Local transformations

The terminology and convention for Gaussian operations used in this work follows standard definitions in continuous-variable quantum information.31,46 To be self-contained, we explicitly list the relevant definitions below.

We use the definitions \(\hat q \equiv (\hat a + \hat a^\dagger )/\sqrt 2\), \(\hat p \equiv (\hat a - \hat a^\dagger )/\sqrt 2 i\), and \([\hat a,\hat a^\dagger ] = 1\). The operator and quadrature transformation matrix of a phase-space rotation are given by

$$\hat R(\theta ) \equiv \exp ( - i\theta \hat a^\dagger \hat a),\quad {\boldsymbol{R}}(\theta ) = \left( {\begin{array}{*{20}{c}} {\cos\theta } & {\sin\theta } \\ { - \sin\theta } & {\cos\theta } \end{array}} \right).$$ (15)

A special rotation operator is the parity operator, i.e., \(\widehat {\cal{P}} \equiv \hat R(\pi )\). A parity operator does not mix q and p-quadratures, but flips their sign, i.e.,

$$\widehat {\cal{P}}^\dagger \hat q\widehat {\cal{P}} = - \hat q,\quad \widehat {\cal{P}}^\dagger \hat p\widehat {\cal{P}} = - \hat p,\quad {\boldsymbol{P}} = \left( {\begin{array}{*{20}{c}} { - 1} & 0 \\ 0 & { - 1} \end{array}} \right).$$ (16)

The operator and quadrature transformation matrix of a single-mode squeezing operation are given by

$$\hat S(r) = \exp \left( { - \frac{r}{2}(\hat a^2 - \hat a^{\dagger 2})} \right),\quad {\boldsymbol{S}}(r) = \left( {\begin{array}{*{20}{c}} {e^r} & 0 \\ 0 & {e^{ - r}} \end{array}} \right).$$ (17)

Our scheme requires tuning interference by modifying scattering amplitudes associated with a particular path; to do this, we consider the generalized amplifying operator \(\hat G(\gamma ) \equiv \hat S(r)\widehat {\cal{P}}^n\) for n ∈ {0, 1}, such that

$$\hat G^\dagger (\gamma )\hat q\hat G(\gamma ) = \gamma \hat q,\quad \hat G^\dagger (\gamma )\hat p\hat G(\gamma ) = \frac{1}{\gamma }\hat p,$$ (18)

$${\boldsymbol{G}}(\gamma ) = \left( {\begin{array}{*{20}{c}} \gamma & 0 \\ 0 & {1/\gamma } \end{array}} \right),$$ (19)

where the amplification factor γ ≡ ±er can be any non-zero real number.

Generally, the transformation matrix of any single mode linear transformation \(\hat U\) must be symplectic to preserve the commutation relation, i.e.,

$${\boldsymbol{U}}{\bf{\Omega }}{\boldsymbol{U}}^{\mathrm{T}} = {\bf{\Omega }},\quad {\mathrm{where}}\,{\bf{\Omega }} \equiv \left( {\begin{array}{*{20}{c}} 0 & 1 \\ { - 1} & 0 \end{array}} \right).$$ (20)

Any single-mode symplectic transformation can be realized by combining squeezing and rotation operations.31 Given some 2 × 2 full-rank real matrix T, it is easy to check the following matrices are symplectic:

$${\boldsymbol{U}} = \sqrt {\det ({\boldsymbol{T}})} {\boldsymbol{T}}^{ - 1}$$ (21)

for det(T) > 0, and

$${\boldsymbol{U}} = \sqrt {|\det ({\boldsymbol{T}})|} {\boldsymbol{T}}^{ - 1}{\boldsymbol{Z}}$$ (22)

$${\mathrm{and}}\,{\boldsymbol{U}} = \sqrt {|\det ({\boldsymbol{T}})|} {\boldsymbol{ZT}}^{ - 1}$$ (23)

for det(T) < 0; Z is the Pauli Z matrix.

Quadrature-diagonalization and classification of two-mode linear transformation

In the main text, we discussed how any two-mode linear transformation can be classified by using the rank and determinant of its transmission and reflection matrices, leading to the five classes listed in Table 1; we also discussed the general procedure of quadrature diagonalization. Here, we provide further technical details on both these topics. Before doing so, we note that our classification scheme may superficially remind one of the classification of Gaussian channels.31,47 We stress though that our problem is fundamentallly different: our problem involves two-mode unitary transformations only, whereas in quantum channel theory, one is usually classifying single mode transmission in the presence of an environment (i.e., the transformation is not unitary).

The general procedure of quadrature-diagonalization discussed in the main text (c.f. Eq. (5)) involves first picking a local transformation which diagonalizes one of the two output modes, i.e.,

$$\begin{array}{*{20}{l}}\left( {\begin{array}{*{20}{c}} {\hat q_i^{{\mathrm{out}}}} \\ {\hat p_i^{{\mathrm{out}}}} \end{array}} \right) &=& {\boldsymbol{L}}_i^{{\mathrm{out}}}{\boldsymbol{T}}_{i1}{\boldsymbol{L}}_1^{{\mathrm{in}}}\left( {\begin{array}{*{20}{c}} {\hat q_1^{{\mathrm{in}}}} \\ {\hat p_1^{{\mathrm{in}}}} \end{array}} \right) + {\boldsymbol{L}}_i^{{\mathrm{out}}}{\boldsymbol{T}}_{i2}{\boldsymbol{L}}_2^{{\mathrm{in}}}\left( {\begin{array}{*{20}{c}} {\hat q_2^{{\mathrm{in}}}} \\ {\hat p_2^{{\mathrm{in}}}} \end{array}} \right) \\ &=& \left( {\begin{array}{*{20}{c}} {\Lambda _{i1}^{(q)}} & 0 \\ 0 & {\Lambda _{i1}^{(p)}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\hat q_1^{{\mathrm{in}}}} \\ {\hat p_1^{{\mathrm{in}}}} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {\Lambda _{i2}^{(q)}} & 0 \\ 0 & {\Lambda _{i2}^{(p)}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\hat q_2^{{\mathrm{in}}}} \\ {\hat p_2^{{\mathrm{in}}}} \end{array}} \right).\end{array}$$ (24)

The commutation relation \({\mathrm{[}}\hat q_i^{{\mathrm{out}}},\hat p_i^{{\mathrm{out}}}{\mathrm{]}} = i\) requires

$$\Lambda _{i1}^{(q)}\Lambda _{i1}^{(p)} + \Lambda _{i2}^{(q)}\Lambda _{i2}^{(p)} = 1.$$ (25)

Because local symplectic transformations do not change the determinant of transmission and reflection matrices, the above equation is equivalent to Eq. (4) by recognizing

$$\det ({\boldsymbol{T}}_{i1}) = \Lambda _{i1}^{(q)}\Lambda _{i1}^{(p)},\quad \det ({\boldsymbol{T}}_{i2}) = \Lambda _{i2}^{(q)}\Lambda _{i2}^{(p)}.$$ (26)

Consider next the output quadratures of the other mode \(\bar i

e i\). In order to commute with output mode i operators, they can only be linear combinations of the following operators:

$$\hat Q_ \bot \equiv \Lambda _{i2}^{(p)}\hat q_1^{{\mathrm{in}}} - \Lambda _{i1}^{(p)}\hat q_2^{{\mathrm{in}}},\quad \hat P_ \bot \equiv \Lambda _{i2}^{(q)}\hat p_1^{{\mathrm{in}}} - \Lambda _{i1}^{(q)}\hat p_2^{{\mathrm{in}}}.$$ (27)

Furthermore, because \(\hat Q_ \bot\) and \(\hat P_ \bot\) behaves as quadrature operators, i.e., \([\hat Q_ \bot ,\hat P_ \bot ] = i\), the linear combination has to be a symplectic transformation. Because \({\boldsymbol{L}}_{\bar i}^{{\mathrm{out}}}\) is not involved in diagonalizing mode i, output mode \(\bar i\) will have the general form

$$\begin{array}{*{20}{l}}\left( {\begin{array}{*{20}{c}} {\hat q_{\bar i}^{{\mathrm{out}}}} \\ {\hat p_{\bar i}^{{\mathrm{out}}}} \end{array}} \right) &=& {\boldsymbol{L}}_{\bar i}^{{\mathrm{out}}}{\boldsymbol{T}}_{\bar i1}{\boldsymbol{L}}_1^{{\mathrm{in}}}\left( {\begin{array}{*{20}{c}} {\hat q_1^{{\mathrm{in}}}} \\ {\hat p_1^{{\mathrm{in}}}} \end{array}} \right) + {\boldsymbol{L}}_{\bar i}^{{\mathrm{out}}}{\boldsymbol{T}}_{\bar i2}{\boldsymbol{L}}_2^{{\mathrm{in}}}\left( {\begin{array}{*{20}{c}} {\hat q_2^{{\mathrm{in}}}} \\ {\hat p_2^{{\mathrm{in}}}} \end{array}} \right) \\ &=& {\boldsymbol{L}}_{\bar i}^{{\mathrm{out}}}{\boldsymbol{U}}\left( {\begin{array}{*{20}{c}} {\Lambda _{i2}^{(p)}} & 0 \\ 0 & {\Lambda _{i2}^{(q)}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\hat q_1^{{\mathrm{in}}}} \\ {\hat p_1^{{\mathrm{in}}}} \end{array}} \right) + {\boldsymbol{L}}_{\bar i}^{{\mathrm{out}}}{\boldsymbol{U}}\left( {\begin{array}{*{20}{c}} { - \Lambda _{i1}^{(p)}} & 0 \\ 0 & { - \Lambda _{i1}^{(q)}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\hat q_2^{{\mathrm{in}}}} \\ {\hat p_2^{{\mathrm{in}}}} \end{array}} \right),\end{array}$$ (28)

