Quantum hypothesis testing for Gaussian states

This paper is a summary of [1].We formulate some problem on quantum hypothesis testing for special Gaussian states, which naturally appear in photonic systems.Moreover, we introduce an optimal test for each quantum hypothesis testing problem.Optimal tests are constructed by using group symmetry of Gaussian states and some results in classical hypothesis testing. 1 Gaussian state We introduce Gaussian states on $\mathcal{H}=L^{2}(\mathbb{R})$ , which represents a single-mode bosonic system and is spanned by sets $\{|k\rangle\}_{k\in \mathbb{Z}_{\geq 0}}$ of the k-th Hermitian functions $|k\rangle.$ $A$ most typical example of a single-mode bosonic system is the one-mode photonic system, in which the vector $|k\rangle$ is called the $k$ -photon number state because it is regarded as the state corresponding to $k$ photons.Then, the set $M_{N}=\{|k\rangle\langle k|\}_{k\in \mathbb{Z}>0}$ , the number measurement, forms a positive-operator-valued measure (POVM).When the vector $|\xi^{-})\in L^{2}(\mathbb{R})(\xi\in \mathbb{C})$ is defined by $| \xi):=e^{-\rfloor}\star^{2}\sum_{k=0}^{\infty}\frac{\xi^{k}}{\sqrt{k!}}|k\rangle,$ This paper is a summary of [1].We formulate some problem on quantum hypothesis testing for special Gaussian states, which naturally appear in photonic systems.Moreover, we introduce an optimal test for each quantum hypothesis testing problem.Optimal tests are constructed by using group symmetry of Gaussian states and some results in classical hypothesis testing. 1 Gaussian state We introduce Gaussian states on $\mathcal{H}=L^{2}(\mathbb{R})$ , which represents a single-mode bosonic system and is spanned by sets $\{|k\rangle\}_{k\in \mathbb{Z}_{\geq 0}}$ of the k-th Hermitian functions $|k\rangle.$$A$ most typical example of a single-mode bosonic system is the one-mode photonic system, in which the vector $|k\rangle$ is called the $k$ -photon number state because it is regarded as the state corresponding to $k$ photons.Then, the set $M_{N}=\{|k\rangle\langle k|\}_{k\in \mathbb{Z}>0}$ , the number measurement, forms a positive-operator-valued measure (POVM).When the vector $|\xi^{-})\in L^{2}(\mathbb{R})(\xi\in \mathbb{C})$ is defined by $| \xi):=e^{-\rfloor}\star^{2}\sum_{k=0}^{\infty}\frac{\xi^{k}}{\sqrt{k!}}|k\rangle,$

May 1, 20130 citations

Asymmetric quantum hypothesis testing with Gaussian states

Gaetana Spedalieri, Samuel L. Braunstein

We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs. In this problem, the aim is to minimize the probability of false negatives and the optimal performance is provided by the quantum Hoeffding bound. After a brief review of these notions, we show how this bound can be simplified for pure states. We then provide a general recipe for its computation in the case of multimode Gaussian states, also showing its connection with other easier-to-compute lower bounds. In particular, we provide analytical formulas and numerical results for important classes of one- and two-mode Gaussian states.We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs. In this problem, the aim is to minimize the probability of false negatives and the optimal performance is provided by the quantum Hoeffding bound. After a brief review of these notions, we show how this bound can be simplified for pure states. We then provide a general recipe for its computation in the case of multimode Gaussian states, also showing its connection with other easier-to-compute lower bounds. In particular, we provide analytical formulas and numerical results for important classes of one- and two-mode Gaussian states.

arXiv:1407.0884Physical Review ANov 6, 201432 citations

\mathcal{H}=L^{2}(\mathbb{R})

, which represents a single-mode bosonic system and is spanned by sets

\{|k\rangle\}_{k\in \mathbb{Z}_{\geq 0}}

of the k-th Hermitian functions

|k\rangle.

A

most typical example of a single-mode bosonic system is the one-mode photonic system, in which the vector

|k\rangle

is called the

k

-photon number state because it is regarded as the state corresponding to

k

photons.Then, the set

M_{N}=\{|k\rangle\langle k|\}_{k\in \mathbb{Z}>0}

, the number measurement, forms a positive-operator-valued measure (POVM).When the vector

|\xi^{-})\in L^{2}(\mathbb{R})(\xi\in \mathbb{C})

is defined by

| \xi):=e^{-\rfloor}\star^{2}\sum_{k=0}^{\infty}\frac{\xi^{k}}{\sqrt{k!}}|k\rangle,