Informationally complete sets of Gaussian measurements
We prove necessary and sufficient conditions for the informational completeness of an arbitrary set of Gaussian observables on continuous variable systems with finite number of degrees of freedom. In particular, we show that an informationally complete set either contains a single informationally complete observable, or includes infinitely many observables. We show that for a single informationally complete observable, the minimal outcome space is the phase space, and the observable can always be obtained from the quantum optical -function by linear postprocessing and Gaussian convolution, in a suitable symplectic coordinatization of the phase space. In the case of projection valued Gaussian observables, e.g., generalized field quadratures, we show that an informationally complete set of observables is necessarily infinite. Finally, we generalize the treatment to the case where the measurement coupling is given by a general linear bosonic channel, and characterize informational completeness for an arbitrary set of the associated observables.
The ability to determine an unknown quantum state produced by some source is central for many applications in quantum information science. The procedure of reconstructing the quantum state, known as quantum state tomography, has therefore been under intense investigations and continues to attract a lot of attention (1); (2). In the continuous variable regime, and in particular its quantum optical realizations, there are two commonly used approaches to quantum tomography. In optical homodyne tomography, the set of rotated quadratures is measured using balanced homodyne detection, thus allowing one to ”scan” the phase space of the system (3); (4). The alternative method uses the Husimi -function which can be measured using a double homodyne detection scheme, and has the advantage that the reconstruction requires the measurement of only a single observable (5).
Both of the above instances fall under the class of Gaussian measurements, i.e., measurements which yield a Gaussian measurement outcome distribution whenever the system is initially in a Gaussian state (6). The purpose of this paper is to present a method for investigating whether or not a given set of such Gaussian observables is informationally complete (7), i.e., allows the reconstruction of an unknown quantum state of the system from the statistics. We consider an -mode electromagnetic field, whose phase space is therefore -dimensional. We show that by measuring a Gaussian observable one obtains the values of the Weyl transform of the state on a linear subspace of the phase space. Therefore, for a set of observables the union of these subspaces needs to be ”sufficiently large” in order for unique state determination to be possible. In particular, we show that if one does not have access to a single informationally complete Gaussian observable, then one necessarily needs infinitely many observables.
After these general results we focus on two specific instances. Firstly, we investigate single informationally complete Gaussian observables in more detail. We show that if we restrict to the smallest possible dimension of the outcome space, then the set of informationally complete Gaussian observables is exhausted, up to linear transformations of the measurement outcomes, by Gaussian observables which are covariant with respect to phase space translations. Furthermore, we show that in a suitable symplectic coordinatization of the phase space, any informationally complete Gaussian observable with a minimal outcome space is a postprocessing of the -function. Secondly, we study commutative Gaussian observables which then include projection valued (also called sharp) Gaussian observables as special cases. We show that no finite set of such observables is informationally complete. For an arbitrary set of generalized field quadratures, i.e., sharp Gaussian observables with one-dimensional outcome space, we prove a further characterization for informational completeness. We also find an interesting connection between the generalized quadratures and informationally complete Gaussian phase space observables. Finally, we consider a more general scenario where the measurement coupling is represented by a general linear bosonic channel. Also in the general case we obtain a characterization for informational completeness and deal explicitly with general covariant phase space observables.
The Hilbert space of an electromagnetic field consisting of bosonic modes is the -fold tensor product , where each single mode Hilbert space is spanned by the number states . The creation and annihilation operators related to the mode are denoted by and . In the coordinate representation where the number states are represented by the Hermite functions. The Hilbert space of the entire field is then .
The states of the field are represented by positive trace one operators acting on , and the observables are represented by positive operator valued measures (POVMs) defined on a -algebra of subsets of some measurement outcome set. In this paper, we will only consider observables taking values in . Each observable is thus represented by a map , where is the Borel -algebra of and denotes the set of bounded operators on , and which satisfies positivity , normalization and -additivity for any sequence of pairwise disjoint sets where the series converges in the weak operator topology. When a measurement of is performed on the system initially prepared in a state , the measurement outcomes are distributed according to the probability measure .