where U is a symplectic matrix. Because symplectic matrices are full rank, one cannot change the rank of a matrix by multiplying by them. Therefore we have Eq. (3), i.e.,

$${\mathrm{rank}}({\boldsymbol{T}}_{\bar i1}) = {\mathrm{rank}}\left( {\begin{array}{*{20}{c}} {\Lambda _{i2}^{(p)}} & 0 \\ 0 & {\Lambda _{i2}^{(q)}} \end{array}} \right) = {\mathrm{rank}}({\boldsymbol{T}}_{i2}),$$ (29)

$${\mathrm{rank}}({\boldsymbol{T}}_{\bar i2}) = {\mathrm{rank}}\left( {\begin{array}{*{20}{c}} {\Lambda _{i1}^{(p)}} & 0 \\ 0 & {\Lambda _{i1}^{(q)}} \end{array}} \right) = {\mathrm{rank}}({\boldsymbol{T}}_{i1}).$$ (30)

For classification, the output mode \(\bar i\) in Eq. (28) can be diagonalized by picking \({\boldsymbol{L}}_j^{{\mathrm{out}}} = {\boldsymbol{U}}^{ - 1}\). In the case where \(\bar i = 2\), and where we assume the experimentally-relevant constraint that mode 2 cannot be squeezed, \({\boldsymbol{L}}_2^{{\mathrm{out}}}\) is limited to being a rotation matrix, so output mode 2 cannot be quadrature-diagonalised. Nevertheless, by using QL decomposition,48 we can express any symplectic matrix U as

$${\boldsymbol{U}} = {\boldsymbol{R}}_U{\boldsymbol{J}}_U{\mathrm{,}}$$ (31)

where R U is a rotation matrix, J U is lower triangular, i.e.,

$${\boldsymbol{J}}_U \equiv \left( {\begin{array}{*{20}{c}} \xi & 0 \\ \cdot & \cdot \end{array}} \right).$$ (32)

\(\hat q_2^{{\mathrm{out}}}\) will be diagonalised if we pick \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{R}}_U^{ - 1}\).

In Eq. (32), the non-vanishing upper left entry, ξ, is the only element of this matrix that is needed for our destructive interference scheme; the remaining entries (indicated with ⋅ above) play no role. Explicitly, our transduction scheme requires only the transformation matrix of the q-quadrature (i.e., Eq. (8)), which is sufficiently obtained from Eqs. (24), (28) and (32).

In the following, we will discuss how Eq. (24) can be constructed in each class given in Table 1, and how each class is convertible to well known two-mode operations. Generally, our strategy is to consider the SVD of the transmission and reflection matrices (c.f. Eq. (2)).

Class [[0, 2]]: Identity

In this class, T 21 = 0 is the null matrix. Therefore output mode 2 involves input mode 2 quadratures only. From Eq. (3), we have T 12 = 0, so output mode 1 involves input mode 1 quadratures only. As a result, class [[0, 2]] transformation is a tensor product of local transformation.

Any class [[0, 2]] transformation can be diagonalised by picking \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{T}}_{11}^{ - 1}\), \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{T}}_{22}^{ - 1}\), \({\boldsymbol{L}}_1^{{\mathrm{in}}}\) and \({\boldsymbol{L}}_2^{{\mathrm{in}}}\) as the identity. The diagonalized transformation gives the relation

$$\hat q_1^{{\mathrm{out}}} = \hat q_1^{{\mathrm{in}}},\quad \hat p_1^{{\mathrm{out}}} = \hat p_1^{{\mathrm{in}}} ,\quad \hat q_2^{{\mathrm{out}}} = \hat q_2^{{\mathrm{in}}} ,\quad \hat p_2^{{\mathrm{out}}} = \hat p_2^{{\mathrm{in}}},$$ (33)

which is an identity operation.

Class [[1, 2]]: QND

We start by diagonalising output mode 1. The SVD of T 12 (c.f. Eq. (2)) can be expressed as either

$${\boldsymbol{T}}_{12} = {\boldsymbol{V}}_{12}\left( {\begin{array}{*{20}{c}} \eta & 0 \\ 0 & 0 \end{array}} \right){\boldsymbol{W}}_{12},$$ (34)

$${\mathrm{or}}\,\quad{\boldsymbol{T}}_{12} = {\boldsymbol{V}}_{12}^\prime \left( {\begin{array}{*{20}{c}} 0 & 0 \\ 0 & \eta \end{array}} \right){\boldsymbol{W}}_{12}^\prime ,$$ (35)

where η ≠ 0; \({\boldsymbol{V}}_{12}^\prime \equiv {\boldsymbol{V}}_{12}{\bf{\Omega }}\) and \({\boldsymbol{W}}_{12}^\prime \equiv {\bf{\Omega }}^{\mathrm{T}}{\boldsymbol{W}}_{12}\); both V 12 , W 12 , \({\boldsymbol{V}}_{12}^\prime\), and \({\boldsymbol{W}}_{12}^\prime\) are rotation matrices.

The freedom in choosing these decompositions allows us to define whether the transmitted quadrature is \(\hat q_2^{{\mathrm{in}}}\) or \(\hat p_2^{{\mathrm{in}}}\). To diagonalize the transmission matrix T 12 , we can pick \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{V}}_{12}^{ - 1}\) and \({\boldsymbol{L}}_2^{{\mathrm{in}}} = {\boldsymbol{W}}_{12}^{ - 1}\) if we want \(\hat q_2^{{\mathrm{in}}}\) transmitted (\(\Lambda _{12}^{(q)} = \eta

e 0\) and \(\Lambda _{12}^{(p)} = 0\) in Eq. (24)), or pick \({\boldsymbol{L}}_1^{{\mathrm{out}}} = ({\boldsymbol{V}}_{12}^\prime )^{ - 1}\) and \({\boldsymbol{L}}_2^{{\mathrm{in}}} = ({\boldsymbol{W}}_{12}^\prime )^{ - 1}\) if we want \(\hat p_2^{{\mathrm{in}}}\) transmitted (\(\Lambda _{12}^{(q)} = 0\) and \(\Lambda _{12}^{(p)} = \eta

e 0\)). Because det(T 11 ) = 1 in this class, T 11 is diagonalized by choosing \({\boldsymbol{L}}_1^{{\mathrm{in}}} = ({\boldsymbol{L}}_1^{{\mathrm{out}}}{\boldsymbol{T}}_{11})^{ - 1}\). Output mode 1 is then diagonalized, and the transformation amplitudes are given by \(\Lambda _{11}^{(q)} = \Lambda _{11}^{(p)} = 1\).

For the purpose of characterization, the output mode 2 can be diagonalised by picking \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{PU}}^{ - 1}\) (note that the phase-flip is added just for clarity), where \({\boldsymbol{U}} = {\boldsymbol{T}}_{22}{\boldsymbol{L}}_2^{{\mathrm{in}}}{\boldsymbol{P}}\). The quadrature-diagonalised transformation is given by

$$\begin{array}{*{20}{l}}\hat q_1^{{\mathrm{out}}} = \hat q_1^{{\mathrm{in}}} + \eta \hat q_2^{{\mathrm{in}}} , \hat q_2^{{\mathrm{out}}} = \hat q_2^{{\mathrm{in}}},\\ \hat p_1^{{\mathrm{out}}} = \hat p_1^{{\mathrm{in}}} , \hat p_2^{{\mathrm{out}}} = - \eta \hat p_1^{{\mathrm{in}}} + \hat p_2^{{\mathrm{in}}},\end{array}$$ (36)

if we pick the decomposition in Eq. (34), or

$$\begin{array}{*{20}{l}}\hat q_1^{{\mathrm{out}}} = \hat q_1^{{\mathrm{in}}}, \hat q_2^{{\mathrm{out}}} = - \eta \hat q_1^{{\mathrm{in}}} + \hat q_2^{{\mathrm{in}}},\\ \hat p_1^{{\mathrm{out}}} = \hat p_1^{{\mathrm{in}}} + \eta \hat p_2^{{\mathrm{in}}}, \hat p_2^{{\mathrm{out}}} = \hat p_2^{{\mathrm{in}}},\end{array}$$ (37)

if we pick the decomposition in Eq. (35).

