Complementary properties of multiphoton quantum states in linear optics networks

We have developed a theory for accessing quantum coherences in mutually unbiased bases associated with generalized Pauli operators in multiphoton multimode linear optics networks (LONs). We show a way to construct complementary Pauli measurements in multiphoton LONs and establish a theory for evaluation of their photonic measurement statistics without dealing with the computational complexity of Boson samplings. This theory extends characterization of complementary properties in single-photon LONs to multiphoton LONs employing convex-roof extension. It allows us to detect quantum properties such as entanglement using complementary Pauli measurements, which reveals the physical significance of entanglement between modes in bipartite multiphoton LONs.


Introduction
Multiphoton multimode linear optics networks (LONs) are the physical platforms for the implementation of possible quantum supremacy in Boson sampling [1]. Experiments of Boson sampling have been realized and rapidly developed in various linear-optics-network systems [2][3][4][5][6][7][8][9][10][11][12][13][14]. Despite the simulation complexity of Boson sampling, statistical characteristics can be exploited to benchmark Boson samplers [15][16][17][18][19][20][21]. For specific linear optics transformations of permutation symmetric states, one can even predict their zero-probability outputs by the suppression laws [22][23][24]. The permutation symmetric states that exhibit the suppression laws are not restricted to Fock states, but also valid for quantum superposition of them [25][26][27]. It implies that quantum coherences play an important role in photon statistics of multiphoton LONs, if we consider their inputs as general multiphoton states. From a different perspective, in this paper, we will consider the characterization of physical properties that related to quantum coherences of multiphoton states by evaluation of the photon statistics in LONs.
To characterize quantum coherences between Fock states, one can employ quantum state tomography to reconstruct the full description of a general multiphoton state in the entire multiphoton Hilbert space of LON systems. The experimental setup of a full quantum state tomography in multiphoton LONs requires either a large number of measurement configurations or a large amount of additional ancillary modes [28], both of which are still very challenging for currently available experimental facilities. In many cases, instead of the full information of a quantum state, one just needs partial information about quantum coherences in measurements of two non-compatible observables. It is therefore meaningful to consider the possibility of accessing quantum coherences in LONs by a reasonable number of measurement configurations associated with non-compatible observables, which can be meanwhile implemented by a set of experimentally available linear optics transforms without any additional ancillary modes and photons.
In single-photon LONs, which are equivalent to qudit systems, mutually unbiased bases (MUBs) [29] are the optimal bases for obtaining maximal quantum coherences [30][31][32][33]. It implies that complementary measurements, which measure quantum states in MUBs, are appropriate for revealing quantum coherences in qudit systems. They can be implemented with the help of generalized Hadamard transforms [29] and serve as coherence quantifiers [34] through the uncertainty relationship of quantum measurements [35][36][37][38]. In multipartite qudit systems, correlations in complementary measurements can be also exploited to detect entanglement [39][40][41][42], as well as the dimensionality of entanglement [43,44]. To open up access to quantum properties associated with quantum coherences in multiphoton LONs, complementary measurements are therefore the desirable keys.
However, in multiphoton LONs, indistinguishability of photons leads to photon bunching in output modes of a LON, which makes the explicit photon statistics of a generalized Hadamard transform #Phard to determine [1]. This phenomenon tangles the complementarity of Pauli operators. In this paper, we will tackle this problem to find the complementary structures of generalized Pauli operators and construct complementary measurements in multiphoton LONs. Our goal is then to establish a theoretical framework for experimental access to complementary properties of multiphoton states in LONs through these complementary measurements. We will show that complementary properties of convex sets of multiphoton states in LONs can be quantified through convex-roof extensions over the subspaces that are well-defined qudit systems and characterized by cyclicly translational mode shifting.
