Stabilizing Preparation of Quantum Gaussian States via Continuous Measurement

We start from reviewing the Wigner phase-space representation [Paris, Illuminati, Serafini, & De Siena, 2003, Wiseman & Milburn, 2010], which is one of the most commonly used representations for quantum Gaussian states.

Denote $\hat{a}_{k}$ and $\hat{a}^{\dagger}_{k}$, $k=1,...,m$, as the canonical bosonic annihilation and creation operators of the $k$-th mode, respectively. The corresponding canonical quadrature operators $\hat{q}_{k}$ and $\hat{p}_{k}$ are defined via

which satisfy the canonical commutation relation

where $i=\sqrt{-1}$ and $\delta_{kj}$ is the Kronecker delta function. Denote the vector of quadrature operators for the $m$ mode system as

where $\top$ is the matrix transpose. The operator vector $\hat{\textbf{X}}$ satisfies the canonical commutation relations [Koga & Yamamoto, 2012, Yamamoto, 2012, Ma, Woolley, Petersen, & Yamamoto, 2014]

where $J=\begin{pmatrix}0&I_{m}\\ -I_{m}&0\end{pmatrix}$, and $I_{m}$ is the $m\times m$ identity matrix.

The joint displacement operator can be described in terms of $\hat{\textbf{X}}$ and $J$ as

where $\bm{\alpha}\in\mathbb{R}^{2m}$ [Paris, Illuminati, Serafini, & De Siena, 2003, Wiseman & Milburn, 2010]. On the basis of $\hat{D}(\bm{\alpha})$, for an arbitrary quantum state operator $\hat{\rho}$, its Wigner characteristic function is defined as

Now the Wigner function $W_{\hat{\rho}}(X)$ of the state $\hat{\rho}$ can be described as

where $X=(q_{1},...,q_{m},p_{1},\cdots,p_{m})^{\top}$ with $q_{i}$ and $p_{j}$ being real numbers [Paris, Illuminati, Serafini, & De Siena, 2003, Wiseman & Milburn, 2010].

A state is called Gaussian if its Wigner function is in the form of

where $\bar{X}=\langle\hat{\textbf{X}}\rangle=\textrm{Tr}(\hat{\textbf{X}}\hat{\rho})$ is the mean value vector whose element $\bar{X}_{i}=\langle\hat{\textbf{X}}_{i}\rangle=\textrm{Tr}(\hat{\textbf{X}}_{i}\hat{\rho})$, $V$ is the covariance matrix whose element $V_{ij}=\frac{1}{2}\langle\Delta\hat{\textbf{X}}_{i}\Delta\hat{\textbf{X}}_{j}+\Delta\hat{\textbf{X}}_{j}\Delta\hat{\textbf{X}}_{i}\rangle$ with $\Delta\hat{\textbf{X}}_{i}=\hat{\textbf{X}}_{i}-\langle\hat{\textbf{X}}_{i}\rangle$, and $\textrm{det}[V]$ denotes the determinant of matrix $V$ [Edwards & Belavkin, 2005, Wiseman & Milburn, 2010]. In addition to the positive semidefinite condition $V\geq 0$, $V$ should satisfy the Heisenberg uncertainty principle [Koga & Yamamoto, 2012, Yamamoto, 2012, Ma, Woolley, Petersen, & Yamamoto, 2014]

It is straightforward to see that a Gaussian state is determined by the mean vector $\bar{X}$ and the covariance matrix $V$.

In this paper, we only focus on the covariance matrix $V$. This is because, on the one hand, that one can always adjust the mean of a Gaussian state to any target mean by a suitable Weyl operator [Parthasarathy, 2010]. On the other hand it is noted that most of the useful properties of Gaussian states such as the purity, entanglement and squeezing are only related with the covariance matrix in practice. To be specific, the purity of a $m$ mode Gaussian state depends only on the covariance matrix as $\textrm{Tr}[\hat{\rho}^{2}]=\frac{1}{2^{m}\sqrt{\textrm{det}[V]}}$ [Paris, Illuminati, Serafini, & De Siena, 2003]. Moreover, for a two-mode pure Gaussian state, its $4\times 4$ covariance matrix contains all the necessary information to determine its entanglement properties for both entanglement criteria and entanglement measures [Rendell & Rajagopal, 2005, Marian & Marian, 2008]. In addition, squeezed coherent states have been extensively studied in quantum information, such as quantum teleportation and ghost imaging. The squeezing properties are also determined by the covariance matrix [Petersen, Madsen, & Mølmer, 2005].