Equations (36) and (37) respectively represent the transformation of the QND gate \(\widehat {\cal{U}} = \exp ( - i\eta \hat p_1\hat q_2)\) and \(\widehat {\cal{U}} = \exp (i\eta \hat q_1\hat p_2)\). These two QND gates are equivalent by interchanging the q- and p-quadratures in both modes. It is thus clear that the decompositions in Eqs. (34) and (35) are equivalent up to simple local rotations.

We note that for any two-mode transformation in this class, the effective QND strength η is not invariant under local transformation. Explicitly, if we choose \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{G}}(\gamma ){\boldsymbol{V}}_{12}^{ - 1}\), the transformation still has the diagonal form of a QND gate, but the QND strength is modified as η → γη.

Recall that we are interested in situations where it is not possible to implement squeezing operations on mode 2. Both output q-quadratures can still be diagonalized by using the QL decomposition as discussed before. The q-quadrature transformation matrix Eq. (8) is

$${\boldsymbol{T}}^{qq} = \left( {\begin{array}{*{20}{c}} 1 & \eta \\ 0 & { - \xi } \end{array}} \right)\quad {\mathrm{or}}\quad {\boldsymbol{T}}^{qq} = \left( {\begin{array}{*{20}{c}} 1 & 0 \\ {\xi \eta } & { - \xi } \end{array}} \right),$$ (38)

if the choice of decomposition is Eq. (34) or (35) respectively. The freedom to choose between these Tqq guarantees that one can always arrange the amplitude of the transmission-transmission path in Eq. (9) to be non-vanishing.

Classes [[2, 2]]

Depending on the sign of determinant of the transmission matrix (and correspondingly that of reflection matrix according to Eq. (4)), different choice of local operation is required to diagonalize the transformation. For clarity, we separately discuss the three cases: 0 > det(T 21 ), 1 > det(T 21 ) > 0, and det(T 21 ) > 1. Interestingly, the equivalent well-known transformation in each case would be different. We note that all transformations in these three classes can be diagonalized while still obeying the practical constraint on mode 2, i.e., mode 2 cannot be squeezed. Besides, all elements in the q-quadrature transformation matrix Eq. (8) is non-vanishing in this class.

0 > det(T 21 ): TMS

Output mode 2 can be quadrature-diagonalized by taking \({\boldsymbol{L}}_2^{{\mathrm{in}}} = {\boldsymbol{W}}_{22}^{ - 1}\), \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{V}}_{22}^{ - 1}\), and \({\boldsymbol{L}}_1^{{\mathrm{in}}} = \sqrt {|\det ({\boldsymbol{T}}_{21})|} {\boldsymbol{Z}}({\boldsymbol{L}}_2^{{\mathrm{out}}}{\boldsymbol{T}}_{21})^{ - 1}\) The coefficients in the diagonalized transformation (c.f. Eq. (24)) are given by

$$\Lambda _{21}^{(q)} = - \Lambda _{21}^{(p)} = \sqrt {|\det ({\boldsymbol{T}}_{21})|} \equiv \sinh r,$$ (39)

$$\Lambda _{22}^{(q)} = D_{22}^{(q)},\quad \Lambda _{22}^{(p)} = D_{22}^{(p)},$$ (40)

where

$$\sqrt {D_{22}^{(q)}D_{22}^{(p)}} = \sqrt {\det ({\boldsymbol{T}}_{22})} \equiv \cosh r.$$ (41)

Next, output mode 1 can be diagonalised by picking \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{U}}^{ - 1}\), where \({\boldsymbol{U}} = ({\boldsymbol{T}}_{12}{\boldsymbol{L}}_2^{{\mathrm{in}}}{\boldsymbol{Z}})/\sinh r\). The q-quadrature transformation matrix is then

$${\boldsymbol{T}}^{qq} = \left( {\begin{array}{*{20}{c}} {D_{22}^{(p)}} & {\sinh r} \\ {\sinh r} & {D_{22}^{(q)}} \end{array}} \right).$$ (42)

For the purpose of characterization, we can pick the same local operations except \({\boldsymbol{L}}_2^{{\mathrm{in}}} = ({\boldsymbol{D}}_{22}{\boldsymbol{W}}_{22})^{ - 1}\). The fully diagonalized transformation becomes

$$\begin{array}{l}\hat q_1^{\mathrm{out}} = \cosh r\hat q_1^{\mathrm{in}} + \sinh r\hat q_2^{\mathrm{in}}, \quad\hat q_2^{\mathrm{out}} = \sinh r\hat q_1^{\mathrm{in}} + \cosh r\hat q_2^{\mathrm{in}},\\ \hat p_1^{\mathrm{out}} = \cosh r\hat p_1^{\mathrm{in}} - \sinh r\hat p_2^{\mathrm{in}}, \quad \hat p_2^{\mathrm{out}} = - \sinh r\hat p_1^{\mathrm{in}} + \cosh r\hat p_2^{\mathrm{in}}.\end{array}$$ (43)

This is the transformation of a two-mode-squeezing operation, i.e., \(\widehat {\cal{U}} = \exp \left( { - ir(\hat q_1\hat p_2 + \hat p_1\hat q_2)} \right)\). Here the TMS strength r is fixed by the determinant of transmission and reflection matrices, and cannot be altered by local operations.

1 > det(T 21 ) > 0: BS

Output mode 2 can be quadrature-diagonalised by picking \({\boldsymbol{L}}_2^{{\mathrm{in}}} = {\boldsymbol{W}}_{22}^{ - 1}\), \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{V}}_{22}^{ - 1}\), and \({\boldsymbol{L}}_1^{{\mathrm{in}}} = \sqrt {\det ({\boldsymbol{T}}_{21})} {\boldsymbol{P}}({\boldsymbol{L}}_2^{{\mathrm{out}}}{\boldsymbol{T}}_{21})^{ - 1}\). The coefficients in the diagonalized form (c.f. Eq. (24)) are then

$$\Lambda _{21}^{(q)} = \Lambda _{21}^{(p)} = - \sqrt {\det ({\boldsymbol{T}}_{21})} \equiv - \sin \theta ,$$ (44)

$$\Lambda _{22}^{(q)} = D_{22}^{(q)},\Lambda _{22}^{(p)} = D_{22}^{(p)}.$$ (45)

We note that

$$\sqrt {D_{22}^{(q)}D_{22}^{(p)}} = \sqrt {\det ({\boldsymbol{T}}_{22})} \equiv \cos \theta .$$ (46)

Output mode 1 can be diagonalized by picking \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{U}}^{ - 1}\), where \({\boldsymbol{U}} = ({\boldsymbol{T}}_{12}{\boldsymbol{L}}_2^{{\mathrm{in}}})/\sin \theta\). The q-quadrature transformation matrix is then

$${\boldsymbol{T}}^{qq} = \left( {\begin{array}{*{20}{c}} {D_{22}^{(p)}} & {\sin \theta } \\ { - \sin \theta } & {D_{22}^{(q)}} \end{array}} \right).$$ (47)

For the purpose of characterization, we can choose the same local transformations except \({\boldsymbol{L}}_2^{{\mathrm{in}}} = ({\boldsymbol{D}}_{22}{\boldsymbol{W}}_{22})^{ - 1}\). The final diagonalized quadrature transformation becomes

$$\begin{array}{l}\hat q_1^{{\mathrm{out}}} = \cos \theta \hat q_1^{{\mathrm{in}}} + \sin \theta \hat q_2^{{\mathrm{in}}},\quad \hat q_2^{{\mathrm{out}}} = - \sin \theta \hat q_1^{{\mathrm{in}}} + \cos \theta \hat q_2^{{\mathrm{in}}},\\ \hat p_1^{{\mathrm{out}}} = \cos \theta \hat p_1^{{\mathrm{in}}} + \sin \theta \hat p_2^{{\mathrm{in}}},\quad \hat p_2^{{\mathrm{out}}} = - \sin \theta \hat p_1^{{\mathrm{in}}} + \cos \theta \hat p_2^{{\mathrm{in}}}.\end{array}$$ (48)

This is the transformation of a BS, i.e., \(\widehat {\cal{U}} = \exp \left( {i\theta (\hat q_1\hat p_2 - \hat p_1\hat q_2)} \right)\). We note that the BS angle θ is fixed by the determinant of the transmission and reflection matrices, and thus cannot be altered by local operations.