In this theoretical framework, we will then derive two approaches for entanglement detection in bipartite multiphoton LON systems employing complementary correlations. Since photons are indistinguishable identical particles, entanglement between photons is only a well-defined concept after exclusion of the "entanglement" arising from particle-label symmetrization in their wavefunctions [45][46][47][48][49][50][51][52][53][54][55][56][57]. In bosonic systems like LONs, entanglement between modes in the second quantization formalism [58], which automatically excludes the "entanglement" arising from particle symmetrization, is therefore a legitimate entanglement concept. In this paper, we therefore assume the perfect indistinguishability of photons in multimode interference of LONs, and consider the entanglement between modes with fix local photon numbers, which is also called entanglement of "particles" [59]. In bipartite multiphoton LONs, entanglement still lacks its physical significance in experiments. In our entanglement detection approaches, the quantities called complementary mutual information and complementary mutual predictability are evaluated in complimentary local measurements. We will extend the threshold of these two complementary correlations for separable states in bipartite qudit systems [39,42] to bipartite multiphoton LONs, such that complementary correlations exceeding these thresholds signify entanglement between modes in a bipartite multiphoton LON system. Our results therefore open up access to the physical significance of entanglement between modes in multiphoton LONs. This paper is structured as follows. In Section 2, we show complementary structures of generalized Pauli operators within the subspaces of multiphoton LONs characterized by a translational modeshifting operator. In Section 3, we show the construction of generalized Pauli measurements, which allows us to access complementary properties within the subspaces specified in the previous section. In Section 4, we show complementary Pauli quantities evaluated in complementary measurements can be exploited to characterize convex sets of quantum states, which leads to measurement uncertainty relationship in multiphoton LONs. In Section 5, we demonstrate an application of the theoretical framework established in previous sections in the detection of entanglement between modes in bipartite multiphoton LONs. Section 6 concludes the paper.

Complementary structures in linear optics networks
A linear optics network (LON) is a multimode interferometer, which is a unitary transform of modes constructed by linear optics elements. In principle, one can construct any unitary transform of modes using Beam splitters [60]. As shown in Fig. 1 (a), each input and output mode of a LON are indexed by m = 0, ..., M − 1. A state transformed by a linear optics interferometer U is measured by photon number resolving detection (PNRD) at each output mode, which resolves a number of photons n m . An output event is then denoted by a Fock number vectors n = (n 0 , ..., n M −1 ), which is associated with a projection onto the Fock state |n n|. Due to energy conservation, linear optics does not change the total photon number |n|. A LON unitary U is therefore diagonal with respect to the subspaces of different total photon numbers. It is therefore legitimate to describe the mechanism of a LON quantum system independently for quantum states with different photon numbers. Consider an N -photon input state ρ N in the LON shown in Fig. 1, the probability of detecting a photon number vector n is given by Note that if the input is a Fock state, Fig. 1 (a) is a Boson sampling scenario. Here we consider a more general scheme which allows an input state to be a superposition of Fock states. In LONs, each input and output mode can be represented by photon creation operators a † m and b † m , respectively. As shown in Fig. 1 (b), under the assumption of perfect indistinguishability of photons in interferometers, a general LON transformation U between input and output modes can be represented by a unitary matrix {u m ,m } m ,m , The unitary matrix {u m ,m } m ,m describes exactly the unitary transformation of single photon Fock states in this LON, while the transformation of a multiphoton Fock state n | U |n is in general #P -hard to calculate in classical computers [1].
In an M -mode single-photon LON system, which is equivalent to an M -dimensional qudit system, two operators that have mutually unbiased eigenbases are complementary for accessing maximal quantum coherences [30][31][32][33]. Generalized Pauli operators are the legitimate candidates for such complementary operators [29]. A generalized Pauli operator Λ i,j is a combination of a mode-shift operator X and a phase-shift operator Z (see Fig. 2), which are called the shift and clock operator, respectively, in qudit systems, The mode-shift operator X shifts a mode to its next neighboring mode translationally and cyclicly, while the phase-shift operator Z adds phases to each mode, where w = exp(i2π/M ) is a phase given by the M -th root of unity and m ⊕ 1 = (m + 1) (mod M ) is the M -modulus sum. Since the Pauli operator Λ i,ij has the same eigenbasis as the Pauli operator Λ 1,j , most of the Pauli-operator eigenspaces can be characterized by the Λ 1,j . In the rest of this section, we therefore focus on the complementary structures in the Λ 1,j eigenspaces. For conciseness, we shorten the notation for the Pauli operator Λ 1,j by Λ j .
In an M -mode multiphoton LON systems, the operator X shifts a Fock state |n translationally and cyclicly, X |n = |n M −1 , n 0 , ..., n M −2 , while Z adds a phase shift µ(n) which is equal to the total mode index of the Fock state |n , The effect of a generalized Pauli operator performed on a Fock state is a combination of the mode shift and phase shift, Λ j |n = w jµ(n) |n M −1 , n 0 , ..., n M −2 .
