Quantum Gaussian states can be considered as the majority of the practical quantum states used in quantum communications and more generally in quantum information. Here we consider their properties in relation to the geometrically uniform symmetry, a property of quantum states that greatly simplifies the derivation of the optimal decision by means of the square root measurements. In a general framework of the $N$-mode Gaussian states we show the general properties of this symmetry and the application of the optimal quantum measurements. An application example is presented to quantum communication systems employing pulse position modulation. We prove that the geometrically uniform symmetry can be applied to the general class of multimode Gaussian states.
Quantum Gaussian states can be considered as the majority of the practical quantum states used in quantum communications and more generally in quantum information. Here we consider their properties in relation to the geometrically uniform symmetry, a property of quantum states that greatly simplifies the derivation of the optimal decision by means of the square root measurements. In a general framework of the N-mode Gaussian states we show the general properties of this symmetry and the application of the optimal quantum measurements. An application example is presented to quantum communication systems employing pulse position modulation. We prove that the geometrically uniform symmetry can be applied to the general class of multimode Gaussian states.
Distinguishing assigned quantum states with assigned probabilities via quantum measurements is a crucial problem for the transmission of classical information through quantum channels. Measurement operators maximizing the probability of correct discrimination have been characterized by Helstrom, Holevo, and Yuen since the 1970s. On the other hand, closed-form solutions are available only for particular situations enjoying high degrees of symmetry. As a suboptimal solution to the problem, measurement operators, directly determined from states and probabilities and known as square-root measurements (SRMs), were introduced by Hausladen and Wootters. These operators were also recognized to be optimal for pure states equipped with geometrical uniform symmetry (GUS). In this paper we discuss the optimality of the SRM and find necessary and sufficient conditions in order that SRM maximize the correct decision probabilities for the set of states formed by several constellations of GUS states. The results are applied to some specific examples concerning double constellations of quantum phase-shift keying and pulse-position modulation states, with possible applications to practical systems of quantum communications.
Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.
Minimum probability of error (MPE) measurements are quantum mechanical measurements that discriminate between a set of candidate states and achieve the minimum error allowed by quantum mechanics. Conditions for a measurement to be an MPE measurement have been derived by Yuen, Kennedy and Lax. But finding explicit measurements that satisfy these conditions is a hard problem in general and even in cases where an explicit measurement is known, calculating the MPE of the measurement is not easy. In some cases, MPE measurements have been found such as when the states form a single orbit under a group action i.e., there is a transitive group action on the states. For such state sets, termed Geometrically Uniform (GU) in [1], it was shown that the `pretty good measurement' (PGM) is optimal. However, calculating the MPE and other performance metrics for the PGM involves inverting large matrices, and it is therefore not easy in general to evaluate. Our first contribution is a formula for the MPE and conditional probabilities of GU sets using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). CGU sets appear in many practical problems in quantum communication and imaging. For example, they include all linear codes formed using pure-state modulation constellations, which are known to achieve the ultimate (Holevo) capacity of optical communication. Optimal (MPE) measurement for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the optimal measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs-which is what is needed to solve for MPE measurements for an arbitrary set of pure states using the Yuen-Kennedy-Lax conditions [2]. We show examples of the evaluation of optimal measurements for CGU states.
Quantum fidelity is a measure to quantify the closeness between two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positive-operator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance both for fundamental interest and practical purposes. We find here the optimal POVMs for quantum fidelity between multimode Gaussian states in a closed analytical form. Our general finding is applied for selected single-mode Gaussian states of particular interest and we identify three types of optimal measurements: an excitation-number-resolving detection, a projection onto the eigenbasis of operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}+\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$, and a quadrature variable detection, each of which corresponds to distinct types of single-mode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation when two arbitrary states are infinitesimally close. It is applied for simplifying the derivations of quantum Fisher information and the associated optimal measurements, exemplified by displacement, phase, squeezing, and loss parameter estimation using Gaussian states.
We determine the optimal method of discriminating and comparing quantum states from a certain class of multimode Gaussian states and their mixtures when arbitrary global Gaussian operations and general Gaussian measurements are allowed. We consider the so-called constant-$\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ displaced states, which include mixtures of multimode coherent states arbitrarily displaced along a common axis. We first show that no global or local Gaussian transformations or generalized Gaussian measurements can lead to a better discrimination method than simple homodyne measurements applied to each mode separately and classical postprocessing of the results. This result is applied to binary state comparison problems. We show that homodyne measurements, separately performed on each mode, are the best Gaussian measurement for binary state comparison. We further compare the performance of the optimal Gaussian strategy for binary coherent states comparison with these of non-Gaussian strategies using photon detections.
We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.