We pause to note an interesting physical consequence of our discussion here and classification scheme: TMS and BS operations cannot be made equivalent using purely local operations, because their determinant of transmission matrix are different. At first glance, this is surprising, as it is well known that both these operations can produce two-mode squeezed vacuum states with appropriately prepared separable input states. In particular, one can either apply TMS on the vacuum state of two modes, or by locally squeezing the vacuum of two modes before sending them through a BS. Our result shows that this correspondence is not general, but only holds for a particular choice of input states.

det(T 21 ) > 1: swapped TMS

Output mode 2 can be quadrature-diagonalized by taking \({\boldsymbol{L}}_2^{{\mathrm{in}}} = {\boldsymbol{W}}_{22}^{ - 1}\), \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{V}}_{22}^{ - 1}\), and \({\boldsymbol{L}}_1^{{\mathrm{in}}} = \sqrt {\det ({\boldsymbol{T}}_{21})} ({\boldsymbol{L}}_2^{{\mathrm{out}}}{\boldsymbol{T}}_{21})^{ - 1}\) The diagonal form (c.f. Eq. (24)) is then determined by:

$$\Lambda _{21}^{(q)} = \Lambda _{21}^{(p)} = \sqrt {\det ({\boldsymbol{T}}_{21})} \equiv \cosh r,$$ (49)

$$\Lambda _{22}^{(q)} = D_{22}^{(q)},\Lambda _{22}^{(p)} = D_{22}^{(p)},$$ (50)

where

$$\sqrt { - D_{22}^{(q)}D_{22}^{(p)}} = \sqrt {|\det ({\boldsymbol{T}}_{22})|} \equiv \sinh r.$$ (51)

Next, output mode 1 can be diagonalised by picking \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{U}}^{ - 1}\), where \({\boldsymbol{U}} = ({\boldsymbol{T}}_{12}{\boldsymbol{L}}_2^{{\mathrm{in}}}{\boldsymbol{P}})/\cosh r\). The q-quadrature transformation matrix is then

$${\boldsymbol{T}}^{qq} = \left( {\begin{array}{*{20}{c}} { - D_{22}^{(p)}} & {\cosh r} \\ {\cosh r} & {D_{22}^{(q)}} \end{array}} \right).$$ (52)

For classification purpose, we take the same local operations except \({\boldsymbol{L}}_2^{{\mathrm{in}}} = ({\boldsymbol{ZD}}_{22}{\boldsymbol{W}}_{22})^{ - 1}\). The quadrature transformation is then

$$\begin{array}{*{20}{l}}\hat q_1^{{\mathrm{out}}} = \sinh r\hat q_1^{{\mathrm{in}}} + \cosh r\hat q_2^{{\mathrm{in}}}, \hat q_2^{{\mathrm{out}}} = \cosh r\hat q_1^{{\mathrm{in}}} + \sinh r\hat q_2^{{\mathrm{in}}},\\ \hat p_1^{{\mathrm{out}}} = - \sinh r\hat p_1^{{\mathrm{in}}} + \cosh r\hat p_2^{{\mathrm{in}}}, \hat p_2^{{\mathrm{out}}} = \cosh r\hat p_1^{{\mathrm{in}}} - \sinh r\hat p_2^{{\mathrm{in}}}.\end{array}$$ (53)

This transformation is equivalent to a composition of a SWAP and a TMS, i.e., \(\widehat {\cal{U}} = \exp \left( { - ir(\hat q_1\hat p_2 + \hat p_1\hat q_2)} \right)\widehat {\Bbb S}\). The TMS strength r is similarly fixed by the determinant of transmission and reflection matrices, and cannot be altered by local operations.

We refer this operation (product of SWAP and TMS) as a swapped TMS operation; it is a less discussed class of two-mode transformation. While both swapped TMS and TMS generate nonlocal excitations, they belong to different classes and so they cannot be made equivalent using local operations only. Further, although it involves a SWAP operation, a swapped TMS operation cannot be employed directly in transduction, because the unwanted TMS part of the transformation cannot be eliminated via local operations only.

Class [[2, 1]]: swapped QND

In this class, we start by diagonalizing output mode 2. We first consider the SVD of T 22 ,

$${\boldsymbol{T}}_{22} = {\boldsymbol{V}}_{22}\left( {\begin{array}{*{20}{c}} 0 & 0 \\ 0 & \eta \end{array}} \right){\boldsymbol{W}}_{22}.$$ (54)

T 22 is diagonalised by picking \({\boldsymbol{L}}_2^{{\mathrm{out}}} = {\boldsymbol{V}}_{22}^{ - 1}\) and \({\boldsymbol{L}}_2^{{\mathrm{in}}} = {\boldsymbol{W}}_{22}^{ - 1}\). Next, T 12 is diagonalised by picking \({\boldsymbol{L}}_1^{{\mathrm{in}}} = ({\boldsymbol{L}}_2^{{\mathrm{out}}}{\boldsymbol{T}}_{21})^{ - 1}\). These processes yield a diagonal form (c.f. Eq. (24)) determined by \(\Lambda _{21}^{(q)} = \Lambda _{21}^{(p)} = 1\), \(\Lambda _{22}^{(q)} = 0\), and \(\Lambda _{22}^{(q)} = \eta\).

The transformation of output mode 1 is given by Eq. (28), so it can be diagonalized by picking \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{PU}}^{ - 1}\). The diagonalized quadrature transformation is given by

$$\hat q_1^{{\mathrm{out}}} = - \eta \hat q_1^{{\mathrm{in}}} + \hat q_2^{{\mathrm{in}}},\quad \hat q_2^{{\mathrm{out}}} = \hat q_1^{{\mathrm{in}}},$$ (55)

$$\hat p_1^{{\mathrm{out}}} = \hat p_2^{{\mathrm{in}}},\quad \hat p_2^{{\mathrm{out}}} = \hat p_1^{{\mathrm{in}}} + \eta \hat p_2^{{\mathrm{in}}}.$$ (56)

As discussed in main text, this is the transformation of a sQND (c.f. Eq. (7)).

We note that the decomposition in Eq. (54) is not unique: we can choose a non-vanishing upper left entry by using \({\boldsymbol{L}}_2^{{\mathrm{out}}} = ({\bf{\Omega }}{\boldsymbol{V}}_{22})^{ - 1}\) and \({\boldsymbol{L}}_2^{{\mathrm{in}}} = ({\boldsymbol{W}}_{22}{\bf{\Omega }})^{ - 1}\). The subsequent procedure of diagonalisation is similar, and the diagonalised transformation remains a swapped QND gate except the q- and p-quadratures are interchanged.

Class [[2, 0]]: SWAP

In this class T 22 = T 11 = 0, so the output mode only contains quadratures from the opposite input mode, and hence the transformation is a SWAP up to local transformation. Explicitly, the transformation can be diagonalized by picking \({\boldsymbol{L}}_1^{{\mathrm{in}}} = {\boldsymbol{T}}_{21}^{ - 1}\), \({\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{T}}_{12}^{ - 1}\), while both \({\boldsymbol{L}}_2^{{\mathrm{in}}}\) and \({\boldsymbol{L}}_2^{{\mathrm{out}}}\) are taken to be the identity. We then have

$$\hat q_1^{{\mathrm{out}}} = \hat q_2^{{\mathrm{in}}},\quad \hat p_1^{{\mathrm{out}}} = \hat p_2^{{\mathrm{in}}},\quad \hat q_2^{{\mathrm{out}}} = \hat q_1^{{\mathrm{in}}},\quad \hat p_2^{{\mathrm{out}}} = \hat p_1^{{\mathrm{in}}},$$ (57)

which indicates a SWAP.

Complete transduction with two TMS

In the main text, we gave an example for how our interference-based scheme could implement perfect transduction by using two incomplete BS operations. Here we analyze a more counter-intuitive example: complete transduction by using two sequential TMS operations. TMS is usually viewed as a process that generates correlated excitations, and is thus not directly related to or suited for state transfer. Nonetheless, our scheme allows TMS operations to be exploited for perfect state transfer, as we now show.

For simplicity, we assume both TMS operations are identical, and that each transforms quadratures as given in Eq. (43). With the arrangement in Fig. 2, we can see that the amplitude associated with two consecutive q-quadrature reflections is \(\cosh ^2r\) (with r the squeezing strength of each TMS). Similarly, the amplitude associated with two consecutive transmissions is \(\gamma \sinh ^2r\). These two paths interfere destructively when \(\gamma = - \coth ^2r\). The overall transformation is then given by

$$\begin{array}{*{20}{l}}\hat q_1^{{\mathrm{out}}} = - \eta \coth ^2r\hat q_1^{{\mathrm{in}}} - \coth r\hat q_2^{{\mathrm{in}}}, \hat q_2^{{\mathrm{out}}} = - \coth r\hat q_1^{{\mathrm{in}}},\\ \hat p_1^{{\mathrm{out}}} = - \tanh r\hat p_2^{{\mathrm{in}}}, \hat p_2^{{\mathrm{out}}} = - \tanh r\hat p_1^{{\mathrm{in}}} + \eta \hat p_2^{{\mathrm{in}}},\end{array}$$ (58)

where \(\eta = 1 + \tanh ^2r.\)

If now appropriate local operations are applied before and after the concatenated TMS, i.e., \({\boldsymbol{L}}_1^{{\mathrm{in}}} = {\boldsymbol{L}}_1^{{\mathrm{out}}} = {\boldsymbol{G}}( - \tanh r)\), the overall transformation becomes the standard form of a sQND in Eq. (6). One-way transduction can be completed by injecting infinitely squeezed state or homodyne detection, as described in main text.