The phase space of the -mode EM-field is and we use the notation for the canonical coordinates. The phase space translations are represented in by the Weyl operators
where the matrix
determines the symplectic form and , with and being the canonical quadrature (i.e., position and momentum) operators acting on the mode. Whenever appropriate, we will emphasize the number of modes in question with a subscript so that, e.g., in the above case we would have . Due to the Stone-von Neumann theorem, the Weyl operators are determined by the relation
up to unitary equivalence. This unitary freedom corresponds to the choice of the canonical coordinates: let denote the group of symplectic transformations of , i.e., real invertible -matrices satisfying . If we change the coordinates as , then the original Weyl operators in the new coordinates are given by . Since is symplectic, the operators indeed also satisfy the Weyl relation (1), and there exists a unitary operator such that
for all . Summarizing, a change of canonical coordinates is effected by the transformation
A set of observables is called informationally complete (7) if an arbitrary unknown state is uniquely determined by the collective measurement outcome statistics , or equivalently, by the corresponding Fourier transforms
For our purposes, the crucial fact is that any state is uniquely determined by its Weyl transform
which in our case turns out to be directly related to the Fourier transforms of the outcome distributions. The Weyl transform is a continuous function, so that is already determined if we know the values of on a dense subset of . The following lemma will then turn out to play a central role. It expresses precisely the intuitive idea that in order to uniquely determine the state, one has to ”scan” the phase space sufficiently well. Here refers to the set of trace class operators on .
Let be a nonempty open set. Then there exists a nonzero such that for all .
Let be such that for all (e.g., the vacuum state). For any define the operator-valued convolution (8) via
so that if and only if . We can now choose to be a nonzero compactly supported -function such that for all . We then obtain via the inverse Fourier transform, and our choice of guarantees that . Thus, is of the required type. ∎
Iii Gaussian observables and their postprocessings
We will next recall the relevant concepts of Gaussian states, channels, and observables, as well as the concept of postprocessing of a measurement. For more details on continuous variable quantum information, we refer the readers to (6); (9) and ((10), Ch. V).
A state is called a Gaussian state, if the Weyl transform is of a Gaussian form. More specifically, the Weyl transform of a Gaussian state is given by
where is the displacement vector whose components are given by the first moments and is the covariance matrix whose elements are where denotes the anticommutator. The covariance matrix is a real symmetric matrix which satisfies the uncertainty relation (11)
A Gaussian channel is a completely positive trace preserving map which maps Gaussian states into Gaussian states. Written in terms of the Weyl transforms of the input and output states, any Gaussian channel can be characterized by
where is a -matrix and is a -matrix which must satisfy the complete positivity condition
We say that an observable is Gaussian, if the measurement outcome distribution is Gaussian whenever the system is initially in a Gaussian state. This can be conveniently expressed in terms of the characteristic function of the POVM
in which case is Gaussian whenever
where is a -matrix and is a -matrix which satisfy the positive definiteness condition
and . Note that our definition differs from the one used, e.g., in (10) where the outcome space of a Gaussian observable is assumed to be a symplectic space, i.e., even dimensional. The physical motivation for this definition will become apparent in the next section where measurement schemes realizing Gaussian observables are discussed.
A postprocessing of a measurement is a fixed transformation performed on the measurement outcome distribution. This is relevant for informational completeness, because information can only get lost in such a process; if a postprocessing is informationally complete, then so is the original measurement. Here we consider two different kinds of postprocessings: linear postprocessings and smearings. To this end, let be a Gaussian observable parametrized by .
Given a matrix we define the observable via
where . Then is clearly Gaussian, and it is called a linear postprocessing of . If is invertible, is called a bijective linear postprocessing of . A straightforward computation shows that the triple of parameters characterizing is . It is clear from the definition that a bijective linear postprocessing of is informationally complete if and only if is.
For any probability measure , we define the observable via
and say that is a smearing of . Note that for any state the corresponding probability distribution is just a convolution of the original one with the measure , i.e., . If is Gaussian, i.e, with , then the smeared observable is also Gaussian and, using the fact that , we find that the parameters of the smeared observable are . Since is nonzero everywhere, the smearing is informationally complete if and only if is.
Iv Unitary dilations of Gaussian observables
We will next construct a unitary measurement dilation for an arbitrary Gaussian observable. This is done by first showing that any Gaussian observable can be measured by applying a Gaussian channel to the field and then performing homodyne detection on the output of the channel. Since unitary dilations of Gaussian channels are known (14), this then allows us to construct the desired measurement dilation.
Suppose first that we have a Gaussian channel determined by the parameters . Let be the canonical spectral measure, i.e., corresponds to multiplication by the indicator function of the set , and define the observable as . The characteristic function of is then where . For any we denote so that , and we obtain
by Eq. (4). Finally, we define the -matrix , the -matrix and the vector by
so that the characteristic function can be expressed as
In other words, the observable is Gaussian.