After M -times Λ j operations, a Fock state |n will be periodically shifted back to its original. Such a periodic operation connects and groups multiphoton Fock states in different orbits, which we call Pauli classes.
Definition 2.1 (Pauli classes and subspaces). A Pauli class E n in a linear optics network is a set of Fock states, whose elements are generated by the mode-shift operator performed on the representative Fock state |n , where d En is the cardinality of the Pauli class. The d En -dimensional Hilbert subspace H En spanned by a Pauli class E n is called a Pauli subspace, Since the operation of a Pauli operator Λ j performed on a multiphoton LON can be described independently within each Pauli class, one can decompose Λ j into diagonal blocks Λ is defined within a Pauli subspace H E . In a Pauli subspace H En , eigenstates of Λ j are constructed by The eigenstate |E n,m (Λ j ) satisfies the eigenequation As a result, the Pauli operator Λ j is a sum of all Λ (En) j constructed within Pauli subspaces, According to Eq. (11), within a Pauli subspace H En , the computational Fock basis E n given in Eq.
If the eigenbases of two operators are MUBs in a Pauli subspace H En , we say these two operators are complementary within the Pauli subspace H En . A pair of complementary operators should therefore define non-degenerated eigenstates within each Pauli subspace. Since eigenstates of Z are degenerated within particular Pauli subspaces, it is not appropriate to represent a physical property in the computational basis that is complementary to the Λ j operator. The degeneracy of Z operator can be seen from the clock-like diagram of the 2-photon 4-mode Z eigenspace as shown in Fig. 3 (a). In this diagram, Fock states |n are grouped by the phases µ(n) of their Z-operator eigenvalues given in Eq. (7), which we call the Z-clock labels. For an operator complementary to Λ j , we need to construct it with non-degenerate labeling in each Pauli subspace as follows,    where Ξ (En) is a clock operator in H En with {|e n,m } m being the computational basis of H En associated with the eigenvalues w m and labeled as follows, Here, σ is an arbitrary permutation in the set {0, ..., d En − 1}. Fig. 3 (b) shows the non-degeneracy of Ξ within 2-photon 4-mode Pauli subspaces for the permutation σ = 1. Compare Fig. 3 (b) and (c), one can see that the operator Ξ and Λ j define non-degenerated eigenstates that are mutually unbiased with each other in each Pauli subspace. For the study of operator complementarity, it is therefore appropriate to refer to the operator Ξ instead of the phase-shift operator Z. Note that in the case gcd(|n|, M ) = 1, which guarantees non-degeneracy of Z, the operator Ξ can be constructed as the phase-shift operator Z according to demands.
Besides the operator pairs { Ξ, Λ j }, two Pauli operators { Λ j , Λ l } can also be complementary. However, their complementarity within a Pauli subspace H En is not guaranteed. The MUB structures of two Pauli operator { Λ j , Λ l } in a Pauli subspace H En depends on the degeneracy of the Z l−j -Pauli operator in H En .

Theorem 2.2 (MUBs within Pauli subspaces). Two Pauli operators
This theorem implies that two Pauli operators Λ j and Λ l can be complementary within a Pauli subspace H En , while non-complementary within the other subspace H E n . The complementarity in Pauli subspaces can be directly seen from the clock diagram of the operator Z l−j . For example, in a 4-mode linear optics network, the clock diagram in Fig. 3    For a prime M , one can therefore construct a complete set of complementary operators in N -photon LONs. If M is a prime power p k , one has to decompose the p k -mode LON into a k-level p-branch treestyle LON and construct MUBs in each p-mode subsystem followed by its extension to higher levels.
In qudit systems, it is shown that a complete set of complementary operators for M = p k exists [29], however the complete set of MUBs in multiphoton LONs with M = p k is not straightforwardly extendible from the qudit system due to the photonic bunching effects. Since characterization of the complete set of MUBs in multiphoton LONs is out of the scope of this paper, we leave this question open.