Imperfect measurement/squeezing and noise-tolerant bosonic codes

Here we discuss the detrimental effect on transduction when imperfect input squeezing and measurement is employed with sQND operation. Without loss of generality, we consider a transduction from mode 1 to 2. The two-mode input state is \(|\Psi ^{{\mathrm{in}}}\rangle \equiv |\psi _0\rangle |\psi _{{\mathrm{anc}}}\rangle\), where |ψ 0 〉 is the mode-1 input state that we wish to transfer, and |ψ anc 〉 is the auxiliary state that is prepared in mode 2. After sQND, the output state becomes

$$|\Psi ^{{\mathrm{out}}}\rangle \equiv \widehat {\cal{U}}_{{\mathrm{sQ}}}|\Psi ^{{\mathrm{in}}}\rangle = e^{i\eta \hat p_1\hat q_2}|\psi _{{\mathrm{anc}}}\rangle |\psi _0\rangle .$$ (59)

We first consider the case where input squeezing is used to mitigate the unwanted QND interaction. The ancilla state in this case would ideally be an infinite squeezed vacuum; we consider the more realistic case of a finitely squeezed vacuum, \(|\psi _{{\mathrm{anc}}}\rangle = \hat S(r)|{\mathrm{vac}}\rangle\). If we trace out mode 1 in the final output state, the resulting state of output mode 2 is given by

$$\rho ^{{\mathrm{out}}} = {\mathrm{Tr}}_{\mathrm{1}}\left\{ {|\Psi ^{{\mathrm{out}}}\rangle \langle \Psi ^{{\mathrm{out}}}|} \right\} = {\int} d p\frac{1}{{\sigma \sqrt \pi }}e^{ - \left( {\frac{p}{\sigma }} \right)^2}e^{ip\hat q_2}|\psi _0\rangle \langle \psi _0|e^{ - ip\hat q_2},$$ (60)

where σ = ηe−r. We thus see that the output state is a ensemble of displaced versions of the input state |ψ 0 〉, where the displacement is of the \(\hat p\) quadrature and follows a Gaussian distribution. The width of the Gaussian distribution depends on the degree of squeezing. It is easy to see that when squeezing is infinite, i.e., r → ∞, the Gaussian distribution becomes a Dirac delta function and so the transduction is perfect, i.e., ρout = |ψ 0 〉〈ψ 0 |.

For the complementary approach where homodyne detection plus feedforward is used to undo the unwanted QND interaction, we first study the ideal case where the measurement is perfect. For simplicity, we assume the auxiliary state is vacuum, i.e., |ψ anc 〉 = |vac〉. Homodyne measurement of output mode 1 will result in a measurement outcome p 1 ; this measurement result is then used to perform a displacement of the mode-2 p-quadrature. This result is a displacement of the mode-2 state that is conditioned on the measurement outcome. Averaging over possible measurement outcomes, the ensemble-averaged output state after measurement plus feedforward becomes

$$\begin{array}{c}\rho ^{{\mathrm{out}}} = {\int} d p_1{\mathrm{Tr}}_{\mathrm{1}}\left\{ {|p_1\rangle \langle p_1|e^{ - i\eta _Dp_1\hat q_2}|\Psi ^{{\mathrm{out}}}\rangle \langle \Psi ^{{\mathrm{out}}}|e^{i\eta _Dp_1\hat q_2}} \right\}\\ = {\mathrm{Tr}}_{\mathrm{1}}\left\{ {e^{ - i\eta _D\hat p_1\hat q_2}|\Psi ^{{\mathrm{out}}}\rangle \langle \Psi ^{{\mathrm{out}}}|e^{i\eta _D\hat p_1\hat q_2}} \right\}\\ = {\int} d p\frac{1}{{(\eta - \eta _D)\sqrt \pi }}e^{ - \left( {\frac{p}{{\eta - \eta _D}}} \right)^2}e^{ip\hat q_2}|\psi _0\rangle \langle \psi _0|e^{ - ip\hat q_2},\end{array}$$ (61)

where η D corresponds to strength of the feedforward operation (i.e., it is the proportionality constant between the measurement outcome and the applied displacement). In the second step above, we employed the fact that \(\hat p|p_1\rangle = p_1|p_1\rangle\).

The transduction is complete (i.e., perfect) when we choose η D = η. Physically it means the unwanted QND gate is exactly cancelled by the measurement plus feedforward operation; mathematically, in Eq. (61) the Gaussian distribution of displacements becomes a Dirac delta function at p = 0, so there is no displacement noise.

Having understood the perfect measurement case, we now turn to an imperfect (inefficient) measurement. This can be modeled by performing a BS operation between mode 1 and an environment mode before homodyne detection is done. For a detection inefficiency \(\epsilon\), the total output state before homodyne detection is

$$|\Psi ^{{\mathrm{HD}}}\rangle \equiv e^{i\eta (\sqrt {1 - \epsilon } \hat p_1 + \sqrt \epsilon \hat p_E)\hat q_2}|{\mathrm{vac}}\rangle |\psi _0\rangle |{\mathrm{vac}}_E\rangle ,$$ (62)

where the subscript E denotes the environment mode (associated with the other BS port); we assume vacuum noise for simplicity. After homodyne detection and the corresponding feedforward displacement, the output state in mode 2 becomes

$$\begin{array}{c}\rho ^{{\mathrm{out}}} = {\int} d p_1{\mathrm{Tr}}_{{\mathrm{1,}}E}\left\{ {|p_1\rangle \langle p_1|e^{ - i\eta _Dp_1\hat q_2}|\Psi ^{{\mathrm{HD}}}\rangle \langle \Psi ^{{\mathrm{HD}}}|e^{i\eta _Dp_1\hat q_2}} \right\}\\ = {\int} d p\frac{1}{{\sigma \sqrt \pi }}e^{ - \left( {\frac{p}{\sigma }} \right)^2}e^{ip\hat q_2}|\psi _0\rangle \langle \psi _0|e^{ - ip\hat q_2},\end{array}$$ (63)

where \(\sigma \equiv \sqrt {\eta ^2 + \eta _D^2 - 2\eta \eta _D\sqrt {1 - \epsilon } }\). The variance of random displacement is minimized when we choose \(\eta _D = \eta \sqrt {1 - \epsilon }\), which gives \(\sigma = \eta \sqrt \epsilon\).

As seen from Eqs. (60) and (63), both finite injected squeezing and inefficient homodyne detection induce the same detrimental effect: the transmitted state is corrupted by a displacement noise in the p-quadrature. One can now ask whether this kind of corrupted transduction is of any practical utility. The answer, not surprisingly, is dependent on the kind of state one is trying to transduce. Consider the very relevant example where we are interested in transferring a logical qubit state which is encoded in the initial mode-1 state. In this case, the fidelity of the transmitted logical qubit is closely related to the type of bosonic code employed and the decoding/recovery procedures.36

Here we give specific examples of how the choice of bosonic code can be exploited to better preserve the qubit state after an imperfect transduction. The mode-1 input state encodes a qubit, and thus has the general form:

$$|\psi _0\rangle = |\vartheta ,\varphi \rangle \equiv \cos \frac{\vartheta }{2}|0_L\rangle + e^{i\varphi }\sin \frac{\vartheta }{2}|1_L\rangle ,$$ (64)

where ϑ and φ parametrize the qubit information, and |0 L 〉, |1 L 〉 are the basis states in the chosen bosonic code. In view of the difficulties of QND measurement and full control in some bosonic platforms (e.g., spin ensembles and optical modes), here we focus on the passive error tolerance of the bosonic code without any decoding and recovery processes. For any pure-state encoding, i.e., |0 L 〉 and |1 L 〉 are pure states, the logical fidelity of the output state is then given by the physical fidelity between the input and output state, i.e.,

$$\begin{array}{*{20}{l}}{\cal{F}}(\vartheta ,\varphi ) &\equiv& \langle \psi _0|\rho ^{{\mathrm{out}}}(\vartheta ,\varphi )|\psi _0\rangle \\ &=& {\int} d p\frac{1}{{\sigma \sqrt \pi }}e^{ - \left( {\frac{p}{\sigma }} \right)^2}\left| {\langle \vartheta ,\varphi |e^{ip\hat q_2}|\vartheta ,\varphi \rangle } \right|^2.\end{array}$$ (65)

The performance of our qubit transduction can be characterized by the average qubit fidelity

$$\overline {\cal{F}} \equiv \frac{1}{{4\pi }}{\int_{0}^{2\pi }} {{\int_{0}^\pi} {\cal{F}} } (\vartheta ,\varphi )\sin \theta d\vartheta d\varphi .$$ (66)