In previous results, the commonly used model in preparing a target Gaussian state is to engineer a dissipative system described by the quantum master equation [Edwards & Belavkin, 2005, Wiseman & Milburn, 2010, Koga & Yamamoto, 2012, Yamamoto, 2012]

where $\hat{\rho}_{t}$ is the density operator, $\hat{H}$ is the effective Hamiltonian of the system and $\hat{L}_{i},~{}i=1,\ldots,m$, are the dissipative channels which describe the interaction between the system and the environment. The dissipative QME (2) can be viewed as an average (unconditional) evolution of the quantum state under the assumption that all the measurement results are discarded. Intuitively if one can utilize the measurement information, the corresponding conditional covariance matrix may be reduced, and thus the useful properties of the Gaussian states can be improved. In this paper, we present how to prepare a desired quantum Gaussian state by utilizing continuous measurement. To account for the randomness of quantum measurement, the evolution of the open quantum system is described in terms of the quantum stochastic master equation.

To proceed, we specify the following notation for clarity. For a matrix $Q=(q_{ij})$, let $Q^{\dagger}$, $Q^{\top}$ and $Q^{*}$ represent the conjugate transpose, transpose and conjugate for all the elements of $Q$, respectively. For example, $Q^{\dagger}=(q^{*}_{ji}),~{}Q^{\top}=(q_{ji}),$ and $Q^{*}=(q^{*}_{ij})=(Q^{\dagger})^{\top}$.

The dynamics of the open system coupled with $m$ independent measurement channels $\{\hat{L}_{i}\}^{m}_{i=1}$ can be described by a family of unitary operators $\{\hat{U}_{t}\}_{t\in\mathbb{R_{+}}}$, which satisfy

where $\hat{K}=i\hat{H}+\frac{1}{2}\sum^{m}_{i=1}\hat{L}^{\dagger}_{i}\hat{L}_{i}$ [Edwards & Belavkin, 2005]. Here we have assumed $\hbar=1$. The differential of the annihilation operators of the bath $d\hat{A}_{i,t}$ obeys the quantum It$\hat{\textrm{o}}$ rules:

For any system operator $\hat{X}$, its time evolution in the Heisenberg picture is represented by

From (3) and the quantum It$\hat{\textrm{o}}$ rules, we have

where

Consider $m$ family of measurement operators $\{Y_{i,t}\}_{t\in\mathbb{R_{+}}}$, where

for $i=1,\cdots,m$. From (3) and the quantum It$\hat{\textrm{o}}$ rules, we have

Once $\hat{Y}_{i,t}=\hat{U}^{\dagger}_{t}(\hat{A}_{i,t}+\hat{A}^{\dagger}_{i,t})\hat{U}_{t}$, $i=1,\cdots,m$, are observed, one can associate a conditional expectation to each observable $\hat{X}$ of the system. The conditional expectation gives the least squares estimator of $j_{t}(\hat{X})$ as $\mathcal{E}[j_{t}(\hat{X})|\hat{Y}^{t}_{0}]$, where $\hat{Y}^{t}_{0}$ is the algebra generated by $\hat{Y}_{i,s\leq t}$, $i=1,\cdots,m$. Since $\hat{Y}^{t}_{0}$ is nondemolition, the observation process is in essence equivalent to $m$ classical stochastic processes

Thus $\mathcal{E}[j_{t}(\hat{X})|\hat{Y}^{t}_{0}]$ actually represents the expectation of $\hat{X}$ conditioned on the classical stochastic observations $\{y_{i,s\leq t}$, $i=1,\cdots,m\}$. This conditional expectation can be conveniently written in the Schrödinger picture as

with $\hat{\rho}^{c}_{t}$ denoting the conditional system state at time $t$. The stochastic master equation of $\rho^{c}_{t}$ is described as [van Handel, Stockton, & Mabuchi, 2005, Edwards & Belavkin, 2005, Wiseman & Milburn, 2010]

where the innovations

are Gaussian, and

It is clear that by taking the expectation of (6), we can obtain the quantum master equation (2).