Complementary Pauli measurements in linear optics networks
Measurements in the MUBs associated with a set of complementary Pauli operators constructed in Corollary 2.3 can be exploited to evaluate complementary properties of multiphoton states in LONs. A trivial measurement is in the computational basis, which is associated with the operator Ξ specified in Eq. (14). For N -photon states, the operator Ξ can be decomposed into the sum of projectors π N,m (Ξ) that project onto eigenspaces with eigenvalues w m labeled by m as The expectation value of Ξ can be then evaluated in the projective measurement { π N,m (Ξ)} m , which we call a Ξ-Pauli measurement. A Pauli operator Λ j can be also decomposed as a sum of eigenvalue projectors π N,m (Λ j ), where the projector π N,m (Λ j ) is called an N -photon Λ j -Pauli projector for the label m, and explicitly defined by In N -photon LONs, a Λ j -Pauli measurement in the eigenspace of Λ j is then a projective measurement represented by the Λ j -Pauli projectors { π N,m (Λ j )} m . For Λ j -Pauli measurements, one needs the corresponding inverse Hadamard transform H † j to transform a Λ j -Pauli eigenbasis to the computational Fock-state basis, such that one can employ photon number resolving detection in the outputs of H † j to measure input states in the Λ j -Pauli eigenbasis. As shown in Eq. (2), an inverse Hadamard transform of modes is determined by its transformation of single-photon states. According to Eq. (11), the transformation of the modes by H † j is then described by }m are evaluated by counting the probability of photon number detection events that satisfy µ(n) = m.
As shown in Fig. 4, the LONs of Hadamard operators can be decomposed into a combination of the standard discrete Fourier transform and a phase shift V as For a single photon, the Pauli operator Λ j performed on inputs of H † j is equivalent to the phaseshift operator Z performed on outputs of H † j up to a phase, In the outputs of a H † j transform of an N -photon input, the additional phase w (M −1)j/2 is added to each photon and leads to a total phase shift w (M −1)j|n|/2 . Applying this relation to an Λ j eigenstate |E n,m (Λ j ) and according to the eigenequation of Λ j given in Eq. (12), the additional phase shift will be eliminated, which leads to the following eigenequation, It means that the H † j transforms a Λ j eigenstate |E n,m (Λ j ) to a Z eigenstate with the eigenvalue w m . As a result, the only possible outputs of the transformation H † j |E n,m (Λ j ) are the Z eigenstates with the eigenvalue w m , which are the Fock states |ν with the Z-clock label µ(ν) = m. This is a suppression law of inverse Hadamard transforms, which is a special case of the suppression law of general permutation invariant states [26,27]. Eq. (22) shows that the eigenspaces of Λ j are transformed to the eigenspaces of Z by the inverse Hadamard H † j , which means that a Λ j -Pauli projector is equivalent to a H j -transformed Z-Pauli projector, From the example in a 2-photon 5-mode system shown in Fig. 5 (a), one can see that the Hadamard transform H † j maps each Λ j eigenspace to its corresponding Z eigenspace without changing the clock labels. Note that from an output event n of a Hadamard transform H † j , one can not distinguish the Pauli subspaces of inputs. We can only distinguish the eigenspaces of the Pauli operator Λ j associated with different labels m by taking all possible outputs satisfying µ(n) = m into account. As a result, one can implement a Λ j -Pauli measurement through PNRD on the output modes of the corresponding inverse Hadamard H † j to obtain the measurement statistics { π N,m (Λ j ) } m according to the following theorem.
Theorem 3.1 (Pauli measurement). Given a quantum state ρ, its expectation value of a Λ j -Pauli projector π N,m (Λ j ) can be evaluated by simply counting the probability of detecting photon number occupations n satisfying µ(n) = m in the output modes of a H † j transform A schematic Λ j -Pauli measurement is shown in Fig. 5  In Fig. 6, Pauli measurement statistics of a 2-photon 5-mode Λ 0 eigenstate |E 11000,0 (Λ 0 ) is demonstrated. Fig. 6 (a) We call such a quantity an L-Pauli quantity. The average of a Pauli quantity Q L obtained in the complementary Pauli measurements configured by the set L can be exploited to quantify complementary properties of quantum states. We call such a quantity a complementary Pauli quantity in the measurement configurations L and define it as follows.
where |L| denotes the cardinality of the operator set L.
Since all Pauli projectors π N,m (·) are block-diagonal with respect to Pauli subspaces by definition (see Eq. (16) and (18) As a consequence, Pauli quantities that evaluated from Pauli measurement statistics { π N,m (·) } m are also this invariance under the Pauli-subspace decoherence.