Conversely, suppose that we have a Gaussian POVM determined by the parameters . We need to find parameters of a Gaussian channel such that Eqs. (8) hold. To this end, first define the matrix by setting and otherwise, and then set so that . Define also the matrix via and otherwise, and the vector in a similar manner as and otherwise. In order to prove the validity of the complete positivity condition (5), we note that by our construction this reduces to showing that which is an immediate concequence of (7) and the definitions of the matrices in question.
In order to reach the desired unitary dilation, we recall that any Gaussian channel can be realized by coupling the -mode system to auxiliary Gaussian modes, and then applying two types of unitary operators to the total system: symplectic transformations with , and displacements with (14) (see Fig. 1). The number of auxiliary modes can always be chosen so that , though the optimal choice depends on the details of the channel (15). The rest of this section is devoted to demonstrating how the parameters of the Gaussian channel, and hence the corresponding observable, are determined by the dilation of the channel.
Now suppose that the auxiliary -mode field is in some Gaussian state with covariance matrix and displacement vector . Since we are interested only in modes, we must discard the other modes after the interaction. The resulting channel is then given by
where denotes partial trace over the Hilbert space of the discarded modes. Without loss of generality we may assume the ordering of the Hilbert spaces to be fixed as
The Weyl transform of the output state is then
so that by denoting we have
If we now write in block form as
where is a -matrix and is a -matrix, and define we have
where , , and . Finally, the parameters of the corresponding Gaussian observable are obtained from these via Eqs. (8).
V Characterization of informational completeness
We will next show a general method for determining the informational completeness of a given set of Gaussian observables. To this end, let be a Gaussian POVM with parameters . For an arbitrary state , the Fourier transform of the probability measure then reads
and by linearity, (9) also holds with replaced by a general trace class operator in which case the measure is complex valued. Since the Gaussian term on the right-hand-side of Eq. (9) is non-zero, we can divide both sides by it. Hence, by measuring we are able to determine the values for all in the set
which is clearly a subspace of . We are now ready to prove the main general result of this paper.
A set of Gaussian observables is informationally complete if and only if is dense in .
If is dense in , then by measuring the observables one can determine the values of on a dense set, so that by the continuity of the Weyl transform, is uniquely determined. Conversely, assume that is not dense. Then there exists a nonempty open set in the complement of the closure of , so by Lemma 1, there exists a nonzero whose Weyl transform vanishes outside . By (9) and the injectivity of the Fourier transform we then have for each , and all . Since the POVM elements are positive, the same holds with replaced by either of the traceless selfadjoint operators and . At least one of these is nonzero, and hence a constant times the difference of two distinct density operators. By definition, this implies that the set is not informationally complete. ∎
The following two results now follow immediately.
If is a finite set, then is informationally complete if and only if is informationally complete for some .
The ”if”-part is trivial. Since each is closed and is finite, also the union is closed. Hence, is informationally complete only if . But this is possible only if for some , because otherwise each subspace would have zero Lebesgue measure and by subadditivity, so would also . An application of Theorem 1 completes the proof. ∎
A Gaussian observable is informationally complete if and only if .
Being a subspace, is closed, so is informationally complete if and only if , i.e., . ∎
We now proceed to investigate the informational completeness of a single Gaussian observable more carefully.
Let be a linear subspace and denote . Then there exists a Gaussian observable such that . In particular, for each there exist both informationally complete an informationally incomplete Gaussian observables.
Let be an orthonormal basis of and define so that is a -matrix. Since is Hermitian, there exists a matrix , such that (7) is satisfied. Then for any the parameters define a Gaussian observable which satisfies . The last claim then follows by choosing and, e.g., . ∎
Prop. 1 shows that if the dimension of is strictly smaller then the dimension of the measurement outcome space of , then from the state reconstruction point of view there is some redundancy in the measurement setup. Indeed, the same information about the state can be obtained with a smaller outcome space and hence, with fewer homodyne measurements. In particular, for an -mode field it is necessary but also sufficient to couple the field to auxiliary modes, thus giving the outcome space dimension . In this case, we say that an observable satisfying the rank condition of Cor. 2 is a minimal informationally complete Gaussian observable. Clearly, this condition is equivalent to being invertible, i.e., .
One immediate consequence of the determinant condition is worth noting explicitly: a triple with -matrices and , if drawn randomly from the subset given by (7), according to any nonsingular probability density, defines a minimal informationally complete Gaussian observable with probability one. Hence, almost all Gaussian observables with minimal outcome space are informationally complete.
Vi Covariant Gaussian observables
We have seen that a minimal informationally complete Gaussian observable has outcome space , which we can identify with the phase space, whose translations and symplectic transformations act as unitary transformations on the range of the observable. We now look at the consequences of this identification.