We are interested in systems of $m$ degrees of freedom with the phase space operator vector

as defined in subsection 2.1. Recall that the operator vector $\hat{\textbf{X}}$ satisfies the canonical commutation relation

The system is coupled to $m$ measurement channels via the operator vector $\hat{\textbf{L}}=\Lambda\hat{\textbf{X}}$, where $\Lambda\in\mathbb{C}^{m\times 2m}$. The Hamiltonian which is quadratic in $\hat{\textbf{X}}$ is represented as

where $G=G^{\top}\in\mathbb{R}^{2m\times 2m}$, $K\in\mathbb{C}^{2m\times m}$ and $u_{t}\in\mathbb{R}^{m}$ is the control. For such a system, from (3)-(5), we have the quantum linear equations in the Heisenberg picture as

where

and the quantum noise increments are given by

For the quantum linear quadratic system, if the initial state $\hat{\rho}$ of the system is Gaussian, then under the nondemolition measurement of the output operators $\hat{\textbf{Y}}_{t}$, the conditional dynamics (6) is also Gaussian. Thus for the linear quadratic Gaussian system, its dynamics can be sufficiently described by the time flow of the conditional mean vector $\bar{X}_{t}=\textrm{Tr}(\hat{\textbf{X}}\hat{\rho}^{c}_{t})$ and the covariance matrix $V_{t}$, whose element

with $\Delta_{c}\hat{\textbf{X}}_{i}=\hat{\textbf{X}}_{i}-\langle\hat{\textbf{X}}_{i}\rangle_{c}$ and $\langle\hat{\textbf{X}}_{i}\rangle_{c}=\text{Tr}(\hat{\textbf{X}}_{i}\rho^{c}_{t})$. Using (6), we have the conditional expectation $\bar{X}_{t}$ of $\hat{\textbf{X}}$ under the measurement of $\hat{\textbf{Y}}_{t}$ as

where

and

is the Gaussian innovation which describes the information gain from the the measurement. The conditional covariance matrix satisfies the matrix Riccati equation

where

It is worth noting that the equation for $V_{t}$ is deterministic and does not depend on the stochastic measurement results.

Taking expectation values of (7) and (8), we have the moment equations for the unconditional approach as

which correspond to the quantum master equation (2). To prepare a desired Gaussian state, most of the previous results are on the basis of the moment equations (9). Note that the final term in (8) causes a reduction in the conditional covariance matrix as compared with that in (9). Recall that most of the useful properties of the Gaussian states are determined by the covariance matrix. Thus intuitively by utilizing the conditional covariance matrix equation (8), one may provide weaker conditions in preparing a desired Gaussian state.

Let us first look at the equation of the conditional expectation (7). We can choose $B$ such that its column space is the same as that of $VC^{\top}+M$, where $V$ is the steady solution of (8). Then we choose $F$ such that it satisfies

Thus, for the Markovian feedback (or direct feedback) $u_{t}dt=Fdy_{t}$ [Wiseman & Milburn, 2010], the equation (7) of the first moment becomes deterministic in the limit $t\rightarrow\infty$ as

Therefore, if $A-MC-VC^{\top}C$ is stable, then the solution of (10) will approach to 0 approximately. In the following we will give necessary and sufficient conditions for the conditional covariance matrix equation (8) to have a steady solution and meanwhile $A-MC-VC^{\top}C$ is stable. From now on we focus on the conditional covariance matrix equation (8).

Above all, we present the following lemma (Theorem 3.4 in [Zhou, Doyle, & Glover, 1995]) concerning the properties of detectability, which is to be utilized in the main results.

The definition of stabilizing steady solution is defined as the following.

Now we provide necessary and sufficient conditions for the Riccati equation (8) to have a unique stabilizing steady solution, which corresponds to a quantum Gaussian state.

Proof: Sufficiency: Firstly, according to Theorem 13.7 in [Zhou, Doyle, & Glover, 1995], if

1. 1.

   $[C,~{}A-MC]$ is detectable, and

2. 2.

   $[CJ^{\top},~{}(A-MC)^{\top}]$ has no unobservable modes on the imaginary axis,

then the Riccati equation (11) has a unique stabilizing solution $V$, i.e., $A-MC-VC^{\top}C$ is stable. Moreover, $V$ is positive semidefinite.

Secondly, let us prove that $[CJ^{\top},~{}(A-MC)^{\top}]$ has no unobservable modes on the imaginary axis if $[C,~{}A-MC]$ is detectable. Suppose, on the contrary, that $[CJ^{\top},~{}(A-MC)^{\top}]$ has an unobservable mode on the imaginary axis. From Lemma 1, there exists $w\in\mathbb{R}$ and a corresponding $x\neq 0$, such that

Thus, we have

Combining this conclusion with $CJ^{\top}x=0$ contradicts with the condition that $[C,~{}A-MC]$ is detectable. Thus, $[CJ^{\top},~{}(A-MC)^{\top}]$ has no unobservable modes on the imaginary axis if $[C,A-MC]$ is detectable. Moreover, it is straightforward to verify that $[C,~{}A-MC]$ being detectable is equivalent to $[C,~{}A]$ being detectable.

