Gaussian Hypothesis Testing and Quantum Illumination

Abstract

Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.

Quantum Estimation Methods for Quantum Illumination

“…The aim of using nonclassical resources is to find a better convergence rate for the error probability Pr err , or to minimize P II by keeping bounded P I (see Refs. [25,26] for this analysis). In the following, we show that both types of error decay faster if an entangled state and the optimal measurement given by the QFI are used.…”

“…Here, from (25) to (26) we have used that the two terms of the commutator contributes positively to G 1 2k+1 . From (26) to (27) we have used the inequality of the arithmetic and geometric means ≤ 1+N B N B , the completeness relation, and then we have consider the antinormal ordering of the 2k+1 k terms contributing to the sum, in the same way as in the previous subsection.…”

Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

“…This allows us to adapt recently developed tools for approximate entanglement transformations in Ref. [20] to study corrections to the asymptotic rates of thermodynamic transformations, the so-called secondorder asymptotics (see, e.g., Refs [21][22][23][24][25][26][27][28] for other recent studies of second-order asymptotics in quantum information). The crucial technical difference between Ref.…”

Quantum illumination assistant with error-correcting codes

“…One of the major applications to enhance the ability of target recognition is quantum illumination [4][5][6][7][8][9], which is the most known protocol for bosonic quantum sensing [10]. Quantum illumination provides us with a potential platform to detect the low-reflectivity object embedded in a bright environment, and it is more efficiently than the way by using classical resources [5,11,12]. Since the pioneering work proposed by Lloyd [4] and its Gaussian version [6,9], many experimental and theoretical schemes have been proposed [13,14], such as quantum illumination in composite optomechanical system [5,15], discrete variable quantum illumination with ancillary degrees of freedom [16], quantum illumination unveils cloaking [14], and quantum illumination based on asymmetric hypothesis testing [17,18].…”