Characterization and optimized engineering of bosonic quantum interfaces under single-mode operational constraints

Controlling the quantum interface between two bosonic modes is essential in countless implementations of quantum information processing. However, full controllability is rarely achieved in most platforms due to specific physical limitations. In this work, we completely characterize the linear two-mode interfaces under the most pessimistic restriction that only single-mode operation is available. When arbitrary Gaussian single-mode operations can be applied to both modes, we find that every interface can be characterized by an invariant transmission strength. Moreover, in the practical situation that squeezing is restricted in one of the modes, we discover two additional quantities, irreducible squeezing and irreducible shearing, that are invariant under the allowable controls. By using this characterization, we develop systematic strategies to engineer an arbitrary linear interface through cascading multiple fixed component interfaces. Without squeezing restriction, our protocol is optimal and requires at most three component interfaces. Under the squeezing constraint, our protocol can be extended to engineer also the additional invariants by using no more than two more rounds of cascade. We also propose the remote squeezing scheme to tackle the squeezing restriction through interfacing with an active auxiliary mode.

To realize the above implementations and applications, it is essential to generate a Gaussian interface with purpose-specific strength and type.Gaussian interfaces are categorized into seven types in Ref. [35] and each type is utilized with specific purposes.They are Beamsplitter (BS), two-mode squeezing (TMS), quantum nondemolition interface (QNDI), SWAP, Identity interface, swapped-TMS (sTMS) and swapped-QNDI (sQNDI).BS type interfaces between modes are required for boson sampling [36,37] and quantum walk simulators [38,39].Two-mode-squeezing (TMS) is the interface underlying spontaneous parametric down-conversion photon generation [40][41][42] and phase-insensitive amplification [43].Quantum non-demolition interface (QNDI) is utilized in generating CV cluster states for measurement-based quantum computing [44,45] and quantum secret sharing [46][47][48].There has been increasing interest in engineering SWAP for transducing information [17,[49][50][51] between different components of hybrid quantum systems [52,53], and rapidly cooling trapped ions [54,55].Identity interfaces are sometimes engineered to avoid unnecessary excitation, such as in heatingless ion separation and merging [56,57].sTMS and sQNDI are newly discovered in Ref. [35].sQNDI allows quantum transduction with less stringent conditions [35,58], while there is no proposed application of sTMS yet.Two-mode linear interfaces have been widely studied by different scientific communities under different names.For examples, in quantum information processing they are usually used as Gaussian logic gates in CV quantum computing, or a description of Gaussian quantum channels in the open quantum system [25].In condensed matter physics, it is commonly known as Bogoliubov transformation [59,60].In mathematics, the transformation introduced by such interface is studied under the name of symplectic transformation [61].In this work, since we focus on the processes' properties in interacting two independent bosonic degrees of freedom, we will consistently denote the process as "interface".
Implementing a two-mode interface with the desired type and strength is generally challenging due to various experimental limitations.First the type of interaction is usually restricted by the physical properties of the platform.For examples, the Stokes and anti-Stokes interactions in electro-optical [62] or optomechanical systems [7] arXiv:2212.05134v3[quant-ph] 26 Feb 2024 typically generate either a TMS or BS interface; QNDI arises naturally between light and atoms ensemble as the interaction is bilinear in the quadrature of each system [17,63].Furthermore, there are limitations in the interaction strength.For example, the strength of the quantum light-atom interface is characterized by the optical depth of the atomic ensemble [64] that cannot be arbitrarily large, since the dense atomic ensemble is avoided to prevent unwanted interaction, such as the dephasing induced by the dipole interaction [65].
One might naively think that the unwanted properties of an imperfect interface can be removed by applying single-mode operations.However, this idea is disproved because the interfaces with different characteristic properties are not inter-convertible by single-mode operations [35].To overcome this fundamental limitation, one approach is to cascade multiple rounds of the restricted interfaces.Earlier proposals suggested that SWAP can be implemented by cascading multiple QNDIs [63,[66][67][68].Recently, Lau and Clerk [35] developed a more general scheme to construct SWAP and sQND cascading up to any six fixed interfaces.Zhang et al. [69] generalized this protocol to implement arbitrary Gaussian operations by applying exponentially many rounds of the same multimode interface.In spite of the possibilities introduced by these protocols, two important practical issues have not been fully addressed: 1. Optimality in rounds of applied interface: all the above protocols have not explored the minimum number of the required interfaces.It is generally beneficial to apply fewer rounds of interface because engineering inter-mode interaction is challenging, especially for the hybrid system.
2. Squeezing: some existing protocols assume that all modes can be squeezed, so they cannot be applied to the platforms involving systems that squeezing is challenging, e.g.optical and spin-ensemble systems.
Our work will address both of these issues.In the first half, we focus on the situation every single-mode operation is allowed.We find that every interface can be characterized by its invariant transmission strength, and every interface sharing the same transmission strength is inter-convertible.We then develop the protocols for engineering an interface with arbitrary transmission strength.By applying single-mode controls between at most three directly available interfaces with predetermined strength, referred to as component interfaces hereafter, our protocol can manipulate the overall transmission strength through amplification and interference.We prove that our protocol is optimal in the sense that it involves the fewest possible number of component interfaces.
In the second half, we extend the study to the restricted scenario that squeezing is available to only one of the modes.We discover two more quantities, irreducible squeezing and irreducible shearing, that are invariant under this additional operational restriction.To engineer an interface with any desired magnitudes of these invariant quantities, we develop a protocol that involves at most four component interfaces.To resolve the squeezing restriction, we also introduce the remote squeezing scheme that squeezes the restricted mode by interfacing it with an active auxiliary mode.
Our paper is organized as follows.We start with the general scenario and review the interfaces' characterization by the transmission strength in Sec.II.Then, we present the optimal interface engineering protocols in Sec.III.In Sec.IV, we discuss the new invariant parameters for characterizing an interface in the squeezing-restricted scenario.In Sec.V, we propose the squeezing-restricted protocols that construct interfaces with any desired characteristic parameters.Particularly, we present the remote squeezing protocol that can serve as a new method to overcome the squeezing restriction in Sec.V F.Sec.VI concludes the paper.

