Abstract A structure theorem is established which shows that an arbitrary multimode bosonic Gaussian observable can be represented as a combination of four basic cases whose physical prototypes are homodyne and heterodyne, noiseless or noisy, measurements in quantum optics. The proof establishes a connection between the descriptions of a Gaussian observable in terms of the characteristic function and in terms of the density of a probability operator-valued measure (POVM) and has remarkable parallels with the treatment of bosonic Gaussian channels in terms of their Choi–Jamiołkowski form. Along the way we give the “most economical” (in the sense of minimal dimensions of the quantum ancilla) construction of the Naimark extension of a general Gaussian observable. We also show that the Gaussian POVM has bounded operator-valued density with respect to the Lebesgue measure if and only if its noise covariance matrix is nondegenerate.
Abstract A structure theorem is established which shows that an arbitrary multimode bosonic Gaussian observable can be represented as a combination of four basic cases whose physical prototypes are homodyne and heterodyne, noiseless or noisy, measurements in quantum optics. The proof establishes a connection between the descriptions of a Gaussian observable in terms of the characteristic function and in terms of the density of a probability operator-valued measure (POVM) and has remarkable parallels with the treatment of bosonic Gaussian channels in terms of their Choi–Jamiołkowski form. Along the way we give the “most economical” (in the sense of minimal dimensions of the quantum ancilla) construction of the Naimark extension of a general Gaussian observable. We also show that the Gaussian POVM has bounded operator-valued density with respect to the Lebesgue measure if and only if its noise covariance matrix is nondegenerate.
A general quantum channel can be represented in terms of a unitary interaction between the information-carrying system and a noisy environment. In this paper the minimal number of quantum Gaussian environmental modes required to provide a unitary dilation of a multimode bosonic Gaussian channel is analyzed for both pure and mixed environments. We compute this quantity in the case of pure environment corresponding to the Stinespring representation and give an improved estimate in the case of mixed environment. The computations rely, on one hand, on the properties of the generalized Choi-Jamiolkowski state and, on the other hand, on an explicit construction of the minimal dilation for arbitrary bosonic Gaussian channel. These results introduce a new quantity reflecting ``noisiness'' of bosonic Gaussian channels and can be applied to address some issues concerning transmission of information in continuous variables systems.
Gaussian operations, as one usually refers to CP-maps that send Gaussian states to Gaussian states, are apt to model a wide range of situations of practical interest and may often be controlled and applied on demand in the laboratory, in a variety of set-ups (that will be the subject matter of Chapter 9 ). Furthermore, they can be classified in a hierarchy of compact, powerful and far-reaching theoretical descriptions, which will be introduced by degrees and discussed in the current chapter. Note that here we are dealing with discrete quantum operations, rather than dynamical equations, which will be dealt with in the next chapter. We shall start from Gaussian unitary operations, corresponding to the symplectic group already introduced in Chapter 3 , and will also clarify how other basic manipulations such as tensoring and partial tracing carry over to the Gaussian picture. Then, we shall broaden our view to include all deterministic (trace-preserving) maps resulting from Gaussian unitary dilations, which are often known as the set of ‘bosonic Gaussian channels’ and encompass open dynamics subject to noise and ‘decoherence’ (the loss of quantum coherence due to the interaction with an environment). Next, we will move on to consider POVMs corresponding to the well-known homodyne and heterodyne detection schemes which, when acting on a subsystem, preserve the Gaussian character of the initial state both when the system is filtered by recording the measurement outcomes, and when acting unconditionally (that is, when the outcome of the measurement is not recorded). Our description will be further enlarged to the wider class of ‘general-dyne’ detections, and we will also introduce methods to account for imperfect detections. The ability to accommodate noise in the analytical description, often daunting with discrete variable and, more generally, outside the Gaussian regime, is probably one of the greatest strengths of the formalism under discussion. Finally, we shall introduce the Gaussian version of the Choi–Jamiolkowski isomorphism, and attain a unified description of all the Gaussian maps previously defined. Throughout our exposition, we will refer to quantum optical systems to exemplify the practical relevance of the operations encountered.
A complete analysis of multi-mode bosonic Gaussian channels is proposed. We clarify the structure of unitary dilations of general Gaussian channels involving any number of bosonic modes and present a normal form. The maximum number of auxiliary modes that is needed is identified, including all rank deficient cases, and the specific role of additive classical noise is highlighted. By using this analysis, we derive a canonical matrix form of the noisy evolution of n-mode bosonic Gaussian channels and of their weak complementary counterparts, based on a recent generalization of the normal mode decomposition for non-symmetric or locality constrained situations. It allows us to simplify the weak-degradability classification. Moreover, we investigate the structure of some singular multi-mode channels, like the additive classical noise channel that can be used to decompose a noisy channel in terms of a less noisy one in order to find new sets of maps with zero quantum capacity. Finally, the two-mode case is analyzed in detail. By exploiting the composition rules of
Gaussian operations, as one usually refers to CP-maps that send Gaussian states into Gaussian states, are apt to model a wide range of situations of practical interest and may often be controlled and applied on demand in the laboratory, in a variety of set-ups. This chapter starts from start from Gaussian unitary operations, corresponding to the symplectic group, and moves on to clarify how other basic manipulations such as tensoring and partial tracing carry over to the Gaussian picture. Then, it includes all deterministic (trace-preserving) maps resulting from Gaussian unitary dilations, which are often known as the set of "bosonic Gaussian channels" and encompasses the open dynamics subject to noise and 'decoherence'. Next, the chapter considers POVMs corresponding to the well-known homodyne and heterodyne detection schemes. Finally, it introduces the Gaussian version of the Choi-Jamiolkowski isomorphism, and attains a unified description of all the Gaussian maps.
We prove the necessary and sufficient conditions for the informational completeness of an arbitrary set of Gaussian observables on continuous variable systems with a 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 corresponding probability distribution can always be obtained from the quantum optical Q -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.