Quantum steering of Gaussian states via non-Gaussian measurements

Quantum steering—a strong correlation to be verified even when one party or its measuring device is fully untrusted—not only provides a profound insight into quantum physics but also offers a crucial basis for practical applications. For continuous-variable (CV) systems, Gaussian states among others have been extensively studied, however, mostly confined to Gaussian measurements. While the fulfilment of Gaussian criterion is sufficient to detect CV steering, whether it is also necessary for Gaussian states is a question of fundamental importance in many contexts. This critically questions the validity of characterizations established only under Gaussian measurements like the quantification of steering and the monogamy relations. Here, we introduce a formalism based on local uncertainty relations of non-Gaussian measurements, which is shown to manifest quantum steering of some Gaussian states that Gaussian criterion fails to detect. To this aim, we look into Gaussian states of practical relevance, i.e. two-mode squeezed states under a lossy and an amplifying Gaussian channel. Our finding significantly modifies the characteristics of Gaussian-state steering so far established such as monogamy relations and one-way steering under Gaussian measurements, thus opening a new direction for critical studies beyond Gaussian regime.

( A (n) j ) 2 = n1 1 n . On the other hand, the sum of squares of expectation values is given by where we use the Cauchy inequality to obtain the inequality in the fifth line. The positivity of the variance and 1 1 2 n = 1 1 n yield the last inequality, which in turn gives the uncertainty relation in equation (8) of main text. The equality holds when a given state is a pure state within the Fock space spanned by {|0⟩, ..., |n − 1⟩}.
(ii) Proof of Theorem: As shown in [1], if the set of observables satisfies uncertainty relations in a sum or product form as Eq. (8) of main text, the correlation of a bipartite quantum state described by the LHS models must satisfy a non-steerablity inequality in a form ∑ is an inferred variance [2] defined by is an estimate based on Bob's outcome B (n ′ ) j [1,2]. With a choice of linear estimate A ⟩ (g j is an arbitrary real number) [1,3] and setting g = g 1 = g 2 = ... = g n , we obtain the non-steerability criterion in equation (9) of main text.
(iii) Proof of Proposition: Similar to the analysis in [4], let us first choose g to make the left-hand side of equation (9) of main text as small as possible, i.e.
Plugging it to the inequality (9) of main text, we obtain the nonsteerability condition as Note that the left-hand side of Eq. (S5) has only the diagonal elements of correlation matrix. We may concentrate the correlation information onto the diagonal terms by taking a singular value decomposition of the correlation matrix C T LOOs nn ′ ( n 2 × n ′2 real matrix) using certain orthogonal matrices O A n and O B n ′ . Under this transformation, TLOOs remain TLOOs as already shown and the left-hand side of Eq. (S5) becomes a trace norm of the new correlation matrix. Using also the invariance of sum of squares of expectation values of TLOOs, the inequality (S5) corresponds to the inequality (10) of main text.

S2. Expectation values of observables in Fock space
In order to calculate the expectation values of Fock-basis observables for a two-mode Gaussian state, the multivariable Hermite polynomials are very useful [5]. The multivariable (4 variables in our case) Hermite polynomials are given by where ⃗ y = (y 1 , y 2 , y 3 , y 4 ) T . The matrix R is a 4 × 4 matrix related to the covariance matrix of the two-mode Gaussian state. With these Hermite polynomials, the matrix elements in Fock basis are given by [5] where R = BU Now let us consider a covariance matrix γ AB in a standard form In this case the R matrix in equation (S7) is given by . (S11) For the covariance matrix γ T M SV AB of a two-mode squeezed vacuum state (TMSV) with squeezing parameter r, the matrix R T M SV is given in a simple form If this TMSV goes through loss in mode B only, the matrix R T M SV LB is given by with η transmittance rate. We can similarly obtain the matrix R T M SV LG for the case of amplification.
Therefore, using the matrix elements in equation (S7), we can calculate the expectation values of any observables A to test our criterion in the main text.

S3. Singular value decomposition of correlation matrix
We here show that violation of the steering criterion inequality (10) of the main text is equivalent to the violation of equation (9) in the main text. In view of Proposition, let us assume that a given two-mode state ρ AB violates the inequality This violation means that there exists a set of TLOOs The sets , respectively [6].
Now, let us consider steering criterion in equation (9) of Theorem and equation (S4) with the observables in equation (S15). We can then derive the following where we used optimal g = − as in equation (S4) and The inequality in the third line is given by equation (S15)