II. CHARACTERIZATION OF TWO-MODE LINEAR INTERFACE
We consider two modes, denoted by the annihilation operators â1 and â2 , interacting through a linear interface.By introducing the Hermitian quadrature operators via â ≡ (q + ip)/ √ 2, the transformation induced by the interface is represented as Here T ij with i, j ∈ {1, 2} is the 2 × 2 sub-matrix of the 4 × 4 real symplectic matrix T; describes the reflection or transmission of quadratures via the interface when i = j or i ̸ = j respectively [25].qin(out) and pin(out) denote the quadrature operators before (after) the interface.We note that Eq. ( 1) is a general description of linear interfaces, i.e. it covers both the ones induced by scattering processes and coherent interaction.For scattering-type, e.g.interaction of travel photons via an optical beam splitter, âin and âout represent respectively the input and output propagating mode operators; for coherent interaction, e.g.coherent mode coupling between superconducting cavities [70], âin and âout are the initial and final time mode operators respectively, i.e., âin ≡ â(0) and âout ≡ Û † â(0) Û for some evolution operator Û.Any interface can be converted by suitable single-mode operations to one of the seven typical interfaces: Identity, QNDI, TMS, BS, sTMS, sQNDI and SWAP [35], i.e.
Here L in(out) i is a 4 × 4 matrix denoting the single-mode Gaussian transformation on the i-th mode and Ū denotes the standard form of the interface that both reflection and transmission matrices are diagonal and the non-zero diagonal entries have equal magnitude.This form is significant not only because the typical interfaces are usually expressed in the standard form in the literature, but it is also useful in interface engineering.The operator and matrix representations of the standard forms are listed in Table I.
Generally, if single-mode operations are applied before and after the interface, the overall transformation matrix will be altered.Moreover, one would expect that an interface can generate inter-mode correlations that cannot be affected by single-mode operations.Indeed, Ref. [35] identified that the ranks of the reflection and transmission matrices, n R ≡ rank(T 22 ) ∈ {0, 1, 2} and n T ≡ rank(T 21 ) ∈ {0, 1, 2}, are invariant under singlemode operations.For the purpose of transduction, they show that each class of interface, as classified by the different combinations of ranks, has different utility in engineering a perfect transduction.
In additional to the ranks, another invariant, the determinant of the transmission matrix T 21 , is recognized in Ref. [35], although it is not involved in the engineering of transduction.For our current purpose of interface engineering, however, we realize that χ is a good classifier because all interfaces that share the same χ and ranks are inter-convertible.Explicitly, any two interfaces A and A ′ with the same χ and ranks have the same standard form according to Eq. ( 2), i.e.
The two interfaces can thus be inter-converted by single mode operations, i.e.
To observe the physical meaning of χ, we consider that for a BS with angle θ ∈ (−π/2, π/2), In this case, χ BS is the magnitude square of the transmittance of the BS.Moreover, we recognize that no information is transmitted through Identity interface, of which χ I = 0.While all information is transmitted through SWAP, this interface has a χ SWAP = 1.From these examples, we can observe that χ can characterize the strength of the transmission through an interface, and hence we will call χ the transmission strength.We now discuss the meaning of χ of the remaining interfaces.For sTMS and TMS, their transmission strengths are given by where r ∈ (−∞, ∞) is the TMS strength.sTMS is equivalent to cascading two operations: amplifying by TMS and transmitting via SWAP.χ sTMS is always larger than 1 and implies that the transmitted information via sTMS is amplified.For TMS, the transmitted mode is associated with noise [43] and the negative χ TMS is the signature of this fundamental incorporated noise.Both QNDI (n T = 1) and Identity (n T = 0) have the zero transmission strength, χ QNDI = χ I = 0, it is because no quantum information is transmitted through either of them.Identity interface obviously transfers no information.Even though QNDI (n T = 1) transmits the information of one quadrature, its quantum capacity vanishes because this transmission can be equivalently implemented by homodyne detection and transmitting the classical measurement outcome.sQNDI (n R = 1), similar to SWAP (n R = 0), has infinite quantum capacity [58] and completely transmits quantum information without adding noise, so it has χ sQNDI = χ SWAP = 1.Interestingly, even though both QNDI and sQNDI involve a non-zero QND strength η ∈ (−∞, ∞), this parameter cannot be used as a classifier since it is not invariant under single-mode transformation.More explicitly, it is easy to show that where S i (γ) denotes the single-mode squeezing with strength γ ∈ (−∞, ∞) of mode i [71], ŪQ (η) and ŪSQ (η) denote respectively the standard form of QNDI and sQNDI with QND strength η.Eqs. ( 7)- (8) show that the QND strength can always be manipulated by applying single-mode squeezing before and afterward the interface.

III. INTERFACE ENGINEERING WITH ARBITRARY SINGLE-MODE OPERATIONS
Each quantum information process requires purposespecific interfaces.According to our characterization in Sec.II, implementing them is equivalent to implement the target interfaces with the specific transmission strength χ tgt and ranks n tgt R and n tgt T .However, physical platforms are usually subjected to practical limitations, so the available interfaces may be restricted.Inspired by previous works [35,69], we develop a systematic scheme that can engineer an interface with arbitrary transmission strength and ranks by cascading multiple component interfaces, i.e. platform-available interfaces.In this section, we consider the situation that any single-mode rotation and squeezing can be implemented on both modes.Our scheme is designed to account for the most general restrictions of the available interface, as we assume that both the ranks and transmission strength of every component interface are fixed but known.
TABLE I. Characterization of interfaces and requirements for interface engineering.The first and second columns respectively show the equivalent well-known operation for each class and the corresponding transmission strength, χ ≡ det(T21).The third column shows the additional characteristic parameters when squeezing is restricted to only one mode.Forth and fifth columns list the operator and matrix representations of the interfaces in the standard form.Here r is the TMS strength, θ is the BS angle, η is the QND strength, I and Z are the 2 × 2 identity and Pauli-Z matrices respectively.The last two columns are the number of fixed interfaces required for engineering a interface in this class under the general and squeezing-restricted situations.
Before discussing the details, it is worth clarifying the differences between our scheme and other interface engineering techniques.First, both the strength and type of an interface can be modified respectively by Hamiltonian amplification [72] and Lloyd-Braunstein protocol [23].However, the required number of component interfaces in these schemes grows with the accuracy and strength of the desired interface.On the contrary, our scheme requires at most three component interfaces to exactly engineer an interface with any desired strength and type.Second, by Bloch-Messiah decomposition [73], any two-mode interface can be decomposed into two BS and a round of single-mode squeezing in between.However, engineering interfaces by this method requires implementing BS with tunable angles, which is in stark contrast to our scheme that can utilize any non-trivial component interfaces with fixed strength.