Necessity: Since the unique stabilizing solution of the Riccati equation (11) can be represented as [Zhou, Doyle, & Glover, 1995]

the matrix $(A-MC)^{\top}-C^{\top}CV$ should be stable. This implies that $[(A-MC)^{\top},~{}C^{\top}]$ is stabilizable, and $[C,~{}A]$ is detectable accordingly.   $\blacksquare$

It can be seen that if $[C,~{}A]$ is detectable, $A-MC-VC^{\top}C$ is stable, and the solution of (10) will approximately approach to 0.

From Theorem 1, if $[C,~{}A]$ is detectable, then there is a unique stabilizing solution of the Riccati equation (11) satisfying $V\geq 0$. But as a covariance matrix depicting quantum Gaussian state, $V$ must satisfy the Heisenberg uncertainty principle,

In fact, $V\geq\pm\frac{i}{2}J$ essentially implies $V>0$ as proved in [Pirandola, Serafini, & Lloyd, 2009].

A necessary and sufficient condition for the solution $V$ of the Riccati equation (11) to satisfy the Heisenberg uncertainty principle is given in Theorem 2.

Since the proof of Theorem 2 involves a Riccati equation in the complex field. We introduce two lemmas for Riccati equations in the complex field.

The proof of Lemma 3.2 is presented in Appendix B by analyzing the Riccati equation in the complex domain (see Appendix A).

Proof: Sufficiency: Suppose, on the contrary, that $[CJ^{\top},~{}(A-MC)^{\top}]$ has an unobservable mode on the imaginary axis. Then there exists $\lambda\in\mathbb{R}$ and a corresponding vector $x\neq 0$, such that

Thus, we have

which contradicts with the condition that $A-MC-\frac{i}{2}JC^{\top}C$ has no eigenvalues on the imaginary axis.

Necessity: Suppose $A-MC-\frac{i}{2}JC^{\top}C$ has an eigenvalue on the imaginary axis. Then there exists $\lambda\in\mathbb{R}$ and a corresponding vector $x\neq 0$, such that

Multiplying the above equation from left by $x^{\dagger}J^{\top}$ yields

Since $(x^{\dagger}Jx)^{\dagger}=-x^{\dagger}Jx$ is a pure imaginary number and $G+\frac{1}{2i}(\Lambda^{\top}\Lambda-\Lambda^{\dagger}\Lambda^{*})$ as well as $C^{\top}C$ are real symmetric matrices, we have $Cx=0$, and $(A-MC)x=i\lambda x$ from (12), accordingly. Then, since $(A-MC)J^{\top}=J(A-MC)^{\top}$, we have

which contradicts with the condition that $[CJ^{\top},~{}(A-MC)^{\top}]$ has no unobservable modes on the imaginary axis.

Thus, $[CJ^{T},~{}(A-MC)^{T}]$ has no unobservable modes on the imaginary axis if and only if $A-MC-\frac{i}{2}JC^{T}C$ has no eigenvalues on the imaginary axis.   $\blacksquare$

Now we give the proof of Theorem 2.

Proof: Sufficiency: From Theorem 1, if $[C,~{}A]$ is detectable, then the Riccati equation $(\ref{conditioncovariance})$ has a unique stabilizing steady solution $V$ which is positive semidefinite. Since $V$ is real and symmetric, and $J$ is antisymmetric, we just need to prove $V\geq\frac{i}{2}J$.

Suppose $Y=V-\frac{i}{2}J$. From $(\ref{riccati})$ we have the Riccati equation for $Y$ as

From the proof of Theorem 1, if $[C,~{}A-MC]$ is detectable, then $[CJ^{\top},~{}(A-MC)^{\top}]$ has no unobservable modes on the imaginary axis. This conclusion combined with Lemma 3.3 implies that $A-MC-\frac{i}{2}JC^{\top}C$ has no eigenvalues on the imaginary axis. Thus, $[0,(A-MC-\frac{i}{2}JC^{\top}C)^{\dagger}]$ has no unobservable modes on the imaginary axis. Moreover, $[C,~{}A-MC-\frac{i}{2}JC^{\top}C]$ is detectable if $[C,~{}A]$ is detectable. According to Lemma 3.2, the Riccati equation (13) for $Y$ has a unique stabilizing solution which is positive semidefinite, i.e., $V\geq\frac{i}{2}J$, if $[C,~{}A]$ is detectable.