Let Ψ S be a set of all the pure states that possess certain property S. If the convex combination of two states in |ψ 1,2 ∈ Ψ S also possesses the property S, then we say the property S is convex-extendible, e.g. separability, entanglement dimensionality, and so on. It is clear that the set of all quantum states with a convex-extendible property is convex. In qudit systems, hyperplanes that separate the convex set of S-property quantum states ρ S from some non-S-property quantum states ρ S can be exploited to characterize the property S. If the corresponding quantity of a quantum state ρ exceeds the bounds of the hyperplanes tangent to the S-property convex set, one can conclude the non-S property of ρ.
If these hyperplanes are defined by a quantity which can be measured in experiments, the property S complement to S is then physically detectible.
In LON systems, quantum states can be characterized in hyperplanes { ρ : C Q,L ( ρ) = q} defined by a complimentary Pauli quantity C Q,L . Since a complementary Pauli quantity is physically accessible by definition, it provides the physical significance of the property S complement to a convex-extendible property S. As a Pauli subspace H En is a well-defined d En -dimensional qudit system, the C Q,Lhyperplane boundaries on a S-property set within the Pauli subspace H En can be determined by well-established theories in qudit systems, According to Corollary 4.2, a complementary Pauli quantity C Q,L (|ψ ) of a pure state |ψ in LONs is given by its Pauli-subspace decoherence, which is a convex combination of pure states |ψ E over Pauli In the case that C Q,L is convex or concave, the C Q,L -hyperplane boundaries on the S-property set in N -photon LONs can be then extended from the bounds determined in Eq. (30) through a convex-roof extension over all Pauli subspaces.
where p E (ρ) = |n ∈E n| ρ|n is the probability of measurement outcomes belonging to a Pauli subspace H E in the computational basis. If a state ρ violates these inequalities, then the state ρ does not possess the property S.
Proof. According to Corollary 4.2, a Pauli quantity is invariant under the decoherence among Pauli subspaces, i.e. C Q, . The upper (lower) bound B S (E) on C Q,L (R E (ρ S )) is then determined by the maximum (minimum) C Q,L for the S-property pure states in the Pauli subspace H E , which is defined in Eq. (30). As a result, Eq. (32) follows.
This theorem allows us to extend well-established hyperplane boundaries on a convex set in qudit system to multiphoton LON systems through convex-roof extension over Pauli subspaces. Since the weight p E (ρ) of a state ρ in a Pauli-subspace H E can be measured in the computational basis, the boundaries given in Theorem 4.3 can be determined adapted to input states. As a result, one can reveal the physical significance of the complement of a convex-extendible property in multiphoton LONs by detecting a complementary Pauli quantity exceeding the bounds determined in Theorem 4.3 in a set of complementary measurements. Since a complementary Pauli quantity takes an average over complementary measurements, the hyperplanes defined by C Q,L becomes finer, if more Pauli measurements are included in the complementary measurement configurations L, which means more non-S-property states can be detected.
As an example, the Shannon entropy H is a concave quantity, which can be exploited to quantify randomness of measurement statistics in qudit systems. Since the property of being a quantum state is by definition convex-extendible, the whole set of quantum states is a convex set. There exists therefore a lower bound B quan. on the average of Shannon entropies of complementary measurement statistics, which implies the uncertainty relation of complementary measurements in qudit systems [34][35][36][37][38]. Such an uncertainty relationship can be extended to multiphoton LON systems according to Theorem 4.3.
Proof. See Appendix.
The lower bound determined in Corollary 4.4 is tight for particular states, e.g. for a Λ 0 eigenstate |ψ 0 given by The statistics of complementary Pauli measurements of this state is shown in Fig. 7 The Pauli measurement statistics of D(|ψ 0 ) is given in Fig. 7

Complementary correlations of entanglement in bipartite LONs
In multipartite qudit systems, complementary correlations have been widely employed to characterize separability and entanglement dimensionality in theory and experiments [39][40][41][42][43][44]. As a straightforward application of Theorem 4.3, one can extend the entanglement criteria that employ complementary correlations in bipartite qudit systems to bipartite multiphoton LON systems, so that we can evaluate entanglement between modes of multiphoton states in LONs theoretically and reveal its physical significance experimentally. In a in each local systems A and B, respectively. A maximally entangled state can be perfectly correlated both in the α 1 ⊗ β 1 -Pauli and α 2 ⊗ β 2 -Pauli measurements at the same time. In each local α l ⊗ β l -Pauli measurements, correlations can be evaluated by certain correlation measures, e.g. mutual information, mutual predictability, Pearson correlation coefficient and so on. The simultaneous correlations in a set of complementary measurements can be evaluated by the average of these correlation measures, which are called complementary correlations. The upper bounds on complementary correlations for separable states specify the hyperplanes that divide the convex set of separable states from particular entangled states. It therefore allows us to detect bipartite entanglement by evaluating complementary correlations exceeding these bounds [39,42].