A. Two-interface setup
The basic setting consists of two cascading component interfaces as illustrated in Fig. 1.This setup is implemented in the following sequence: 1) two modes couple via the first component interface A; 2) they are transformed by controllable single-mode operations, L 1 and L 2 ; 3) they couple again via the second component interface B. After the sequence, the two modes are effectively transformed under the resultant interface AB.
For simplicity, we assume that the component interfaces A and B have already been converted to the standard form by appropriate single-mode operations.Then, the resultant transformation matrix T AB is given by, Our aim is to find the suitable single-mode operations L 1(2) , for constructing a resultant interface T AB with the desired rank and transmission strength χ AB .
In principle, there are 49 combinations of component interfaces because A and B can belong to any of the 7 classes in Table I.Fortunately, as explained below, only 6 combinations are independent and need to be considered individually.
First, for engineering a non-trivial interface, it is legitimate to assume component interfaces A and B do not belong to Identity and SWAP, because these two classes do not generate any non-trivial correlation between the involving modes.
Second, any combination involving sTMS or sQNDI components, which can be treated respectively as a TMS or QNDI followed by a SWAP, needs not be considered individually.We note that applying a SWAP after an interface with strength χ ′ will result in an overall interface with strength 1 − χ ′ .It is because SWAP exchanges the transmitted and reflected quadratures, and hence their strengths, and the sum of transmission and reflection strengths always equal to unity according to the canonical commutation relations, i.e. det(T 22 ) + det(T 21 ) = 1 [35].If the combination of A and B can engineer a resultant strength χ AB , then by using the same single-mode operations, an interface with strength χ , where χ B is the transmission strength of B. As a result, the combination involving sTMS or sQNDI components can be deduced from the combinations of TMS or QNDI respectively.
Third, the order of interfaces does not affect the transmission strength.It can be understood by considering the inverse of Eq. ( 9), i.e.
and recognizing that any interface has the same transmission strength as its inverse.For any non-trivial interface, its inverse can be constructed by applying single-mode operations before and after itself, i.e.
where R i and F i ≡ R i (π/2) denote respectively the rotation and Fourier gate for the i-th mode [71].The standard form of BS, TMS and sTMS interfaces are denoted as Ūχ .If there exists a protocol that engineers a desired interface AB by first applying A then B (denoted by combination A + B hereafter), one can substitute Eqs. ( 11)-( 13) into Eq.( 10) to obtain a protocol to generate AB by first applying B then A.

B. Interference mechanism
The principle of the cascading interface setup is to use the controllable single-mode operations L 1(2) to interfere and hence manipulate the transmission.As a pedagogical example, we consider a simple setting with only one controllable squeezing in between, i.e.L 1 = S 1 (γ) and L 2 = I.Furthermore, we assume the component interfaces A and B are respectively BS and TMS in the standard form.Since both interfaces and single-mode operations are quadrature-diagonal, the overall transmission matrix is given by where As illustrated in Fig. 2, the q-transmission amplitude of q1 to q2 is the sum of the amplitudes of two paths: the upper path (blue) reflects in A, passes through the squeezer and transmits in B; and the lower path (red) transmits in A and reflects in B. The squeezer amplifies the amplitude of the first path, i.e. the first term in Eq. ( 15), while the amplitude of the second path is not modified, i.e. the second term in Eq. ( 15).A similar process happens for the p-transmission in Eq. ( 16), except that the first path is deamplified by the squeezer.It is clear that adjusting the squeezing strength γ can tune the resultant transmission strength χ AB = T 21,q AB T 21,p AB .χ AB can be arbitrarily small near destructive interference, i.e. a γ is chosen that T 21,q AB or T 21,p AB is close to zero, while it can be arbitrarily large by exerting a large γ such that T 21,q AB is large.As a result, the whole spectrum of χ AB can be covered by manipulating γ.

C. General Two-interface Protocols
Next, we present the complete protocols for all six combinations of component interfaces.We will consider the setting that consists of mode-1 squeezing and rotations on both modes, i.e.
where ϕ 1 , ϕ 2 ∈ (0, 2π) are the rotation angles.A straightforward calculation shows that the resultant transmission strength is given by where A is determined only by the transmission strength of the component interfaces, and Z accounts for the interference that is controllable by the in-between single-mode operations.Our strategy is to tune Z to adjust χ AB to the desired value, χ tgt .

Amplify Interface B (TMS)
Interface A (BS) Interference of q-quadrature transmission for the cascading interfaces setup.There are two paths for transmitting qin 1 to qout 2 .Due to the squeezing of mode 1 (triangle), the blue path is amplified and the red one is unchanged.The overall transmission amplitude is determined by the interference of the two paths and thus controllable by the squeezing strength γ.
Config.Z takes different forms for different combinations.We will discuss each combination as follows, and the result is summarized in Table II.

BS+TMS, BS+QNDI or TMS+QNDI
When two interfaces of different classes are combined, we turn off all the rotations, i.e. ϕ 1 = ϕ 2 = 0, and the value of Z is solely determined by the squeezing strength γ.For BS+TMS, Z is given by which can be tuned to arbitrary values.For BS+QNDI and TMS+QNDI, their Z are expressed as which can be arbitrary non-zero value [74].Under these settings, the resultant interface can be any non-trivial interface, i.e.BS, TMS, sTMS, QNDI or sQNDI, but not Identity and SWAP.We will discuss more about the latter two in Sec.III D.

BS+BS or TMS+TMS
For the combination BS+BS or TMS+TMS with χ A ̸ = χ B , 1 − χ B , we can set ϕ 2 = 0, and obtain Z as where Although |(γ 2 + 1)/γ| is lower bounded by 2, Z covers all real values thanks to the rotation ϕ 1 .More explicitly, we turn off the squeezing, i.e. γ = 1, and adjust the value of Z by ϕ 1 .
Under these settings, we can engineer any non-trivial interface.
We now discuss two special cases.First, for χ A = χ B , we will obtain Identity instead of QNDI when χ AB = 0. To engineer QNDIs, a rotation is needed to mix the q-and p-quadratures to avoid the complete destructive interference of both quadratures in the transmission.We show in Appendix B that QNDIs can be constructed by setting ϕ 1 ̸ = 0, ϕ 2 = 0 and γ = tan ϕ 1 − sec ϕ 1 .
Another special case is χ A = 1 − χ B , which may happen in BS+BS combination.Tuning χ AB = 1 by the above protocol will result in a SWAP but not sQNDI.To construct sQNDIs, we need a modified setting to prevent reflection from completely destructively interfering in both quadratures.We show in Appendix B that this can be achieved by setting ϕ 1 ̸ = 0, ϕ 2 = 0 and γ = − tan ϕ 1 − sec ϕ 1 .

QNDI+QNDI
If one of the available component interfaces is a QNDI, it can be used to engineer a QNDI with arbitrary QND strength by applying single-mode squeezing, i.e.Eq. (7).For other non-trivial interfaces, we can pick ϕ 1 = ϕ 2 = π/2 and obtain which can be adjusted by γ to cover all non-zero values [75].We are also able to engineer an Identity by changing the setting to ϕ 1 = ϕ 2 = 0 and γ = −η A /η B , then the two component QNDIs will be mutually cancelled.

