The momentum and position observables in an [Formula: see text]-mode boson Fock space [Formula: see text] have the whole real line [Formula: see text] as their spectrum. But the total number operator [Formula: see text] has a discrete spectrum [Formula: see text]. An [Formula: see text]-mode Gaussian state in [Formula: see text] is completely determined by the mean values of momentum and position observables and their covariance matrix which together constitute a family of [Formula: see text] real parameters. Starting with [Formula: see text] and its unitary conjugates by the Weyl displacement operators and operators from a representation of the symplectic group [Formula: see text] in [Formula: see text], we construct [Formula: see text] observables with spectrum [Formula: see text] but whose expectation values in a Gaussian state determine all its mean and covariance parameters. Thus measurements of discrete-valued observables enable the tomography of the underlying Gaussian state and it can be done by using five one-mode and four two-mode Gaussian symplectic gates in single and pair mode wires of [Formula: see text]. Thus the tomography protocol admits a simple description in a language similar to circuits in quantum computation theory. Such a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states permit a tomography of the channel parameters. However, in our procedure the number of counting measurements exceeds the number of channel parameters slightly. Presently, it is not clear whether a more efficient method exists for reducing this tomographic complexity. As a byproduct of our approach an elementary derivation of the probability generating function of [Formula: see text] in a Gaussian state is given. In many cases the distribution turns out to be infinitely divisible and its underlying Lévy measure can be obtained. However, we are unable to derive the exact distribution in all cases. Whether this property of infinite divisibility holds in general is left as an open problem.
The momentum and position observables in an [Formula: see text]-mode boson Fock space [Formula: see text] have the whole real line [Formula: see text] as their spectrum. But the total number operator [Formula: see text] has a discrete spectrum [Formula: see text]. An [Formula: see text]-mode Gaussian state in [Formula: see text] is completely determined by the mean values of momentum and position observables and their covariance matrix which together constitute a family of [Formula: see text] real parameters. Starting with [Formula: see text] and its unitary conjugates by the Weyl displacement operators and operators from a representation of the symplectic group [Formula: see text] in [Formula: see text], we construct [Formula: see text] observables with spectrum [Formula: see text] but whose expectation values in a Gaussian state determine all its mean and covariance parameters. Thus measurements of discrete-valued observables enable the tomography of the underlying Gaussian state and it can be done by using five one-mode and four two-mode Gaussian symplectic gates in single and pair mode wires of [Formula: see text]. Thus the tomography protocol admits a simple description in a language similar to circuits in quantum computation theory. Such a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states permit a tomography of the channel parameters. However, in our procedure the number of counting measurements exceeds the number of channel parameters slightly. Presently, it is not clear whether a more efficient method exists for reducing this tomographic complexity. As a byproduct of our approach an elementary derivation of the probability generating function of [Formula: see text] in a Gaussian state is given. In many cases the distribution turns out to be infinitely divisible and its underlying Lévy measure can be obtained. However, we are unable to derive the exact distribution in all cases. Whether this property of infinite divisibility holds in general is left as an open problem.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup E2(H) of bounded operators in Γ(H). This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position–momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(P)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in H, ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C, and [αrs] is a symmetric matrix in Mn(C); (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its E2(H)-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in Γ(Cn) by the estimation of its E2(Cn) parameters using O(n2) measurements with a finite number of outcomes.
We present two projects concerning the main part of my PhD work. In the first one we study quantum channels, which are the most general operations mapping quantum states into quantum states, from the point of view of their divisibility properties. We introduced tools to test if a given quantum channel can be implemented by a process described by a Lindblad master equation. This in turn defines channels that can be divided in such a way that they form a one-parameter semigroup, thus introducing the most restricted studied divisibility type of this work. Using our results, together with the study of other types of divisibility that can be found in the literature, we characterized the space of qubit quantum channels. We found interesting results connecting the concept of entanglement-breaking channel and infinitesimal divisibility. Additionally we proved that infinitely divisible channels are equivalent to the ones that are implementable by one-parameter semigroups, opening this question for more general channel spaces. In the second project we study the functional forms of one-mode Gaussian quantum channels in the position state representation, beyond Gaussian functional forms. We perform a black-box characterization using complete positivity and trace preserving conditions, and report the existence of two subsets that do not have a functional Gaussian form. The study covers as particular limit the case of singular channels, thus connecting our results with the known classification scheme based on canonical forms. Our full characterization of Gaussian channels without Gaussian functional form is completed by showing how Gaussian states are transformed under these operations, and by deriving the conditions for the existence of master equations for the non-singular cases.
Non-Gaussianity is a key resource for achieving quantum advantages in bosonic platforms. Here, we investigate the : a non-Gaussianity monotone that satisfies remarkable operational and resource-theoretic properties. Mathematically, the symplectic rank of a pure state is the number of symplectic eigenvalues of the covariance matrix that are strictly larger than those of the vacuum. Operationally, it (i) is easy to compute, (ii) emerges as the smallest number of modes onto which all the non-Gaussianity can be compressed via Gaussian unitaries, (iii) lower bounds the non-Gaussian gate complexity of state preparation independently of the gate set, (iv) governs the sample complexity of quantum tomography, and (v) bounds the computational complexity of bosonic circuits. Crucially, the symplectic rank is nonincreasing under postselected Gaussian operations, leading to new no-go theorems for Gaussian conversion. Remarkably, this allows us to show that the resource theory of non-Gaussianity is irreversible under exact Gaussian operations. Finally, we show that the symplectic rank is a robust non-Gaussian measure, explaining how to witness it in experiments and how to exploit it to meaningfully benchmark different bosonic platforms. In doing so, we derive lower bounds on the trace distance (resp. total variation distance) between arbitrary states (resp. classical probability distributions) in terms of the norm distance between their covariance matrices, which may be of independent interest.
We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position and momentum, we efficiently interpret the properties of such tomograms, being consequent with Heisenberg's uncertainty principle. We also obtain, by means of the dual tomographic symbols, the expectation value for the volume operator, which coincides with the values reported in the literature. We expect that our findings result interesting as the introduced tomographic representation may be further benefited from the well-developed measure techniques in the areas of Quantum optics and Quantum information theory.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
Operator-sum or Kraus representations for single-mode bosonic Gaussian channels are developed, and several of their consequences explored. The fact that the two-mode metaplectic operators acting as unitary purification of these channels do not, in their canonical form, mix the position and momentum variables is exploited to present a procedure which applies uniformly to all families in the Holevo classification. In this procedure the Kraus operators of every quantum-limited Gaussian channel can be simply read off from the matrix elements of a corresponding metaplectic operator. Kraus operators are employed to bring out, in the Fock basis, the manner in which the antilinear, unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel, denoted $\mathcal{D}(\ensuremath{\kappa})$ in the Holevo classification. The matrix transposition channels $\mathcal{D}(\ensuremath{\kappa})$, $\mathcal{D}({\ensuremath{\kappa}}^{\ensuremath{-}1})$ turn out to be a dual pair in the sense that their Kraus operators are related by the adjoint operation. The amplifier channel with amplification factor $\ensuremath{\kappa}$ and the beam-splitter channel with attenuation factor ${\ensuremath{\kappa}}^{\ensuremath{-}1}$ turn out to be mutually dual in the same sense. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasiprobabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. The semigroup property of the amplifier and attenuator families leads in both cases to a Zeno-like effect arising as a consequence of interrupted evolution. In the cases of entanglement-breaking channels a description in terms of rank 1 Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that almost every noisy Gaussian channel can be realized as a product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for these noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.