For any encoding bases {|0 L 〉, |1 L 〉}, we find that the logical fidelity can be improved by simply using squeezed versions of the original encoding states \(\{ |0_S\rangle \equiv \hat S|0_L\rangle ,|1_S\rangle \equiv \hat S|1_L\rangle \}\), i.e., \(|\psi _0\rangle = \hat S|\vartheta ,\varphi \rangle\). The average fidelity of the squeezed encoding is readily computed as

$${\cal{F}}_S(\vartheta ,\varphi ) = {\int} d p\frac{1}{{\sigma e^r\sqrt \pi }}e^{ - \left( {\frac{p}{{\sigma e^r}}} \right)^2}\left| {\langle \vartheta ,\varphi |e^{ip\hat q_2}|\vartheta ,\varphi \rangle } \right|^2.$$ (67)

When comparing with Eq. (65), the squeezed encoding effectively modifies the variance of the random displacement distribution, i.e., σ → σer. The variance is reduced if r < 0, i.e., the squeezing \(\hat S\) is a p-quadrature amplification. The intuition behind this strategy is simple: for a fixed amount of p-quadrature noise, its relative significance would be reduced if the information encoded in p-quadrature is amplified. In Fig. 5, we show an explicit numerical result for the cat code,49 i.e.,

$$|0/1_L\rangle = \frac{1}{{{\cal{N}}_ \pm }}\left( {|i\alpha \rangle \pm | - i\alpha \rangle } \right),$$ (68)

where |±iα〉 is a coherent state with purely imaginary amplitude ±iα; \({\cal{N}}_ \pm \equiv \sqrt {2(1 \pm e^{ - 2|\alpha |^2})}\). We note that appropriately squeezing a bosonic state can also improve the tolerance of encoded quantum information against channel loss.37,38

Fig. 5 Error tolerance of bosonic codes. a Schematic of various single-mode bosonic codes of a logical qubit. Solid orange (green) eclipse denotes the pure-state basis states |+ L 〉 (|− L 〉) of each encoding. Dotted eclipses denote the basis state after random displacement in the p-quadrature. The shaded area for the mixed-state QSP encoding denotes the region within which a physical state can represent the mixed-state logical basis. b State-averaged logical qubit fidelity as a function of the variance σ of p-quadrature displacement noise. The cat code has displacement α = 2 (blue), and the squeezed cat code has α = 2 and er = 1/2 (red). c State-averaged logical qubit fidelity versus displacement noise for cat code state with α = 2 (blue), and for the case where the final state is treated as a QSP encoded qubit (red with dots). We note that these fidelities quantify the uncorrected noise tolerance of the bosonic codes, but not their ultimate performance under optimal decoding and recovery Full size image

Apart from choosing another encoding, the logical fidelity can also be improved by using noiseless subsystems (NS).39,40,41 The idea of NS is to encode a qubit information not by two states, but by any state in two subspaces. A consequence is that if an encoding state is corrupted by noise, the logical information is not lost if the corrupted state remains in the same subspace.

In bosonic systems, adapting NS allows a pure logical state to be represented by a mixed physical state.42 To illustrate how NS can enhance the noise tolerance, we consider a variant of the recently proposed quadrature-sign parity (QSP) encoding,43 which represents the logical computational basis by the parity of the bosonic state, and the qubit coherence by the sign of the quadrature of the bosonic wave function. It has been shown that the basis states of cat code lie within the encoding subspace of QSP encoding.

The intuition of improved noise tolerance could be understood from Fig. 5a. Under displacement fluctuation, each coherent state component of a cat state roughly remains in the same side of the phase space, i.e., its wavefunction has the same sign of p-quadrature. Therefore, although the encoding state is transformed by displacement noise, the logical information is retained. Qualitatively, we compute the QSP logical fidelity for a noisy cat code qubit as43

$$\begin{array}{*{20}{l}}{\cal{F}}_{\Bbb L}(\vartheta ,\varphi ) &=& \frac{1}{2}\left( {1 + \sin \theta \cos \phi \langle \hat X_M\rangle } \right.\\ &&\left. { + \sin \theta \sin \phi \langle \hat Y_M\rangle + \cos \theta \langle \hat Z_M\rangle } \right)\end{array}$$ (69)

where \(\langle \hat O\rangle \equiv {\mathrm{Tr}}\left\{ {\hat O\rho } \right\}\); the logical operators are

$$\hat X_M = {\int} d p\Theta (p)|p\rangle \langle p|;\quad \hat Z_M = e^{i\pi \hat a^\dagger \hat a};\quad \hat Y_M \equiv i\hat X_M\hat Z_M;$$ (70)

Θ(p) is the sign function. The average fidelity is still obtained by Eq. (66). As shown in Fig. 5c, QSP encoding can improve the logical fidelity of a noisy cat-code qubit.

In summary, even with finite injected squeezing and imperfect measurement, our scheme produces a very specific kind of noise (random displacement noise in one quadrature only); this allows us to develop noise-tolerant bosonic codes of qubit states. We note that the logical fidelity can be further improved by conducting active error-correction;36 this kind of single-quadrature noise also allows simpler error-correction protocols (c.f. Supplementary Information in ref. 9).

Perfect two-way transduction via interference

Here, we provide more details on the procedure presented in Eq. (14) of the main text, where interference is used to accomplish perfect transduction without any need for injected squeezing or a homodyne measurement. The first and the last swapped QND gates, \(\widehat {\cal{U}}_{{\mathrm{sQ}}}(\eta _1)\) and \(\widehat {\cal{U}}_{{\mathrm{sQ}}}(\eta _3)\) can be realized by the procedure described in main text. The second swapped QND gate, \(\widehat {\cal{U}}_{{\mathrm{sQ}}}^\dagger (\eta _2)\), can be realized by implementing \(\widehat {\cal{U}}_{{\mathrm{sQ}}}(\eta _2)\), and applying local rotations before and afterwards, i.e.,

$$\widehat {\cal{U}}_{{\mathrm{sQ}}}^\dagger (\eta _2) = \hat R_1( - \frac{\pi }{2})\hat R_2( - \frac{\pi }{2})\widehat {\cal{U}}_{{\mathrm{sQ}}}(\eta _2)\hat R_1(\frac{\pi }{2})\hat R_2(\frac{\pi }{2}),$$ (71)

where \(\hat R_l\) denotes the rotation at mode l.

In the circuit diagram Fig. 4, we have derived a circuit (Fig. 4b) that is equivalent to the original transformation in Eq. (14). Our strategy is to sandwich each QND gate by an amplification (\(\hat G\)) and a deamplification (\(\hat G^\dagger\)) of the same strength. This arrangement will modify the QND gate strength as45

$${\hat G_1}^{\dagger} (\gamma )\exp (i\eta {\hat p_1}\hat q_2){\hat G_1}(\gamma ) = \exp (i\frac{\eta }{\gamma }{\hat p_1}\hat q_2) = {\hat G_2}(\gamma )\exp (i\eta {\hat p_1}\hat q_2){\hat G_2}^{\dagger} (\gamma),$$ (72)

where the subscript of \(\hat G\) denotes the mode it is applying on.

The three QND gates can be merged to be a single QND gate with strength

$$\frac{1}{{\gamma _1}}\eta _3 + \frac{1}{{\gamma _1\gamma _2}}( - \eta _2) + \frac{1}{{\gamma _2}}\eta _1 = \frac{{\gamma _2\eta _3 - \eta _2 + \gamma _1\eta _1}}{{\gamma _1\gamma _2}}.$$ (73)

This strength vanishes if local amplification γ 1 and γ 2 are chosen to satisfy η 2 = γ 1 η 1 + γ 2 η 3 , which is always possible for any non-vanishing η 1 , η 2 , η 3 . This will eliminate the QND gate, and so the circuit in Fig. 4c is reduced to a perfect SWAP.

Noise in lossy beam splitter

We here present the analysis of the performance of our scheme when one of the mode is lossy. This situation can describe a practical application where a quantum state is transferred between a lossy quantum processor and an essentially lossless quantum memory. By directly integrating Eq. (13), the solution for the final-time mode operators is given by

$$\left( {\begin{array}{*{20}{c}} {\hat a_1(t)} \\ {\hat a_2(t)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\tilde T_{11}} & {\tilde T_{12}} \\ {\tilde T_{21}} & {\tilde T_{22}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\hat a_1(0)} \\ {\hat a_2(0)} \end{array}} \right) + \mathop {\sum}\limits_{k \ge 3} {\left( {\begin{array}{*{20}{c}} {\tilde T_{1k}\hat A_k^{{\mathrm{in}}}} \\ {\tilde T_{2k}\hat A_k^{{\mathrm{in}}}} \end{array}} \right)} .$$ (74)

The environmental noise is injected through the temporal bath modes:

$${\hat{A}}_k^{\mathrm{in}} \equiv \int_{0}^{t} {\mathcal{A}}_{k} (t^{\prime})\hat {B}^{\mathrm{in}}(t^{\prime})dt^{\prime},$$ (75)

where \(\hat B^{{\mathrm{in}}}(t)\) is the bath operator that satisfies \([\hat B^{{\mathrm{in}}}(t),\hat B^{{\mathrm{in}}\dagger }(t{\prime})] = \delta (t - t{\prime})\) and \([\hat B^{{\mathrm{in}}}(t),\hat B^{{\mathrm{in}}}(t{\prime})] = 0\). \({\cal{A}}_k(t)\) are functions of t that describe the temporal profile of the bath modes; they are orthogonal such that each temporal bath mode behaves as an independent bosonic mode, i.e., \([\hat A_k^{{\mathrm{in}}},{\mathrm{ }}\hat A_l^{{\mathrm{in}}\dagger }] = \delta _{kl}\) and \([\hat A_k^{{\mathrm{in}}},{\mathrm{ }}\hat A_l^{{\mathrm{in}}}] = 0\). We use k = 1, 2 to denote the system source and target modes as always, and use k≥3 to label additional temporal modes of the environment.