D. Three-interface Protocols for Identity and SWAP
In view of the success of two-interface protocol in engineering non-trivial interface, one might be tempted to use such protocol to also engineer Identity and SWAP.Engineering these interfaces requires eliminating the amplitudes of both q-and p-quadratures in either the transmission (for Identity) or reflection (for SWAP).However, in the two-interface configuration, there is only one round of squeezing between the two component interfaces.Intuitively, it seems impossible to simultaneously destructively interfere both q-and p-quadratures as the squeezing of one quadrature will inevitably anti-squeeze the other.
To verify this intuition, we exploit all available singlemode controls in Appendix B, and discover that Identity and SWAP cannot be engineered by the two-interface protocol unless the two interfaces follow stringent relations.For engineering Identity, the component interfaces must have the same strength, χ A = χ B .In this situation, interface A can always be converted to the inverse of B by appropriate single-mode operations (see Eqs. ( 11)-( 13)), then the two component interfaces mutually cancel and the resultant interface becomes an Identity.
To engineer SWAP, the strengths of the component interfaces must be complemented, i.e. χ A = 1 − χ B .This condition implies that interface A can be converted to the swapped-inverse of interface B, i.e.
where ŪS is the standard form of SWAP.By eliminating T B , we obtain a SWAP.
In general, the available component interfaces may not satisfy the above stringent conditions.To engineer Identity and SWAP, our strategy is to cascade more interfaces and single-mode controls, such that additional rounds of interference can be implemented to destructively interfere both quadratures.Inspired by the above understandings, we develop systematic protocols that require the minimum rounds of component interfaces, i.e. using three interfaces, A, B and C, as illustrated in Fig. 3(a).
For engineering Identity, our strategy is to apply the two-interface protocol in Sec.III C to combine two component interfaces, A and B, to construct an intermediate interface AB that is the inverse of C (Fig. 3(b)).Similarly, to engineer SWAP, AB can be engineered to be the swapped-inverse of C, i.e.T AB = T −1 C ŪS , so that the resultant interface ABC becomes a SWAP (Fig. 3(c)).
To summarize this section, any non-trivial interface, BS, TMS, sTMS, QNDI and sQNDI, can be engineered by cascading two component interfaces with suitable single-mode controls.Moreover, engineering Identity and SWAP is generally impossible with only two arbitrary interfaces, so we have introduced a three-interface protocol.Comparing with the existing schemes that require six arbitrary component interfaces to engineer a SWAP [35], and four identical component interfaces for engineering an arbitrary two-mode interface [69], our protocols are optimal because the number of component interface in- volved is the minimum possible.

IV. CHARACTERIZING SQUEEZING RESTRICTED INTERFACE
So far we have considered the situation that all singlemode operations are available on both modes.However, this situation is not always relevant in practice because actively squeezing an unknown state is known to be challenging in many platforms, such as optics and spin ensembles.If one involved platform in a hybrid quantum system suffers from such squeezing restriction, one would expect that the interfaces with different squeezing on the restricted mode cannot be inter-converted by the available single-mode controls.
Apart from hybrid quantum systems, such squeezing restriction can also be found even when all modes have the same physical nature, but each mode plays a different role in the application.For example, in collision-based trapped ion cooling [76], which aims to cool data ions that encoded information in their internal states, squeezing can only be applied to the ancillary coolant ions that do not encode information.It is because squeezing ion motion would require coupling the motional and internal states [77], and thus will corrupt the encoded information.
Although the squeezing restriction is not uncommon, its implications on interface characterization and engineering are not well understood.In the relevant literature, Ref. [35] studies this issue only for the engineering of a specific interface (i.e.SWAP).The interface engineering scheme in Ref. [69], Moreover, heavily relies on the phase-sensitive amplification of every mode and thus cannot be applied under this practical restriction.We will fill this missing gap in the following sections.In this section, we will first complete the interface characterization, and its implication in interface engineering will be discussed in Sec.V.
A. Diagonalizing mode-2 reflection matrix T 22   In the previous sections, we illustrate that each interface can be classified by its transmission strength and sub-matrix ranks under the assumption that arbitrary Gaussian single-mode operation can be applied.Under the squeezing restriction, however, these classifiers are no longer sufficient.Here is an example: consider a linear interface that introduces only single-mode squeezing to the restricted mode, this interface falls into the class of Identity.Obviously, this interface cannot be converted to some other interface belongs to the same class, e.g. an operation that is identity on both modes, because it requires squeezing of the restricted mode.Thus a modified scheme is needed to classify the interfaces under squeezing restriction, and we will introduce it as follows.
Without loss of generality, we assume mode 2 is squeezing-restricted, the allowable operation in this setup will consist of any single-mode operation on mode 1 but only rotation on mode 2. We first consider the mode-2 reflection matrix T 22 since it is the part in the transformation matrix T that is mostly affected by this restriction, i.e.T 22 can be altered by rotation only.According to the property of singular value decomposition (SVD) [78], the singular values of a matrix are invariant under rotation operations.The SVD of T 22 is given by where Σ is a 2 × 2 diagonal matrix with the singular values.We thus realize that the singular values of T 22 can be used to construct the invariant classifier under squeezing restriction.We will discuss its physical meaning for each class of interface.
1. Interfaces with χ ̸ = 0, 1 For interfaces with χ ̸ = 0, 1, the corresponding Σ can always be written as where the + and − signs are respectively for the interfaces with χ < 1 and χ > 1.The additional classifier Λ characterizes the discrepancy between the two singular values.We identify Λ as the irreducible squeezing strength since it will not be altered unless squeezing is applied on mode 2. We can denote Λ as the squeezing of mode 2, i.e.Λ ≥ 1, by setting Σ χ with the positive and larger upper diagonal element, since the squeezed and anti-squeezed quadratures of mode 2 can always be exchanged by applying Fourier gates before and after the interface.
In Appendix C, we show that the transformation matrix T of any interface in this class can always be transformed, by the allowable single-mode operations, into one of the following two simplest forms that consist of only two irreducible parts: a standard-form interface and a mode-2 squeezing.They are the pre-squeezing form, i.e. mode-2 squeezing is applied before a standard-form interface, and the post-squeezing form, i.e. mode-2 squeezing is applied after a standard-form interface, They are useful for constructing the squeezing-restricted protocols in Sec.V-V F and show clearly that each interface in this class is characterized by two invariant parameters: the transmission strength χ ̸ = 0, 1, and the irreducible squeezing Λ ≥ 1.

