 √ d k ω km |k ⊗ |k + l . Our noise model is defined as The following properties of the noise are essential for us. For each i and j and the second property is We have the same relation for observables based on intensities. Noise resistance.-Applying this model we get that entanglement detection is possible with the Cauchy-like condition for observables based on rates, in the case of Ψ d BSV mixed with the noise, if the threshold visibility v crit fulfills In case of observables based on intensities, we get (77)

3 × 3 mode bright squeezed vacuum
In case of observables based on rates, the respective terms in (76) are as follows: For observables based on intensities in (77) we have and The first equality of (81) can be obtained as (here, x = tanh 2 Γ and c = cosh 6 Γ): The threshold visibility in function of the amplification gain, v crit (Γ), for the macroscopic singlet Ψ 3 BSV is presented in Fig. 3. [q k (m)] ll Π a † l a l NΠ , whereq k (m) are the qudit operators (40). Therefore, we have [q k (m)] ll [q † k (m)] nn = dδ ln δ l n .
All that, and [a i , a † j ] = δ ij , allow one to perform the following calculation: [dδ ln δ l n − δ ll δ nn ]a † l a l a † n a n 1 NΠ =Π 1 N − ln a † l a l a † n a n + d ll a † l a l a † l a l 1 NΠ =Π 1 Thus, (83) holds. An analogue relation for the observables involving intensities, which reads can be obtained by similar steps. It is a generalization of (28).

B. Formula 2
We here calculate the expressions which enter of Cauchy-Schwartz-like separability conditions based on rates (14) and intensities (16) in the main text for a d × d mode bright squeezed vacuum. Some of the formulas are also used in the discussion of noise resistance.
Let us consider first the condition (16) in the main text: its LHS and RHS read To get the formula for RHS we used (91) The action ofχ B k (m = 0) on an unnormalized |ψ n d of (71), which we put as |φ n = U rs (m ) We have = δ s,s δ t,t .