The transformation matrix of the system modes can be obtained analytically as

$$\left( {\begin{array}{*{20}{c}} {\tilde T_{11}} & {\tilde T_{12}} \\ {\tilde T_{21}} & {\tilde T_{22}} \end{array}} \right) = \frac{e^{ - \theta \sin \Gamma }}{\cos \Gamma }\left( {\begin{array}{*{20}{c}} {\cos ((\theta \cos \Gamma ) + \Gamma )} & {\sin (\theta \cos \Gamma )} \\ { - \sin (\theta \cos \Gamma )} & {\cos ((\theta \cos \Gamma ) - \Gamma )} \end{array}} \right).$$ (76)

The coherent and dissipative BS angles, θ and Γ, are respectively

$$\theta \equiv gt,\quad e^{i\Gamma } \equiv \sqrt {1 - \left( {\frac{\kappa }{{4g}}} \right)^2} + i\frac{\kappa }{{4g}}.$$ (77)

We note that the definition of coherent BS angle θ is the same as in the lossless case, i.e., when κ → 0.

Obtaining the system-bath transformation amplitudes, \(\tilde T_{ik}\), from direct integration would be tedious. Instead, we employ a general result which shows that any linear, dissipative two-mode transformation can be represented as a sub-system of a four-mode unitary transformation.50 Furthermore, because the our evolution is passively linear and purely dissipative (i.e., no gain), the system-bath transformation amplitudes can be uniquely determined from the transformation matrix of the system modes alone.

Consider the SVD of the system transformation matrix:

$$\widetilde {\boldsymbol{T}} \equiv \left( {\begin{array}{*{20}{c}} {\tilde T_{11}} & {\tilde T_{12}} \\ {\tilde T_{21}} & {\tilde T_{22}} \end{array}} \right) = \widetilde {\boldsymbol{U}}{\mathrm{ }}\left( {\begin{array}{*{20}{c}} {\lambda _1} & 0 \\ 0 & {\lambda _2} \end{array}} \right)\widetilde {\boldsymbol{W}},$$ (78)

where \(\widetilde {\boldsymbol{U}}\) and \(\widetilde {\boldsymbol{W}}\) are 2 × 2 matrices that represent lossless BS operations. We necessarily have λ 1 , λ 2 ≤ 1 because the system is purely dissipative (i.e., no gain). The evolution in Eq. (74) can be written as50

$$\begin{array}{*{20}{l}}\left( {\begin{array}{*{20}{c}} {\hat a_1(t)} \\ {\hat a_2(t)} \end{array}} \right) &=& \widetilde {\boldsymbol{U}}\left( {\begin{array}{*{20}{c}} {\lambda _1} & 0 \\ 0 & {\lambda _2} \end{array}} \right)\widetilde {\boldsymbol{W}}\left( {\begin{array}{*{20}{c}} {\hat a_1(0)} \\ {\hat a_2(0)} \end{array}} \right)\\ &&+ \widetilde {\boldsymbol{U}}\left( {\begin{array}{*{20}{c}} {\sqrt {1 - \lambda _1^2} } & 0 \\ 0 & {\sqrt {1 - \lambda _2^2} } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\hat a_3^{{\mathrm{in}}}} \\ {\hat a_4^{{\mathrm{in}}}} \end{array}} \right),\end{array}$$ (79)

where \(\hat a_3^{{\mathrm{in}}}\) and \(\hat a_4^{{\mathrm{in}}}\) are two orthogonal temporal bath modes; the system-bath transformation amplitudes \(\tilde T_{i3}\) and \(\tilde T_{i4}\) can be obtained from the second line. The physical meaning of this method is that if the temporal bath modes are defined appropriately, only two independent modes are needed to describe all the effects of the environment on the transduction. We note that the temporal profile of \(\hat a_3^{{\mathrm{in}}}\) and \(\hat a_4^{{\mathrm{in}}}\) is unrelated to our analysis if we assume a vacuum bath.

To execute our scheme, two lossy BS are implemented by evolving the system according to Eq. (13) twice, each with duration τ/2. In between the two BS, mode 1 is squeezed with strength γ. Because there are two lossy BS, the environmental noise can be described by at most four temporal bath modes, i.e., \(\hat a_3^{{\mathrm{in}}}\) and \(\hat a_4^{{\mathrm{in}}}\) (\(\hat a_5^{{\mathrm{in}}}\) and \(\hat a_6^{{\mathrm{in}}}\)) are responsible for the loss in the first (second) BS.

Because \(\widetilde {\boldsymbol{T}}\) in Eq. (76) is real, both the singular values (λ 1 and λ 2 ) and the matrices \(\widetilde {\boldsymbol{U}}\) and \(\widetilde {\boldsymbol{W}}\) are real. If we decompose the mode operators into real and imaginary parts (quadratures), i.e., \(\hat a_k = (\hat q_k + i\hat p_k)/\sqrt 2\), they do not mix with each other. In other words, each lossy BS is quadrature-diagonal for both system and bath modes. Furthermore, the mode-1 squeezing at t = τ/2 is also quadrature-diagonal. As a result, the overall transformation (from t = 0 to τ) is quadrature-diagonal, i.e.,

$$\hat x_i^{{\mathrm{out}}} = T_{i1}^{xx}\hat x_1^{{\mathrm{in}}} + T_{i2}^{xx}\hat x_2^{{\mathrm{in}}} + \mathop {\sum}\limits_{k = 3}^6 {T_{ik}^{xx}} \hat x_k^{{\mathrm{in}}}$$ (80)

where i ∈ {1, 2} and x ∈ {q, p}; we have extended the T matrix in Eq. (1) to include temporal bath modes.

The transformation amplitudes are given by

$$T_{ik}^{qq} = \gamma \tilde T_{i1}\tilde T_{1k} + \tilde T_{i2}\tilde T_{2k}$$ (81)

$$T_{ik}^{pp} = \frac{1}{\gamma }\tilde T_{i1}\tilde T_{1k} + \tilde T_{i2}\tilde T_{2k}$$ (82)

for i = 1, 2 and k = 1, 2, 3, 4, and

$$T_{i5}^{qq} = T_{i5}^{pp} = \tilde T_{i3},T_{i6}^{qq} = T_{i6}^{pp} = \tilde T_{i4}.$$ (83)

For quantum memory write-in (one-way transduction from mode 1 to 2), our scheme starts by impedance-matching one quadrature by destructive interference. We pick this to be the q-quadrature without loss of generality; this requires \(T_{22}^{qq} = 0\). This can be achieved by choosing the mode-1 squeezing strength as \(\gamma = - (\tilde T_{22})^2/\tilde T_{21}\tilde T_{12}\).

To complete the transduction, the p-quadrature reflection noise should also be suppressed. We have discussed a strategy to first measure the p-quadrature of mode-1’s output state, and then perform a conditional displacement on mode 2. When there is no environmental noise, we have discussed that the conditional displacement strength η D should be the same as the unwanted QND strength, i.e., η D = η.

On the other hand, in the lossy case the measurement outcome p 1 contains both the reflected quadrature noise and environmental noise. Here the choice of η D is more subtle, because a full suppression of reflection noise might conversely enhance the environmental noise. Therefore an optimal choice of η D should minimize the sum of these two noises. We note that the conditional displacement is still assumed to be linearly proportional to the measurement outcome for simplicity; we leave it as an open question whether it is advantageous to use a more complicated non-linear dependence of the feedforward displacement on measurement outcome.