sQNDIs (
For interfaces with χ = 1 and n R = 1, their Σ is of rank 1 and given by Here the non-zero singular value Λ can be recognized as the QND strength of the sQNDI.Different from Λ ≥ 1 for interfaces with χ ̸ = 0, 1 (Sec.IV A 1), Λ for sQNDIs should denote both squeezing or anti-squeezing of mode 2, i.e.Λ > 0. It is because only one quadrature is presented in the reflection through a sQNDI, applying Fourier gates cannot switch the squeezing to antisqueezing of this quadrature nor vice versa.Moreover, in contrast to the situation that both modes can be squeezed, the QND strength is invariant under the squeezing restriction.This can be understood from Eq. ( 8) that altering the QND strength of a sQNDI requires squeezing both modes, but since it is forbidden in this situation, the QND strength becomes invariant.We show in Appendix C that any interface in this class can be transformed into the pre-and post-squeezing forms, Both forms consist of a sQNDI with a unity strength and a mode-2 squeezing with strength Λ.The latter can be recognized as an irreducible squeezing and it can be used to characterize a sQNDI.
3. QNDIs (χ = 0, nT = 1) For interfaces with χ = 0 and n T = 1, their T 22 can always be diagonalized by SVD as Same as Sec.IV A 1, Λ ≥ 1 can be recognized as irreducible squeezing and can be used to characterize QNDIs.Surprisingly, in addition to Λ, we discover another parameter from the transmission matrix T 21 that is invariant under the allowable operations.We show in Appendix C that the transmission matrix can be decomposed as 11 by QL decomposition [78], where Here η is the QND strength, which can be manipulated by squeezing mode 1.Moreover, the parameter κ is invariant under the available single-mode operations.We call κ the irreducible shearing [25] since it can be recognized as an additional mode-2 shearing applied after the standard-form QNDI.Since a shearing operation can be decomposed into rotation and squeezing [25], in Appendix C we show that any QNDI can be transformed to the following pre-and post-squeezing forms [79], 34) Here tan ϕ R ≡ κ and tan ϕ L ≡ −κΛ 2 .Overall, any QNDI is characterized by three parameters, χ = 0, Λ and κ.Any interface with χ = 0 and n T = 0 is equivalent to single-mode operations applying to both modes.Under the squeezing restriction, any operation on the unrestricted mode 1 can be removed by single-mode controls, while that on the squeezing-restricted mode 2 can be removed up to a squeezing operation, i.e.
In other words, T 22 can be diagonalized according to Eq. ( 24) with Σ = S 22 2 (Λ).As such, any interface in this class is characterized by the irreducible squeezing strength Λ.Any interface with χ = 1, n R = 0 can always be converted to a standard-form SWAP by the available singlemode operations, i.e.
There is no irreducible squeezing for this class since any squeezing applying on mode 2 can be swapped to mode 1, i.e.ŪS S 2 (Λ) = S 1 (Λ) ŪS , and be cancelled by mode-1 squeezing controls.We summarize all additional characteristic parameters due to the squeezing restriction for every class of interface in Table .I. We note that, in analogous to the unrestricted case, any two interfaces A and A ′ that share the same set of characteristic parameters are interconvertible.It is because they can be transformed by the allowable single-mode operations to the same standard form in conjunction with the same irreducible local operations (Eqs.( 26), ( 27), ( 29), ( 30), ( 33), ( 34), ( 35), (36) Inter-conversion can then be done by inverting the singlemode operations.

V. SQUEEZING-RESTRICTED PROTOCOLS
Because each interface possesses additional invariant properties under the squeezing restriction, one might expect that the protocols in Sec.III are no longer general in engineering arbitrary interfaces in this situation.Indeed, there are two main reasons that a modified interface engineering protocol is needed: 1) in addition to ranks and transmission strength, the protocol should also generate a target interface with the desired irreducible squeezing and shearing; 2) the irreducible properties of the component interfaces have to be considered.In the following, we will present the modified protocols that take these two issues into account.Intuitively, in order to cope with the additional restrictions, the number of the required component interfaces will increase; such requirement is summarized in Table I.

A. Two-interface module
We start with introducing the two-interface module that is required in the protocols.Assuming a compo-  nent interface A is applied before B, the first step is to convert A and B into respectively the pre-and postsqueezing forms, as illustrated in Fig. 4(a), such that the irreducible mode-2 operations are placed at the very beginning and very end of the whole sequence.By considering the standard-form parts of A and B, we can then apply the two-interface protocols in Sec.III C to engineer an intermediate interface AB with the desired χ AB , which is not affected by the irreducible squeezing and shearing of A and B. After that, we can convert AB into either the pre-or post-squeezing form (Fig. 4(b)), depending on the context of the protocol.

B. SWAP
We first discuss the protocol for SWAP, which has no additional characteristic parameter.We realize that the protocol in Sec.III D can be applied with minor mod- ifications to take care of the irreducible squeezing and shearing of the component interfaces.Our protocol is illustrated in Fig. 5(a).An intermediate interface AB with χ AB = 1 − χ C is first constructed by the two-interface module in the last subsection.Next, the component interface C is converted into the post-squeezing form and applied after AB, as shown in Fig. 5(b).By converting AB into the swapped-inverse of C up to some irreducible mode-2 operations L AB , i.e.T AB = Ū−1 C ŪS L AB , the non-trivial interface C will be cancelled.A SWAP will be remained together with mode-2 operations (Fig. 5(c)), which can be eliminated by swapping squeezing and rotating from mode 1.At the end, we will have a SWAP in the standard form.

C. Interfaces with χ
The next protocol is for engineering an interface with χ ̸ = 0, 1.Our goal is to obtain the target interface with the transmission strength χ tgt and the irreducible squeezing Λ tgt .Our protocol involves four non-trivial component interfaces A, B, C and D. The strategy is to match the characteristics parameters sequentially: two interme-diate interfaces AB and CD are first engineered that their combined transmission strength matches χ tgt , then Λ tgt is matched through controlling the squeezing in between.

Tuning χABCD
To match the transmission strength, our first step is to combine A with B and C with D to form two intermediate interfaces AB and CD (Fig. 6(a)) by applying the two-interface module.χ AB should be chosen according to the criteria [80] : and χ CD should be tuned according to The reason for the above choice is to guarantee the combination of intermediate interfaces is either BS+BS, TMS+TMS or TMS+sTMS, otherwise the resultant irreducible squeezing Λ ABCD will be bounded (details in Appendix D).Secondly, we convert AB and CD into respectively preand post-squeezing forms, as illustrated in Fig. 6(b).The mode-1 operation in between AB and CD is chosen as As illustrated in Fig. 6(c), the purpose of R α 1 and R β 1 is to introduce a controllable rotations, R α ′′′ 2 and R β ′′′ 2 , between the standard-form interfaces and the irreducible mode-2 operations through the following equivalent circuits, where the relation between the rotations can be found in Appendix B. As will be discussed in Sec.V C 2, this setting will be helpful to engineer the desired irreducible squeezing.Other rotations listed in Eqs.(39)(40) will be cancelled by the controllable rotations, such that the overall transmission strength is determined by Ūχ AB , which is labelled in Fig. 6(c).As illustrated in Fig. 7, the purpose of F 1 is to exchange the q-and p-quadratures of mode 1 after passing through AB, so the quadrature interference between AB and CD can be switched off.This can be seen from the transmission matrix of T ζ , where Ūij α ≡ diag( Ū ij,q α , Ū ij,p α ) for α = AB, CD.The amplification and deamplification due to the controllable mode-1 squeezing S 1 (γ) appear only in the off-diagonal   39)-( 40), the controllable mode-1 rotations R α 1 and R β 1 effectively introduces controllable mode-2 rotations between the standardform transformation and the irreducible squeezing.(c) The equivalent circuit after the rearrangement.R2 is chosen to cancel all rotation between AB and CD, while mode-1 control follows the non-interfering setup in Fig. 7.(d) The resultant interface ABCD.Its transmission strength χABCD can be engineered to the desired value according to Eq. ( 42), and the resultant irreducible squeezing ΛABCD, which is determined by ΛAB, ΛCD and the controllable Λa can be tuned to the target value according to Eq. ( 43) or (44).entries of Eq. ( 41), so their effects in the transmission strength compensate each other, i.e. χ ABCD ≡ det(T 21 ζ ) is independent of γ.We find that the overall transmission strength is then given by which is determined by the strengths of AB and CD only.Then, the choice of χ AB and χ CD guarantees that χ ABCD matches χ tgt (c.f.Eq. ( 38)).
) is applied on mode 1 to interchange the q-and p-quadratures.
After passing through intermediate interface CD, each output quadrature will contain transmitted information from both input quadratures, each coming from an independent path.There is thus no interference, and the squeezing on mode 1 (triangle) does not alter the resultant transmission strength.