Here, we consider bipartite multiphoton LON systems with the same number of modes M A = M B = M . For multiphoton states with N A and N B photons in each local system, complementary operators can be constructed locally with separable operators where { α l } l and { β l } l are complementary Pauli operators in the N A -photon and N B -photon local system, respectively, which are constructed according to Corollary 2.3. We call L a set of complementary separable Pauli operators in (N A , N B )-photon (M, M )-mode LON systems. In this section, we will derive experimentally accessible criteria for entanglement between modes in bipartite multiphoton LON systems using complementary mutual information and complementary mutual predictability.

Complementary mutual information (CMI)
For an entangled state that has correlations in the Pauli measurements configured by a set of complementary separable Pauli operators L given in Eq. (37), complementary mutual information (CMI) is a good quantity for entanglement detection. It takes the average of the mutual information in all α l ⊗ β l -Pauli measurements where where In the following example, we demonstrate this entanglement criterion in a (5, 5)-mode bipartite LON system. According to Corollary 2.3, one can construct complementary separable Pauli operators L with α l , β l ∈ { Ξ, Λ 0 , ..., Λ M −1 }. Since the Z operator is non-degenerate in all 5-mode Pauli subspaces, we can construct the Ξ operator as Ξ = Z. For entanglement detection of quantum states that have correlations in Z ⊗ Z and Λ j ⊗ Λ j eigenbases, one can construct measurement configurations L as follows, An entangled state, which is an eigenstate of every Pauli operator Λ j ⊗ Λ j , has perfect correlations in all measurement configurations L ∈ L. An example of such entangled states with (3 A , 2 B ) photons can be generated using beam splitters and single photon sources [25], The state |φ 3 A ,2 B satisfies the following eigenequations According to Theorem 3.1, an ( α l ⊗ β l )-Pauli measurement of |φ 3 A ,2 B has perfect correlations in the Z-clock labels of local photon-number-occupation-vector outputs, The ( α l ⊗ β l )-Pauli measurement statistics of |φ 3 A ,2 B is shown in Fig. 8. Local photon number vectors n A,B are sorted by their Z-clock labels µ(n). One can see that the measurement outcomes (n A , n B ) are perfectly correlated in (µ A , µ B ) blocks in each measurement configuration as given in Eq. (44). Fig. 8 (a) is the trivial measurement in the computational basis. The collective probability . The mutual information in this measurement is therefore I Z:Z = log(5). Fig. 8 (b) is the Λ 0 ⊗ Λ 0 -Pauli measurement. The collective probability of (µ A , µ B ) is Pr(µ A , µ B ) = 0.2 δ −µ B µ A , and hence I Λ0:Λ0 = log(5). Fig. 8 (c)-(f) show the measurement statistics in the configuration Λ 1 ⊗ Λ 1 , ..., Λ 4 ⊗ Λ 4 , respectively. The (µ A , µ B ) probabilities in each non-zero block are all 0.2. In each measurement configuration α l ⊗ β l ∈ L one therefore obtains a mutual information I α l :β l = log (5). As a result, the complementary mutual information of the state |φ 3 A ,2 B in the complementary measurement configurations L is If we implement all the six complementary measurements, the upper bound on C I,L for separable states determined in Corollary 5.1 is log(5/3), which is much smaller than the CMI of the entangled state |φ 3 A ,2 A .

