A measure of non-Gaussianity for quantum states

Ivan, J. Solomon; Kumar, Manoranjan; Simon, Richard

doi:10.1007/s11128-011-0314-2

Cited by 59 publications

(64 citation statements)

References 44 publications

(54 reference statements)

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“…In the case of Hilbert-Schmidt degree (2.11) we were able to give an analytic result, while for the fidelity-based degree and the relative-entropy measure we performed numerical evaluations. We found that they depend on the thermal mean occupancȳ n unlike the Wehrl-entropy measure [18]. Our evaluations have also shown a consistent relation between these three non-Gaussianity measures.…”

Section: A Case Study: Damped Photon-added Thermal Statessupporting

confidence: 73%

“…The non-Gaussianity of the state (4.1) was recently evaluated in Ref. [18] by employing the Wehrl-entropy measure and found to coincide with the non-Gaussianity of the number state |M M|, being thus independent of the thermal mean occupancyn. We employ here Eq.…”

Section: A Case Study: Damped Photon-added Thermal Statesmentioning

confidence: 96%

“…Another approach to quantifying non-Gaussianity is based on the Q function and leads to a measure expressed by the difference between the Wehrl entropies of the Gaussian stateρ G and the given non-Gaussian stateρ [18]. It is also worth noting that the non-Gaussianity measures [14,15] were used to check on the relation between non-Gaussianity and the possibility of extending Hudson's theorem to mixed states [19,20].…”

Section: Introductionmentioning

confidence: 99%

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Gaussification through decoherence

2013

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We investigate the loss of nonclassicality and non-Gaussianity of a single-mode state of the radiation field in contact with a thermal reservoir. The damped density matrix for a Fock-diagonal input is written using the Weyl expansion of the density operator. Analysis of the evolution of the quasiprobability densities reveals the existence of two successive characteristic times of the reservoir which are sufficient to assure the positivity of the Wigner function and, respectively, of the $P$ representation. We examine the time evolution of non-Gaussianity using three recently introduced distance-type measures. They are based on the Hilbert-Schmidt metric, the relative entropy, and the Bures metric. Specifically, for an $M$-photon-added thermal state, we obtain a compact analytic formula of the time-dependent density matrix that is used to evaluate and compare the three non-Gaussianity measures. We find a good consistency of these measures on the sets of damped states. The explicit damped quasiprobability densities are shown to support our general findings regarding the loss of negativities of Wigner and $P$ functions during decoherence. Finally, we point out that Gaussification of the attenuated field mode is accompanied by a nonmonotonic evolution of the von Neumann entropy of its state conditioned by the initial value of the mean photon number.Comment: Published version. Comments are welcom

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Section: A Case Study: Damped Photon-added Thermal Statessupporting

confidence: 73%

Section: A Case Study: Damped Photon-added Thermal Statesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Gaussification through decoherence

2013

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“…Our method works by reduction to a constrained optimization problem (even for solving the unconstrained SR inequality), so it can be simply adapted to find the MS with an extra constraint on Gaussianity. Several measures of non-Gaussianity have been used in the literature [15,16,[33][34][35]], but we instead suggest using a parameter g capturing the degree of Gaussianity, inspired from our former work on non-Gaussian states with a positive Wigner function [36,37]. Denoting asρ G the Gaussian state that has the same covariance matrix γ (and same mean values x and p ) as stateρ, we define the Gaussianity ofρ as…”

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confidence: 99%

Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

Mandilara¹,

Cerf

2012

Phys. Rev. A

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We rederive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation. The Heisenberg uncertainty principle [1] captures the difference between classical and quantum states, and sets a limit on the precision of incompatible quantum measurements. It has been introduced in the early days of quantum mechanics, but its form has evolved with the understanding and formulation of quantum physics throughout the years. The first rigorous mathematical proof of Heisenberg's uncertainty relation for the canonical operators of positionx and momentump [7] for more details), while the properties of the states saturating this inequality were also progressively unveiled. The original uncertainty relation (1) only concerned the operatorsx andp, but it was generalized to any pair of Hermitian operators by Schrödinger [8] and Robertson [9], in the case of pure states. In the same works, the anticommutator ofx andp was also included in Eq. (1), yielding a stronger uncertainty relation, (1) and (2) inx andp and the fact that we deal with the quadratic Hamiltonian of a harmonic oscillator. Then, we move on to find bounded uncertainty relations [15], which give stronger bounds than Eq. (2) for states on which some a priori information is known, such as their purity [7] or entropy [16]. Specifically, we derive a Gaussianity-bounded uncertainty relation, depending on the degree of Gaussianity of the state as measured by a parameter g that we introduce. We identify its corresponding set of MSs and find among them all the eigenstates of the harmonic oscillator. This yields a fundamental set of non-Gaussian minimum-uncertainty states, going beyond the common knowledge on the Heisenberg principle.Although the uncertainty relations, being at the root of quantum mechanics, have been investigated in various situations, such as multidimensional [17,18] or mixed states [7,16,19], our results imply that there is more to gain by analyzing them under the perspective of the Gaussian character of a state. Non-Gaussian states of light can now be handled in the laboratory [20][21][22][23][24][25] and have been proven essential in the field of continuous-variable quantum information [26][27][28][29][30], but they remain hard to classify. Identifying states of minimum uncertainty among them may lead to a better understanding of the structure of the state space in infinite dimension and, since the Heisenberg principle is at the heart of the limitations on measurement...

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“…In view of these limitations, it will not come as a surprise that a substantial effort has been put into developing a consistent resource theory of non-Gaussianity. Many non-Gaussianity measures have been proposed and studied in the past decade [31][32][33][34][35][36], that can be applied e.g. to bound the conversion rates between arbitrary states by means of Gaussian operations [35].…”

Section: Introductionmentioning

confidence: 99%

All phase-space linear bosonic channels are approximately Gaussian dilatable

2018

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We compare two sets of multimode quantum channels acting on a finite collection of harmonic oscillators: (a) the set of linear bosonic channels, whose action is described as a linear transformation at the phase space level; and (b) Gaussian dilatable channels, that admit a Stinespring dilation involving a Gaussian unitary. Our main result is that the set (a) coincides with the closure of (b) with respect to the strong operator topology. We also present an example of a channel in (a) which is not in (b), implying that taking the closure is in general necessary. This provides a complete resolution to the conjecture posed in Sabapathy and Winter (2017 Phys. Rev. A 95 062309). Our proof technique is constructive, and yields an explicit procedure to approximate a given linear bosonic channel by means of Gaussian dilations. It turns out that all linear bosonic channels can be approximated by a Gaussian dilation using an ancilla with the same number of modes as the system. We also provide an alternative dilation where the unitary is fixed in the approximating procedure. Our results apply to a wide range of physically relevant channels, including all Gaussian channels such as amplifiers, attenuators, phase conjugators, and also non-Gaussian channels such as additive noise channels and photon-added Gaussian channels. The method also provides a clear demarcation of the role of Gaussian and non-Gaussian resources in the context of linear bosonic channels. Finally, we also obtain independent proofs of classical results such as the quantum Bochner theorem, and develop some tools to deal with convergence of sequences of quantum channels on continuous variable systems that may be of independent interest.

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A measure of non-Gaussianity for quantum states

Cited by 59 publications

References 44 publications

Gaussification through decoherence

Gaussification through decoherence

Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

All phase-space linear bosonic channels are approximately Gaussian dilatable

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