Manipulating ΛABCD
Although the resultant transmission strength is independent of the in-between mode-1 squeezing S 1 (γ), the resultant irreducible squeezing does depend on γ.This gives us the ability to engineer the resultant interface with the desired irreducible squeezing, i.e.Λ ABCD = Λ tgt .Explicitly, thanks to the controllable rotations R α ′′′ 2 and R β ′′′ 2 introduced by the equivalent circuit in Eqs. ( 39)-( 40), the intermediate interface T ζ can be quadrature-diagonalized, as illustrated in Fig. 6(d).The resultant irreducible squeezing is then given by a simple product of those of the intermediate interfaces AB, CD, and T ζ , i.e.
or the division of them, i.e., Λ AB and Λ CD are by-products of engineering AB and CD and assumed to be untunable; Λ a ≥ 1 is determined by the singular values of T 22 ζ , which is controllable by the mode-1 squeezing γ.In Appendix D, we show that Λ a , and hence Λ ABCD , can be tuned to arbitrary values.Overall, this protocol can engineer an interface with arbitrary χ tgt ̸ = 0, 1 and Λ tgt .

D. sQNDIs
Our protocol to engineer arbitrary sQNDIs involves four non-trivial component interfaces.As illustrated in Fig. 8(a), by using the two-interface module in Sec.V A, we first combine the component interfaces A and B to form a standard-form sQNDI, and similarly C with D to form a QNDI in the post-squeezing form.Next, the QND strength of CD is manipulated by using the mode-1 squeezing according to Eq. (7).Finally, by recognizing that a sQNDI is equivalent to a QNDI followed by a SWAP, i.e.ŪSQ (Λ) = ŪQ (Λ) ŪS , and applying two standard-form QNDIs in sequence will result in a QNDI with the sum of their strengths, i.e.ŪQ (η) ŪQ (η ′ ) = ŪQ (η + η ′ ), cascading AB with CD will result in a sQNDI with a controllable QND strength, and equivalently a controllable irreducible squeezing.By considering the explicit expression of the mode-2 reflection matrix, including the irreducible squeezing and shearing of CD, we show in Appendix D that the resultant irreducible squeezing strength is given by where tan ϕ CD ≡ κ CD Λ 2 CD .By controlling γ, Λ ABCD can be tuned to any desired value.

E. QNDIs
Surprisingly, any QNDI can be engineered by using also four non-trivial interfaces, even though there is one more parameter to be matched.Our strategy is to reverse engineer the required single-mode controls by using the sQND engineering protocol in Sec.V D. Explicitly, we first construct two standard-form sQNDIs by combining the component interfaces A with B and C with D according to the two-interface-module.Our aim is to explore the single-mode controls, L (n) (n = 1, 2, 3), that can generate the desired QNDI when cascading with the two intermediate sQNDI components, i.e.
where κ tgt ≡ tan ϕ tgt and Λ tgt are respectively the target irreducible shearing and squeezing.We note that the QND strength η is unimportant as it is adjustable by mode-1 operation.By rewriting Eq. ( 46) as it coincides with the configuration of sQNDI engineering in Sec.V D, i.e. a sQNDI with strength Λ AB is constructed by cascading a sQNDI with strength Λ CD and a QNDI with irreducible shearing κ tgt and squeezing Λ tgt .By applying our protocol in Sec.V D, the required L (n) can be identified. 1 1 ( −1 ) FIG. 8. Four-interface protocol for constructing arbitrary sQND under squeezing restriction.(a) Component interfaces A and B are combined to form a sQNDI while C and D are forming a QNDI.The QND strength of the ŪQ (ηCD) is modified by controllable mode-1 squeezing before and afterwards according to Eq. ( 7).Subsequently, it is combined with ŪQ (ΛAB) to form a standard-form QNDI with the controllable strength γηCD + ΛAB.(b) The resultant interface is a sQNDI.Its QND strength, which is determined by both the controllable strength γηCD + ΛAB and the irreducible squeezing and shearing of CD, is given by Eq. ( 45).

F. Remote squeezing
Finally, we discuss the protocol to engineer an arbitrary Identity class interface, i.e. an interface with n T = 0, χ = 0 and a controllable Λ.Under the squeezing restriction, a general interface in this class is no longer an Identity operation but an irreducible squeezing in mode 2. Constructing this interface is thus equivalent to "remotely" squeeze the restricted mode with any desired strength Λ tgt through interfacing with an active mode; we thus call this type of protocol remote squeezing.Remote squeezing can be straightforwardly implemented by using two rounds of SWAP.After the first SWAP, the mode-2 initial state is transferred to mode 1 and then directly squeezed.With the second SWAP, the squeezed initial state is sent back to mode 2. By using the threeinterface SWAP engineering protocol in Sec.V B, this double-swap method can be implemented with any six non-trivial component interfaces.
Moreover, we developed a simplified remote squeez-  ing scheme that requires only five non-trivial component interfaces, as illustrated in Fig. 9. Following the fourinterface protocols in Sec.V C-V E, our idea is to use the first four components to engineer an intermediate interface ABCD that is the inverse of the fifth component E preceded with the desired mode-2 squeezing, i.e.T ABCD = T −1 E S 2 (Λ tgt ).After cascading with E, the non-trivial transformations are eliminated and the resultant interface becomes a remote squeezing at our desired value, S 2 (Λ tgt ).
As a remark, we have also presented other remote squeezing schemes in Appendix E that could use fewer component interfaces except a few special cases.