An observable is a covariant phase space observable if it satisfies
According to Cor. 2, every covariant Gaussian phase space observable is thus informationally complete. This result was previously obtained, e.g., in (16); (8). As the next proposition shows, any other minimal informationally complete Gaussian observable is connected to a covariant one via linear postprocessing.
A Gaussian observable is informationally complete if and only if it is a bijective linear postprocessing of a covariant Gaussian observable.
Let be a covariant Gaussian observable, parametrized by , and any invertible matrix. The postprocessing is then parametrized by , and is thus informationally complete by Corollary 2. Conversely, suppose that is an informationally complete Gaussian observable, parametrized by . Then is invertible, so we can define an invertible matrix , and further define , . Then the triple determines a covariant Gaussian observable for which . ∎
Next we take into account the symplectic structure of the phase space. The change (3) of the canonical coordinates via a symplectic matrix transforms a phase space observable into the phase space observable given by
In particular, a covariant Gaussian observable parametrized by transforms into the covariant Gaussian observable given by , because . Since the positivity condition (7) reduces to , we can use Williamson’s theorem (17) to choose the symplectic matrix such that it diagonalizes , i.e., , where the symplectic eigenvalues of satisfy . Letting be the Gaussian probability measure such that , the transformed covariant observable is thus
where is the is the -function of the state, i.e., the covariant Gaussian observable with . To summarize, each covariant Gaussian observable is a Gaussian smearing of the -function, up to the choice of the canonical coordinates of the phase space. Combining this with Prop. 2, we have the following result:
A Gaussian observable is informationally complete if and only if there exist an invertible matrix , a symplectic matrix , and a Gaussian probability measure , such that
In other words, any minimal informationally complete Gaussian observable coincides with a postprocessing of , up to the choice of the canonical coordinates.
Notice that the any covariant Gaussian observable can be measured with the following setup: the signal -mode field is coupled to auxiliary parameter modes in a Gaussian state by means of 50:50 beam splitters, and then rotations of are performed on half of the output modes (the case of a single mode is illustrated in Fig. 2). By varying the Gaussian state of the parameter field, one can obtain an arbitrary covariant Gaussian observable. In particular, a measurement of the -function is obtained by choosing the parameter field to be the vacuum (i.e., V=I, m=0).
Vii Commutative and sharp Gaussian observables
Recall that an observable is commutative, if for all , and sharp if . It is well-known that an observable is sharp if and only if it is projection valued, i.e., a spectral measure.
Clearly, sharp observables are always commutative. Conversely, every commutative observable is obtained from some sharp observable by postprocessing with a Markov kernel (see ((18), Thm. 4.4)). Smearing with a convolution kernel is a special case, which however turns out to be sufficient to get all commutative Gaussian observables from sharp ones.
In terms of characteristic functions, is sharp if and only if the map is a unitary representation of , i.e.,
Similarly, is commutative if and only if
This follows by standard approximation arguments from the fact that holds for any integrable , where is the (suitably normalized) Fourier transform.
In the case of Gaussian observables, the relationship between commutativity and sharpness can now be analyzed explicitly. By (13), is a commutative if and only if all the Weyl operators commute, which happens exactly when
Note that this condition is exactly the one that makes a unitary representation, and thus a characteristic function of a sharp Gaussian observable which we denote by . Now if (14) holds, then by the positivity condition (7). Hence, there exists a Gaussian probability measure , such that , and we have
In conclusion, we have proved that a Gaussian observable is commutative exactly when it is a Gaussian smearing of a sharp Gaussian observable.
Since sharp observables are commutative, we can use the (15) along with condition (12) to characterize sharp Gaussian observables. Indeed, since now and the operators are unitary, we have that is sharp if and only if for all . But since is Gaussian, this reduces to the condition . This implies , and since by (7), we have . Hence, sharp Gaussian observables are parametrized by with (14). Note that this can be obtained from the unbiased sharp Gaussian observable by shifting the outcomes with the vector . Here ”unbiased” refers to the fact that for any state with zero expectation for all canonical quadratures , , also the expectation of vanishes.
It is known that a single commutative observable is never informationally complete ((19), Thm. 2.1.2). For commutative Gaussian observables with minimal outcome space this follows immediately from Cor. 2, because we have . For a general commutative Gaussian observable this can still be seen directly by noting that the rank of is at most , as can be seen by applying the Frobenius inequality ((20), Eq. 4.3.3(9), p. 61):
Cor. 1 then tells that no finite set of commutative Gaussian observables is informationally complete.