Necessity: It can be obtained from the necessity part of Theorem 1.   $\blacksquare$

From Theorem 2 we can see that if $[C,~{}A]$ is detectable, then the stabilizing steady state solution of (8) corresponds to a conditional covariance matrix of some Gaussian state. Since pure Gaussian states of continuous variable systems are the key ingredients of secure optical communications and Heisenberg limited interferometry, the characterization of pure Gaussian states has attracted much attention.

Note that the purity of a $m$ mode Gaussian state is simply

In [Koga & Yamamoto, 2012], by utilizing a dissipation-induced approach on the basis of the model (9), an explicit algebraic characterization of the purity of Gaussian state was given. For the model (9), to have a unique stabilizing steady pure state with covariance matrix $V_{s}$, besides $A$ being stable, $V_{s}$ must satisfy the following matrix equations:

However, these matrix equations are not easy to verify without knowing the solution $V_{s}$.

A natural question is whether the unique stabilizing steady solution $V$ of the Riccati equation (8) satisfying $V\geq\pm\frac{i}{2}J$ corresponds to some pure Gaussian state. The following Theorem 3 gives an affirmative answer.

Proof: To prove the steady state being pure, we resort to the correspondence of Gaussian states between the Schr$\ddot{\textrm{o}}$dinger picture and the Heisenberg picture.

Consider the following stochastic master equation of $\hat{\rho}_{t}$ starting from an initial pure Gaussian state $\hat{\rho}_{0}$,

where the innovations

are Gaussian, and

It is straightforward to verify that

This combined with the initial state $\hat{\rho}_{0}$ being pure implies that the state $\hat{\rho}_{t}$ is pure for all time $t$. Physically this means that if the initial state is pure and there is no information loss during the measurement process, then the state will always be pure.

Moreover, the evolution (15) preserves the Gaussian property. Thus (15) can be fully depicted by the equations of the first- and second-moment in the Heisenberg picture whose dynamics are the same as (7) and (8), while the initial values should correspond to the pure state $\hat{\rho}_{0}$, respectively. Recall that the purity of the Gaussian state only depends on the second moment. Hence, from the correspondence between the Schr$\ddot{\textrm{o}}$dinger picture and the Heisenberg picture for Gaussian states, if (8) has a unique stabilizing steady solution, it must correspond to a pure Gaussian state. Thus, from Theorem 2, if $[C,~{}A]$ is detectable, then the Riccati equation (8) has a unique stabilizing steady solution $V$, which can be considered as a conditional covariance matrix of some pure Gaussian state.    $\blacksquare$

To prepare a Gaussian state, the approach via continuous measurement has the following advantages:

(i) From Theorem 3, to prepare a pure Gaussian state, the conditions that need to be satisfied via the continuous measurement are much weaker as compared with conditions given by the dissipation-induced approach.

(ii) For the unconditional approach, if $V_{s}$ is a unique steady pure state of (9), then it must also be a unique steady pure state of the conditional covariance matrix equation (8). This is because that $V_{s}$ being a unique steady pure state of (9) is equivalent to that (14) holds. This further yields

Thus, (8) and (9) have the same steady pure state.

(iii) Under the same system Hamiltonian and coupling operators, the conditional covariance matrix $V_{\text{c}}$ obtained via continuous measurement approach evolving under (8) takes the form [Giovanni, Brunelli, & Genoni, 2021]

where $V_{\text{unc}}$ obeys (9), and

with $\bar{X}_{c}$ obeying (7). Note that the Gaussian state which corresponds to $V_{\text{unc}}$ may be mixed. Since most of the useful properties of a Gaussian state are determined by the covariance matrix, generally the smaller the covariance matrix (in the matrix partial order) is, the more useful the Gaussian state is. Thus, continuous measurement approach can prepare more useful Gaussian states under the same system parameters.

To better illustrate the advantage of the conditional method in preparing pure Gaussian states, we give a single mode example.

Example 1. Consider the Hamiltonian as

with $\kappa$ the damping rate of the cavity. The corresponding $G$ matrix is $G=\begin{pmatrix}2\kappa&0\\ 0&0\end{pmatrix}$. It can be verified that there does not exist a positive definite matrix $V$ such that it satisfies the second equation of (14). In other words, if the system Hamiltonian is $\hat{H}=\kappa\hat{q}^{2}$, no matter how to choose the coupling operator $\hat{L}$, no pure Gaussian states can be prepared by utilizing the dissipation-induced (unconditional) approach.

Now we employ the conditional approach, and verify that pure Gaussian states can be prepared via continuous measurement. We take $V=\frac{1}{2}\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ as an example. It is clear that $V$ corresponds to a pure Gaussian state. To prepare it, select the measurement operator as