To obtain the optimal η D , we first consider the total output state \(\rho _{{\mathrm{tot}}}^{{\mathrm{out}}}\) of all system and bath modes after the overall transformation (80). After homodyne detection and conditional displacement, the output state at mode 2 is

$$\begin{array}{*{20}{l}}\rho _2^{{\mathrm{out}}} &=& {\int} d p_1{\mathrm{Tr}}_{\backslash 2}\left\{ {e^{ - i\eta _Dp_1\hat q_2}\langle p_1|\rho _{{\mathrm{tot}}}^{{\mathrm{out}}}|p_1\rangle e^{i\eta _Dp_1\hat q_2}} \right\}\\ &=& {\mathrm{Tr}}_{\backslash 2}\left\{ {e^{ - i\eta _D\hat p_1\hat q_2}\rho _{{\mathrm{tot}}}^{{\mathrm{out}}}e^{i\eta _D\hat p_1\hat q_2}} \right\}.\end{array}$$ (84)

In the second step, we have employed the identity

$$\begin{array}{*{20}{l}}|p_1\rangle \langle p_1| \otimes e^{ \pm i\eta _Dp_1\hat q_2} &=& e^{ \pm i\eta _D\hat p_1\hat q_2}\left( {|p_1\rangle \langle p_1| \otimes \widehat {\Bbb I}_2} \right)\\ &=& \left( {|p_1\rangle \langle p_1| \otimes \widehat {\Bbb I}_2} \right)e^{ \pm i\eta _D\hat p_1\hat q_2},\end{array}$$ (85)

where \(\widehat {\Bbb I}_2\) is the identity of mode 2. The physical intuition is that applying a conditional displacement after a homodyne detection is equivalent to applying a QND gate before the homodyne detection.

As such, the result of homodyne detection and post-processing can be accounted for by simply considering an effective transformation, which an extra QND gate is applied on the output modes. The effective quadrature transformation remains in the form of Eq. (80), i.e.,

$$\hat x_2^{{\mathrm{out}}} = {\cal{T}}_{21}^{xx}\hat x_1^{{\mathrm{in}}} + {\cal{T}}_{22}^{xx}\hat x_2^{{\mathrm{in}}} + \mathop {\sum}\limits_{k = 3}^6 {{\cal{T}}_{2k}^{xx}} \hat x_k^{{\mathrm{in}}},$$ (86)

but the transformation amplitudes are modified as

$${\cal{T}}_{2k}^{qq} = T_{2k}^{qq},{\cal{T}}_{2k}^{pp} = T_{2k}^{pp} - \eta _DT_{1k}^{pp}.$$ (87)

We note that the output mode 1 can be neglected in this effective transformation (Eq. (84)).

For the standard approach (where one just evolves once under the BS Hamiltonian for a fixed time), the effective transformation is also given by Eq. (86), except the parameters are replaced by γ = 1 (i.e., no amplification at t = τ/2) and η D = 0 (i.e., no measurement and post-processing).

We quantify the performance of the transducer by the total added noise (in units of quanta) in both quadratures,

$$\overline {\cal{N}} \equiv {\cal{N}}_q + {\cal{N}}_p$$ (88)

where the added noise in x-quadrature is34

$${\cal{N}}_x \equiv \frac{1}{2}\mathop {\sum}\limits_{k \ge 2}^6 {\frac{{({\cal{T}}_{2k}^{xx})^2\left\langle {(\hat x_k^{in})^2} \right\rangle }}{{({\cal{T}}_{21}^{xx})^2}}} ,$$ (89)

for x ∈ {q, p}. With this metric, the optimal displacement strength \(\eta _D^{{\mathrm{opt}}}\) should be that minimizes \(\overline {\cal{N}}\), i.e.,

$$\frac{{\partial {\cal{N}}_p}}{{\partial \eta _D}}|_{\eta _D \to \eta _D^{{\mathrm{opt}}}} = 0.$$ (90)

We note that \({\cal{N}}_q\) does not depend on η D .

The total added noise in Eq. (88) is not invariant under local transformation of the input state. Specifically, because of the local squeezing and the effective QND gate, the added noise is generally different for the q and p-quadratures. If the input state is amplified in the more noisy quadrature, the added noise could be reduced.34 Generally, for an initial phase sensitive amplification step that transforms the input mode as \(\hat q_1^{{\mathrm{in}}} \to \gamma _0\hat q_1^{{\mathrm{in}}}\) and \(\hat p_1^{{\mathrm{in}}} \to \frac{1}{{\gamma _0}}\hat p_{\mathrm{1}}^{{\mathrm{in}}}\), the added noise is modified as

$$\overline {\cal{N}} \to \frac{1}{{\gamma _0^2}}{\cal{N}}_q + \gamma _0^2{\cal{N}}_p.$$ (91)

It is easy to find that \(\overline {\cal{N}}\) is the minimum when the initial amplification is chosen as \(\gamma _0^2 = \sqrt {{\cal{N}}_q/{\cal{N}}_p}\), then

$$\overline {\cal{N}} _{\min } = 2\sqrt {{\cal{N}}_q{\cal{N}}_p} .$$ (92)

We note that the definition of added noise in Eq. (89), which follows that in34 for quantifying noise of linear amplifiers, is given by taking the total fluctuations at the output that did not originate at the input, and dividing by the transmission probability. In amplifier terminology, this corresponds to referring the output fluctuations back to the input. We stress that reflected noise at the input contributes to this added noise. In the case of Fig. 3, because the BS interaction is applied for only τ 0 /10, the transmission amplitude is small. Therefore, although the reflected input field is in vacuum, its contribution is still large on the scale of the weakly transmitted signal. This translates to an added noise of ≈20 quanta.

Apart from added noise, the performance of a one-way transducer can also be quantified by its effective quantum channel capacity.31 This metric is particularly useful when the transducer is applied in quantum communication, e.g., quantum memory transfer inside a quantum repeater. For simplicity, we compute the quantum capacity, \({\cal{Q}}\), with a single use of channel and transmitting pure Gaussian state;51,52 this quantity is a lower bound of the general quantum channel capacity. \({\cal{Q}}\) can be calculated from the channel transmissivity,47 τ C , and the noise number n C ,

$${\cal{Q}} = \max \left\{ {0,\log |\frac{{\tau _{\mathrm{C}}}}{{1 - \tau _{\mathrm{C}}}}| - G(n_{\mathrm{C}})} \right\},$$ (93)

where

$$G(n_{\mathrm{C}}) \equiv (n_{\mathrm{C}} + 1)\log (n_{\mathrm{C}} + 1) - n_{\mathrm{C}}\log (n_{\mathrm{C}}).$$ (94)

For a one-way transduction from mode 1 to 2 that transforms the quadratures as in Eq. (80) (effectively as in Eq. (86) because of measurement and feedforward), the effective channel transmissivity and noise number are given by

$$\tau _{\mathrm{C}} = {\cal{T}}_{21}^{qq}{\cal{T}}_{21}^{pp},$$ (95)

$$n_{\mathrm{C}} = \frac{{\sqrt {\left( {\mathop {\sum}\limits_{k = 2}^{6} {({\cal{T}}_{2k}^{qq})^2} \langle (\hat q_k^{\mathrm{in}})^{2}\rangle } \right)\left( {\mathop {\sum}\limits_{k = 2}^{6} {({\cal{T}}_{2k}^{pp})^2} \langle (\hat p_k^{\mathrm{in}})^{2}\rangle } \right)} }}{{|1 - \tau _{\mathrm{C}}|}} - \frac{1}{2}.$$ (96)

We note that when τ C ≤ 1/2, the channel is anti-degradable,31 and Eq. (93) gives the exact channel capacity, i.e., \({\cal{Q}} = 0\).

We note that although the above analysis focuses on the write-in process (transduction from mode 1 to 2), the readout process (transduction from mode 2 to 1) can be studied similarly after making two changes. First, instead of destructively interfering the q 2 reflection (i.e., \(T_{22}^{qq} = 0\)), the local squeezing between two BS should be adjusted to destructively interfere the p 1 reflection, i.e., \(T_{11}^{pp} = 0\). Both conditions are simultaneously satisfied when there is no loss, which is a property of sQND. However, this is not generally true in the lossy case. Second, instead of using homodyne detection to remove unwanted quadrature noise, infinitely squeezed vacuum is injected into input mode 1. As usual, this is because we have assumed that mode 2 is less controllable than mode 1.

Without measurement and post-processing, it is not necessary to construct an effective transformation, and so the transduction is fully characterized by Eq. (80). The x-quadrature added noise in the readout scheme is given by

$${\cal{N}}_x \equiv \frac{1}{2}\mathop {\sum}\limits_{k \ge 3}^6 {\frac{{(T_{1k}^{xx})^2\langle (\hat x_k^{{\mathrm{in}}})^2\rangle }}{{(T_{12}^{xx})^2}}} .$$ (97)

We note that \((T_{11}^{qq})^2\langle (\hat q_1^{{\mathrm{in}}})^2\rangle = 0\) due to the injected infinite squeezing. The channel capacity can be computed by the effective channel parameters:

$$\tau _{\mathrm{C}} = T_{12}^{qq}T_{12}^{pp},$$ (98)

$$n_{\mathrm{C}} = \frac{{\sqrt {\left( {\mathop {\sum}\limits_{k = 3}^{6} {(T_{1k}^{qq})^2} \langle (\hat q_k^{\mathrm{in}})^{2}\rangle } \right)\left( {\mathop {\sum}\limits_{k = 3}^{6} {(T_{1k}^{pp})^2} \langle (\hat p_k^{\mathrm{in}})^{2}\rangle } \right)} }}{{|1 - \tau _{\mathrm{C}}|}} - \frac{1}{2} .$$ (99)

We have computed, but not shown, that the performance of readout is similar to that of write-in in Fig. 2.