Complementary mutual predictability (CMP)
If a state ρ is close to a target entangled state |φ N A ,N B , which is an eigenstate of all complementary Pauli separable operators α l ⊗ β l with eigenvalues wμ l mutual predictability [42] can be exploited to quantify the specific complementary correlations close to the target entangled state |φ N A ,N B . The mutual predictability F φ ( α l , β l ) of a quantum state ρ for a target entangled state |φ N A ,N B is the probability of measuring the expected correlated outputs specified by µ A + µ B =μ, where Pr α l ,β l (µ A , µ B | ρ) is the probability of (µ A , µ B ) outputs in the α l ⊗ β l -Pauli measurement of ρ. According to Theorem 3.1, it is equivalent to the expectation value of Pauli projectors that project onto the specific correlations m A + m B =μ, Mutual predictability F φ ( α, β| ρ) quantifies the closeness of a testing state ρ to a target state |φ in an α ⊗ β-Pauli measurement. Complementary mutual predictability (CMP) for a target state |φ N A ,N B therefore quantifies the closeness of ρ to |φ N A ,N B by taking the average of mutual predictability in the complementary Pauli measurements configurations L, If a state ρ is close enough to the target entangled state |φ N A ,N B such that its CMP is above the threshold for separable states, then one can confirm the entanglement of ρ. The threshold for entanglement determination can be derived analogous to Corollary 5.1 by the convex-roof extension over Pauli subspaces according to Theorem 4.3.
where The corresponding CMP of separable states is upper bounded by The Λ 0 ⊗ Λ 0 -measurement of ρ φ ( ) with = 5/6. Neither CMI nor CMP can detect its entanglement.
Proof. See Appendix.
Note that this bound is tight and achievable for example by the separable state |11100 |11000 .
In [25], entanglement detection criterion using CMP has been derived for the specific multiphoton entangled states withμ l = 0 in two complementary measurement configurations { Z ⊗ Z, Λ 0 ⊗ Λ 0 }. Corollary 5.2 is a generalization of the criterion in [25] for more general target entangled states and complementary measurement configurations. For entanglement detection of the exemplary entangled state |φ 3 A ,2 B in Eq. (42), the mutual predictability for |φ 3 A ,2 B in each ( α ⊗ β)-Pauli measurement with α ⊗ β ∈ L constructed in Eq. (41) is given by The |φ 3 A ,2 B -targeting CMP C F φ ,L of separable states is upper bounded by (|L|+4)/(5|L|) according to Corollary 5.2. It is obvious that CMP of the target entangled state C F φ ,L (|φ 3 A ,2 B ) has the maximum value, which is much larger than the separable bounds,

Entanglement detection under errors
In either generation or measurements of a target entangled state, errors are unavoidable. To estimate the robustness of entanglement detection of |φ 3 A ,2 B against totally random errors, we assume the white noise model, where 1 N = |n A |=3,|n B |=2 |n A , n B n A , n B | is the identity operator in the (3, 2)-photon subspace.
Since the background random noise is added uniformly to every possible (3, 2)-photon outputs, the probability distributions Pr(µ A , µ B ) under the white noise are then in the computational basis; The corresponding mutual information and mutual predictability is therefore uniform in every measurement configuration. As a consequence, different choices of the measurement configurations L ⊆ { Z ⊗ Z, Λ 0 ⊗ Λ 0 , ..., Λ 4 ⊗ Λ 4 } do not change the CMI and CMP of ρ φ ( ), but change the upper bounds on the CMI and CMP for separable states. According to Corollary 5.1 and 5.2, entanglement of ρ φ ( ) is still detectable by CMI, if while it is still detectable by CMP, if There exist therefore thresholds |L| for white-noise errors, upon which entanglement is not detectable by CMI or CMP in the complementary measurement configurations L. In Fig. 9 (a), the CMI and CMP of the noisy state ρ φ ( ) are plotted with a blue solid line and an orange dashed line, respectively. The white noise thresholds for entanglement detection using CMI and CMP are marked by blue and orange points, respectively. One can see that the more configurations a complementary measurement setting has, the more robust an entanglement detection is against white noises. Entanglement is not detectable for > 5/6 either by CMI or CMP. As an example, the Λ 0 ⊗ Λ 0 -measurment statisics of ρ φ ( = 5/6) is shown in Fig. 9 (b). Compare these two approaches, one can see that entanglement detection using CMP is more robust against white noises than entanglement detection using CMI. The intuition behind this is that CMP is tailor-made for the particular correlations µ A + µ B =μ l of the target entangled state |φ 3 A ,2 B , while CMI can also detect other entanglement correlations. This intuition can be confirmed as follows. If we introduce a phase