VI. CONCLUSION
We have completely characterized the linear bosonic two-mode interfaces under single-mode operational constraints.We have studied two situations: the general situation that any single-mode operation is available on both modes, and the restricted situation that squeezing is available on one mode only.We recognized the classifiers that are useful in identifying interfaces: interfaces with different classifiers are not inter-convertible by any allowable single-mode control.Under the general situation, any non-trivial interface can be classified by its transmission strength χ.For the restricted situation, we discovered two additional invariant classifiers: irreducible squeezing Λ and irreducible shearing κ.The main results of characterization are listed in Table .I.
Guided by the characterization, we developed the protocols for engineering arbitrary linear two-mode interfaces with any available interface in the platform.Our protocols incorporate the available interfaces as components, and cascade multiple of them with suitable singlemode controls.For the general situation, our protocol can construct any non-trivial interface by using only two rounds of component interface, and SWAP and Identity by using only three rounds.We prove that our protocols are optimal because they involve the fewest possible rounds of component interface.Under the squeezing restriction, we also developed the modified protocols to engineer the additional invariant classifiers.The required number of component interface remains three for engineering SWAP, while four is needed for engineering other interfaces.To resolve the squeezing restriction, we introduce the remote squeezing protocol to squeeze a restricted mode through interfacing with an active mode.We summarize the number of required interfaces in Table I.
Our characterization can help benchmarking experiment platforms and optimizing their designs for implementing specific applications.Our interface engineering protocols can overcome the limitations of physical platforms in implementing interfaces.They thus facilitate the realization of quantum technologies in a wider range of platforms, such as implementing universal CV logic gates in passive systems, reducing noise in quantum transduction, and conducting interferometry between hybrid quantum systems.Moreover, our remote squeezing scheme can help generating resourceful states for sensing, communication and computation, such as spin squeezed state [81] and squeezed optical state [82].
Now, we are able to determine L ′ 1 and L ′ 2 for different combination of component interfaces.As discussed in Sec.III A, it is legitimate to assume component interfaces A and B do not belong to Identity and SWAP, and the combination involving sTMS or sQNDI components can be deduced from the combinations of TMS or QNDI respectively, so we need to consider only six combinations: BS+BS, TMS+TMS, BS+TMS, BS+QNDI, TMS+QNDI and QNDI+QNDI.
For BS+BS, TMS+TMS and BS+TMS, we are always able to find the suitable L B,u 1 , L B,v 2 , L A,x 1 and L A,y 2 to simplified L ′ 1 as R 1 (ϕ 1 )S 1 (γ) and L ′ 2 as I 2 respectively according to Eqs. (B3) and (B4).Therefore, these combinations have only 2 independent single-mode operational parameters, and the corresponding simplified configuration is given by For BS+QNDI or TMS+QNDI, the rearrangement of single-mode transformations around the QNDI, which is the interface B, is restricted according to Eqs. (B14), (B15) and (B16).We find that L ′ 2 can still be chosen as identity by setting L B,v 2 = I and L A,y 2 = L 2 .However, under this setting, L ′ 1 cannot be simplified, i.e. it is still characterized by the general rotation-squeezing-rotation sequence, L ′ 1 = R 1 (ϕ 1 )S 1 (γ)R 1 (ϵ).Then, the simplified configuration is given by that contains 3 independent single-mode operational parameters.
For QNDI+QNDI, we discover that even though L B,v 2 and L A,y 2 are restricted, L ′ 2 can still be reduced to R 2 (ϕ 2 ).Explicitly, we can choose L B,v This choice is based on the QR decomposition [78], i.e. any real matrix, [L 2 ] 22 , can be decomposed into an orthogonal matrix, [R(ϕ 2 )] 22 , and a triangular matrix, [L A,y 2 ] 22 .Similar as BS+QNDI or TMS+QNDI, L ′ 1 cannot be simplified, therefore the most simplified configuration is given by that contains 4 independent single-mode operational parameters.
With the above simplified configurations, we can determine the sufficient and necessary conditions for engineering Identity and SWAP by brute force calculation based on Eqs.(B17)-(B19).To engineer Identity interfaces, it requires the overall transmission matrix to be null.We list the transmission matrices for all combinations in Table.III.By direct calculation, we determine that a null transmission matrix can be generated if and only if two interfaces are of the same type with the identical transmission strength, i.e. χ A = χ B .There are three possible combinations, BS+BS, TMS+TMS and QNDI+QNDI.Specifically, for BS+BS, two BS angles are required to be equal, i.e. |θ A | = |θ B |, and hence two modes are split by the first interface but recombined by the second interface.For TMS+TMS, two TMS strength are equal, |r 1 | = |r 2 |, so the modes are amplified and then deamplified sequentially.For QNDI+QNDI, the QND strength of one QNDI is manipulated by applying single-mode squeezing before and after the interface as discussed in Sec.II, such that the two QNDIs will have QND strengths with the same magnitude but opposite sign.Then, the two QNDIs can cancel each other.
For constructing SWAP, the overall reflection matrix should be engineered to null matrix.We list the explicit expression of the reflection matrix of all combinations on Table .IV.By straight forward calculation, we determine that except for BS+BS combination under the stringent condition, χ A = 1−χ B , it is impossible to engineer a null reflection matrix with any γ, ϵ, ϕ 1 and ϕ 2 .This stringent condition is nothing but the case that the two BS are complementary, i.e. |θ A | + |θ B | = π/2 + nπ for integer n, so that the combined interface will be a BS with resultant angle θ AB = |θ A | + |θ B |, which is essentially a SWAP.

Supplement for two-interface protocol of BS+BS and TMS+TMS
In Sec.III C 2, we discussed two special cases.First, when χ A = χ B , we are able to engineer QNDI and Iden-tity.To construct the QNDI, it is required to engineer a rank 1 transmission matrix.According to the results in Table III, we can verify that γ = tan ϕ 1 − sec ϕ 1 and ϕ 1 ̸ = 0 is a solution for engineering a QNDI from BS+BS and TMS+TMS combinations.For other combinations of γ and ϕ 1 , e.g. the protocol discussed in the beginning of Sec.III C 2, we will obtain a rank 0 transmission matrix and construct the Identity.
Second, when we are considering BS+BS combination and χ A = 1 − χ B , there are two possible resultant interfaces, sQNDI or SWAP.The requirement for engineering sQNDI is to have a rank 1 reflection matrix.According to the results in Table .IV, we can verify that a sQNDI is engineered when γ = − tan ϕ 1 − sec ϕ 1 .Any other combinations of γ and ϕ 1 gives SWAP as the resultant interface.