It is worth noting that in the case of a single mode (), and minimal outcome space () we have
so that is informationally complete if and only if is noncommutative. In other words, informational incompleteness is equivalent to the observable being a smearing of a sharp observable. Note that this is not true for multiple modes. Indeed, using the above result that for commutative observables, we see that if , any given -matrix with , together with a chosen to satisfy (7), determine a Gaussian observable which is informationally incomplete and noncommutative.
Before proceeding to more specific cases, we make one more observation. It follows from (15) that a set of commutative Gaussian observables with matrices , is informationally complete if and only if the set of the corresponding sharp observables is such. In other words, a set of commutative observables can always be reduced to a set of sharp observables, without affecting informational completeness. Accordingly, we concentrate mostly on sharp observables in the concrete examples.
Consider now the special case of a one dimensional outcome space. Then for any Gaussian observable , the matrix is, in fact, a vector in , and we let . The condition (14) is automatically satisfied, so we conclude that every Gaussian observable is commutative, and thus a Gaussian smearing of the corresponding sharp observable . Furthermore, if is a Gaussian observable with matrix , then is a Gaussian smearing of with , exactly when .
The observable has the characteristic function so that is simply the spectral measure of the selfadjoint operator
These operators are sometimes referred to as a generalized field quadratures, and they have also previously been considered in the context of quantum tomography (21). Thm. 1 now reduces to the following result:
Let , be Gaussian observables with corresponding vectors . Then is informationally complete if and only if is dense in the surface of the unit ball of .
We now look at two particular examples in the case of a single mode ().
By setting and we obtain the well known rotated quadrature operators , the corresponding observables being denoted by . From Prop. 4 we immediately see that we can restrict our attention to the values , and that a set of the form where is informationally complete if and only if is dense in . This therefore sharpens the result of, e.g., (22) by proving that density is indeed also necessary. Explicitly, the line on which the Weyl transform can be determined from the measurement of a quadrature , is given by
As a slight modification, we set and so that we obtain the squeezed rotated quadratures . In this case we have
so that for any change in the value of the squeezing parameter causes a change in the slope of the line. For instance, we can fix only two values and consider a set of squeezing parameters. By Prop. 4, the set is informationally complete if and only if
is dense in the unit circle, which happens exactly when is dense in . Indeed, for each of the four choices of signs, the map bijectively parametrizes the part of the unit circle lying in the interior of the -quadrant. In a similar manner we can consider any finite number of values for . The benefit of adding more rotations is that for a fixed one only has to find a set of squeezing parameters which is dense in some interval in order to guarantee informational completeness.
We could also have looked at smearings of sharp quadratures. As already mentioned, this does not add any more structure from the point of view of informational completeness. However, smearings often appear more naturally than sharp observables; we close this section with a particular case involving postprocessing.
Consider the linear postprosessing of a phase space observable with matrix , given by a matrix . Such a postprocessing is called a (nonnormalized) marginal of . Now has one-dimensional outcome space, and is a commutative Gaussian smearing of the generalized quadrature given by the vector . Using this equality, together with Cor. 2, we now get the following characterization of informational completeness of phase space observables in terms of quadratures.
Let be a Gaussian phase space observable. Then the following are equivalent:
is informationally complete;
There is a one-to-one correspondence between generalized quadratures , , and linear maps , such that is a Gaussian smearing of .
Viii Measurements involving general linear bosonic channels
As we have noted before, the physical motivation for the definition of Gaussian observables comes from their measurement realizations: any Gaussian POVM can be measured by first applying a Gaussian channel to the system, and then performing homodyne detection. Now Gaussian channels are special cases of what are known as linear bosonic channels, that is, channels which map Weyl operators according to
where is a matrix and is a complex valued function which must again satisfy suitable complete positivity conditions (12); (13). The corresponding generalization for Gaussian observables is then given by POVMs satisfying
again with some restrictions on and .
We immediately notice that the only difference in the treatment of informational completeness when compared to the Gaussian case is that the function may be zero at some points. Hence, by measuring we can always determine the values of on the set
where is the subspace related to any Gaussian observable with the parameter . In particular, we have but this is typically not a subspace. The proof of the following characterization of informational completeness is identical to the proof of Thm. 1 and is therefore omitted.
Let , , be observables satisfying (16) for some set of parameters . Then is informationally complete if and only if is dense in .
Note that in this more general scenario, a finite set of observables may be informationally complete even though no single observable is such. For instance, suppose that