shift Z θ in the local system B to the target entangled state |φ 3 A ,2 B , the modified state |ψ(θ) is still maximally entangled within the Pauli subspaces H E11100 ⊗ H E11000 and H E11010 ⊗ H E01001 , but its correlations are changed. To detect entanglement of |ψ(θ) , we choose The CMI and the |φ 3 A ,2 B -targeting CMP of |ψ(θ) are plotted in a blue solid line and an orange dashed line, respectively, in Fig. 10 (a). Compare these two approaches, one can see that CMI is sensitive to entanglement of the state |ψ(θ) with θ close to the values {0, 1, 2, 3, 4}, while the |φ 3 A ,2 B -targeting CMP can only detect entanglement close to θ = 0. The correlations of |φ 3 A ,2 B with µ A + µ B = 0 as shown in Fig. 8 (b) are transformed into the other type of correlations, e.g. µ A + µ B = 3 for θ = 1 as shown in Fig. 10 (b). In this case, the perfect correlations of the entangled state |ψ(θ) can be detected by CMI, but not by the |φ 3 A ,2 B -targeting CMP. From the comparison between entanglement detection using CMI and CMP in Fig. 9 (a) and Fig. 10 (a), one can see that CMI can detect entangled states of different types of correlations, while CMP is more robust against white noises than CMI.

Conclusion
In this paper, we have studied the complementary structures of generalized Pauli operators in multiphoton LONs, and found that their MUBs are constituted within Pauli subspaces that are characterized by a cyclicly translational mode shift (Theorem 2.2). Accordingly, a set of complementary Pauli operators in fixed photon number LON systems has been constructed (Corollary 2.3).
It has been shown that, in a Pauli measurement, which is the projective measurement associated with a Pauli operator, the probability distribution over its Pauli-operator eigenspaces is given by the statistics of Z-clock labels in the outputs of its corresponding Hadamard transform (Theorem 3.1). Although the explicit Hadamard transformation of multiphoton states are #P -hard to calculate, this result lifts the computational complexity of Boson sampling in the simulation of Pauli measurement statistics. It therefore allows us to predict the probability distribution of Z-clock labels in a Pauli measurement of a given state, and vice versa to access complementary properties of an unknown state from Pauli measurement statistics.
Assessment of complementary properties from complementary Pauli measurement statistics has been shown to be invariant under decoherence over Pauli subspaces (Corollary 4.2). As a result, we can exploit such assessed quantities, which we call complementary Pauli quantities, to characterize the convex set of quantum states of a specific property S in multiphoton LONs through the convexroof extension of its hyperplane boundaries over Pauli subspaces (Theorem 4.3). It therefore allows us to detect the non-S property of quantum states in multiphoton LON systems experimentally in complementary Pauli measurements. Evaluation of measurement uncertainty relations in a multiphoton LON is a straightforward application of this theory (Corollary 4.4).
Exploiting this theory, we have shown that entanglement between modes in bipartite multiphoton LON systems can be physically detected by complementary correlations in complementary Pauli measurements. We have demonstrated entanglement detection in bipartite multiphoton LON systems with the detection approaches employing complementary mutual information (Corollary 5.1) and complementary mutual predictability (Corollary 5.2).
Our results open up physical access to desired quantum coherences in the MUBs in multiphoton LONs without falling into the computational complexity in Boson samplings. It allows us to predict and reveal the physical significance of entanglement between modes in bipartite multiphoton LONs in complementary Pauli measurements. The developed theory provides a general theoretical framework for the problems of hyperplane characterization of convex sets of multiphoton states in LON systems. Besides the detection of bipartite entanglement, it could be further employed in multipartite entanglement detection and entanglement dimensionality characterization in multiphoton LON systems. It therefore paves a way to extend quantum information processing in multipartite single-photon LONs to the multiphoton regime. Although our analysis is carried on in LONs, which encode paths in modes, it is general enough for any bosonic multimode system that allows generalized Hadamard transforms.

The Proof of Corollary 5.2
Proof. It is shown in [42] that the upper bounds on C F φ ,L for separable states σ E in a Pauli subspace H E is determined by where d E A ,E B := min(E A , E B ). Since CMP is linear, one can extend these upper bounds to (N A , N B )photon LON systems through convex-roof extension according to Theorem 4.3.