Any non-trivial interface except for QNDI
Our aim is to show that there always exists a set of single-mode controls to implement the conversion, i.e.L out 1 R out 2 TL in 1 R in 2 = ŪS 2 or S 2 Ū.We will show that this identity is satisfied for every block matrix, by illustrating how the suitable single-mode operations are constructed.First, we can always identify the required mode-2 rotations by the SVD of T 22 , i.e.Then, by the commutation relation between all the output quadratures, we determine that the output mode-1 quadratures must be the linear combination of (λ p  For QNDIs, their pre-or post-squeezing form has the non-diagonal mode-2 reflection matrix, so the required single-mode operations are generally different from that determined in the Sec.C 1. We first show that any QNDI can be transformed into the semi-quadrature-diagonal form, i.e.L out 1 R out 2 TL in 1 R in 2 = W, where Then, we show that W can always be converted into the pre-or post-squeezing form by suitable single-mode operations. For QNDI, T 22 can still be diagonalized according to Eq. (C1) with λ q 22 = (λ p 22 ) −1 = Λ.However, Eq. (C4) is not guaranteed, since T 21 is of rank 1.Instead, we consider [R out 2 ] 22 T 21 ≡ J, where J is a general rank-1 matrix and can always be expressed as To determine the mode-1 operation L in 1 for diagonalizing T 21 , we decompose J as following .We note that Λ AB(CD) has no bound and hence Λ ABCD can be tuned to arbitrary values.To summarize, these two alternative protocols requires at most four component interfaces, but they have limitations, both of them requires particular component interfaces.The first protocol considers the special case of the protocols in Sec.V C-V E, and the second protocol works when the components are BS, TMS or sTMS.If we have ability to choose the type of component interfaces in a platform, these alternative protocols are preferred since they requires fewer number of interfacing.It may reduce the implementation time and the chance of the operation error.
Possibility of engineering each class of interface with the six combinations of two component interfaces.For the resultant interface being BS/TMS/sTMS, it is always possible to engineer χ to any desired value.* It is generally not possible except the special case that the two component interfaces have the same strength, χA = χB.† It is generally not possible except the special case that the two component interfaces have complementary strengths, χA = 1 − χB.

1 FIG. 3 .
FIG. 3. Three-interface protocol for engineering Identity and SWAP.(a) The scheme involves cascading three component interfaces A, B, and C and suitable single-mode controls.(b) For engineering Identity, A and B are utilized to engineer an intermediate interface AB to be the inverse of C. (c) For constructing SWAP, AB is engineered to be the swappedinverse of C.

FIG. 4 .
FIG. 4. Two-interface module for engineering an intermediate interface under squeezing restriction.(a) Component interfaces A and B are first converted to respectively the pre-and post-squeezing forms, and then combined to form an intermediate interface AB with strength χAB by using the protocol in Sec.V A. (b) With suitable single-mode operations, interface AB can be transformed to the pre-squeezing (or postsqueezing) form for further processing.

FIG. 5 . 1 C
FIG. 5. Three-interface protocol for engineering SWAP under the squeezing restriction.(a) Component interface A is combined with B to form an intermediate interface AB with the transmission strength χAB = 1 − χC .(b) Such interface AB can be converted to a swapped-inverse of interface C, i.e.Ū−1 C ŪS , up to irreducible mode-2 operations, LAB.(c) After eliminating ŪC , the irreducible mode-2 operations, LAB and LC , can be removed by swapping operations from mode 1.Finally, the standard-form SWAP is formed.

orFIG. 6 .
FIG.6.Four-interface protocol to engineer an interface with χ ̸ = 0, 1 under squeezing restriction.(a) Each pair of two component interfaces are combined to form the intermediate interfaces AB and CD.They are then converted to preand post-squeezing form.(b) According to Eqs. (39)-(40), the controllable mode-1 rotations R α 1 and R β 1 effectively introduces controllable mode-2 rotations between the standardform transformation and the irreducible squeezing.(c) The equivalent circuit after the rearrangement.R2 is chosen to cancel all rotation between AB and CD, while mode-1 control follows the non-interfering setup in Fig.7.(d) The resultant interface ABCD.Its transmission strength χABCD can be engineered to the desired value according to Eq. (42), and the resultant irreducible squeezing ΛABCD, which is determined by ΛAB, ΛCD and the controllable Λa can be tuned to the target value according to Eq. (43) or(44).

FIG. 7 .
FIG.7.Illustrating the shutdown of transmission amplitude interference, i.e.Z = 0 for any γ.After passing through the intermediate interface AB, a Fourier gate F ≡ R(π/2) is applied on mode 1 to interchange the q-and p-quadratures.After passing through intermediate interface CD, each output quadrature will contain transmitted information from both input quadratures, each coming from an independent path.There is thus no interference, and the squeezing on mode 1 (triangle) does not alter the resultant transmission strength.

FIG. 9 .
FIG. 9. Five-interface protocol for engineering remote squeezing.(a) By using the four-interface protocol for engineering arbitrary non-trivial interface, component interfaces A, B, C, D can always be combined to become the inverse of the component E preceded with an extra mode-2 squeezing S(Λtgt).(b) Cascading these interfaces will result in a remote squeezing with strength Λtgt.

Finally, by considering the explicit expression of λ q 22 and λ p 22
where Σ ≡ diag(λ p 22 , λ q 22 ), D ≡ diag(−λ p 21 , −λ q 21 ) and V is a symplectic matrix representing the linear combination.Since the transformation V Σ, the first term in Eq. (C8), represents the mode-1 reflection, it should be equal to the diagonalized T 11 , i.e.V Σ = Ū11 .The second term in Eq. (C8) represents the mode-1 transmission, and henceV D = [L out 1 ] 12 T 12 [R in 2 ]12 .By using V Σ = Ū11 , we can determine V and then obtain V D = Ū11 −λ p 21 for each interface, for the pre-squeezing form, Eq. (C9) becomes 11 , (C14) another component is sQNDI.Then we can directly cascade them according to Fig.8(b).Similar for engineering a QNDI as the intermediate interface, it requires only 2 component interfaces if we can prepare two sQNDIs as components.The second alternative scheme requires four non-trivial component interfaces A, B, C and D that are neither QNDI nor sQNDI.The strategy is to engineer two intermediate interfaces AB and CD with the same χ such that AB and CD are inverse of each other up to the mode-2 operation.The explicit protocol is 1.Converting A (C) and B (D) into respectively the pre-and post-squeezing forms, same as the first step of the two-interface module in Sec.V A; 2. Combining interfaces A with B and C with D to form respectively the intermediate interface AB and CD with the same transmission strength, i.e. χ AB = χ CD ≡ χ int according toT AB(CD) ≡ S 2 (Λ B(D) ) Ūχ B(D) S 1 Ūχ A(C) S 2 (Λ A(C) ); (E1)3.Applying suitable single-mode controls on AB according to Eqs. (11)-(13), such that AB is engineered to be the inverse of CD with the specific mode-2 squeezing, i.e.T ′ AB = T −1 CD S 2 (Λ ABCD ), where Λ ABCD = Λ AB Λ CD ; 4. Tuning χ int such that the resultant remote squeezing Λ ABCD = Λ tgt .According to Eq. (E1), the irreducible squeezing of the intermediate interface is given byΛ AB(CD) = Λ A(C) Λ B(D) |1 − χ int | 2 − χ A(C) − χ B(D) − χ int + (χ int − X f,+ )(χ int − X f,− ) ,(E2) where X f,± ≡ ±2 χ A(C) χ B(D) (1 − χ A(C) )(1 − χ B(D) )+ χ A(C) + χ B(D) − 2χ A(C) χ B(D)