S. Hamieh, R. Kobes, H. Zaraket
We study the problem of optimization over positive valued-operator measure to extract classical correlation in a bipartite quantum system. The proposed method is applied to binary states only. Moreover, to illustrate this method, an explicit example is studied in details.
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We compute the informational power for the Hoggar SIC-POVM in dimension 8, i.e. the classical capacity of a quantum-classical channel generated by this measurement. We show that the states constituting a maximally informative ensemble form a twin Hoggar SIC-POVM being the image of the original one under a conjugation.
Circuit optimization of qubit IC-POVMs for shadow estimation
本文针对随机测量阴影估计需频繁调整线路的问题，利用维度扩张框架优化单量子比特 IC-POVM/SIC-POVM 编译，将 CNOT 数分别降至至多 2 个和 1 个，并验证其抗噪性与灵活性。
The concepts of `conditional entropy' and `information' retain their validity for quantum systems, but their properties differ somewhat from those of their classical counterparts; specifically, some equalities and inequalities of classical information theory are in general violated. In this paper the concepts are generalized to include arbitrary indirect measurements (POVMs). Though the generalization is straightforward, it is important to ascertain that the basic relationships between the generalized quantitites remain the same for the POVMs as for direct measurements.
We present a classical protocol for simulating correlations obtained by bipartite POVMs on an EPR pair. The protocol uses shared random variables (also known as local hidden variables) augmented by six bits of expected communication.
Similarly to quantum states, also quantum measurements can be "mixed", corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are "indecomposable", i. e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each…
Exploiting overcompleteness of Platonic-solid POVMs for shadow estimation
该研究利用柏拉图立体POVM的超完备性，结合k局部最优对偶框架，通过经典代理态联合优化POVM取向与权重，发现其在分子哈密顿量估计中可优于标准随机泡利测量，为近期量子硬件的可观测量估计提供了改进途径。
A quantum measurement can be described by a set of matrices, one for each possible outcome, which represents the positive operator-valued measure (POVM) of the sensor. Efficient protocols of POVM extraction for arbitrary sensors are required. We present the first experimental POVM reconstruction that takes explicit advantage of a quantum resource, i.e., nonclassical correlations with an ancillary state. A POVM of a photon-number-resolving detector is reconstructed by using strong quantum correlations of twin beams generated by parametric down-conversion. Our reconstruction method is more stati…
It is a well-known fact that the optimal POVM for quantum state tomography is the symmetric, informationally complete, positive operator valued measure (SIC-POVM). We investigate the same problem only in the case when there are some a priori information about the state, specifically when some parameters are known. In this paper we mainly focus on solving a 3-dimensional optimization problem, which gives us a non-trivial example for the so-called conditional SIC- POVMs, a straightforward generalization of the concept of SIC-POVMs. We also present other special cases to show further applications…
Observations of quantum systems carried out by finite observers who subsequently communicate their results using classical data structures can be described as "local operations, classical communication" (LOCC) observations. The implementation of LOCC observations by the Hamiltonian dynamics prescribed by minimal quantum mechanics is investigated. It is shown that LOCC observations cannot be described using decoherence considerations alone, but rather require the \textit{a priori} stipulation of a positive operator-valued measure (POVM) about which communicating observers agree. It is also show…
Post-processing optimization and optimal bounds for non-adaptive shadow tomography
本文提出通过凸minimax问题优化后处理，以获得非适应影成像中固定POVM和可观察量下的最优方差界，并通过数值实验展示了其在减少采样复杂度和改善系统规模缩放方面的优势。
We propose a scheme to implement general quantum measurements, also known as Positive Operator Valued Measures (POVMs) in dimension $d$ using only classical resources and a single ancillary qubit. Our method is based on the probabilistic implementation of $d$-outcome measurements which is followed by postselection of some of the received outcomes. We conjecture that the success probability of our scheme is larger than a constant independent of $d$ for all POVMs in dimension $d$. Crucially, this conjecture implies the possibility of realizing arbitrary nonadaptive quantum measurement protocol o…
We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits o…
The expectation value <O> of an arbitrary operator O can be obtained via a universal measuring apparatus that is independent of O, by changing only the data-processing of the outcomes. Such a ``universal detector'' performs a joint measurement on the system and on a suitable ancilla prepared in a fixed state, and is equivalent to a positive operator valued measure (POVM) for the system that is ``informationally complete''. The data processing functions generally are not unique, and we pose the problem of their optimization, providing some examples for covariant POVM's, in particular for SU(d) …
SeCQC is an open-source program code which implements a Numerical Search for the classical Capacity of Quantum Channels (SeCQC) by using an iterative method. Given a quantum channel, SeCQC finds the statistical operators and POVM outcomes that maximize the accessible information, and thus determines the classical capacity of the quantum channel. The optimization procedure is realized by using a steepest-ascent method that fol- lows the gradient in the POVM space, and also uses conjugate gradients for speed-up.
In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements, or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory. This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes. We show that on $\mathbf{C}^d$, the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM, the latter having only $O(d…
In the present paper, an exact analytic solution for the optimal unambiguous state discrimination (OPUSD) problem involving an arbitrary number of pure linearly independent quantum states with real and complex inner product is presented. Using semidefinite programming and Karush-Kuhn-Tucker convex optimization method, we derive an analytical formula which shows the relation between optimal solution of unambiguous state discrimination problem and an arbitrary number of pure linearly independent quantum states.
We discuss the problem of implementing generalized measurements (POVMs) with linear optics, either based upon a static linear array or including conditional dynamics. In our approach, a given POVM shall be identified as a solution to an optimization problem for a chosen cost function. We formulate a general principle: the implementation is only possible if a linear-optics circuit exists for which the quantum mechanical optimum (minimum) is still attainable after dephasing the corresponding quantum states. The general principle enables us, for instance, to derive a set of necessary conditions f…
We study the nature of the information preserved by a quantum channel via the observables which exist in its image (in the Heisenberg picture), and can therefore be simulated on the receiver's side. The sharp observables preserved by a channel form an operator algebra which can be characterized in terms of the channel's elements. The effect of the channel on these observables can be reversed by another physical transformation. These results generalize the theory of quantum error correction to codes characterized by arbitrary von Neumann algebras, which can represent hybrid quantum-classical in…
On the sampling complexity of coherent superpositions
研究对纯态叠加施加POVM的采样问题，提出一种将强经典模拟归约为弱模拟的算法，适用于连续结果POVM，并给出了高斯态叠加下的时间复杂度。
In this study for particular states of bipartite quantum system in 2n?2m dimensional Hilbert space state, similar to m or n-qubit density matrices represented in Bloch sphere we call them generalized Bloch sphere states(GBSS), we give an efficient optimization procedure so that analytic evaluation of quantum discord can be performed. Using this optimization procedure, we find an exact analytical formula for the optimum positive operator valued measure (POVM) that maximize the measure of the classical correlation for these states. The presented optimization procedure also is used to show that f…
We present the first complete optimization of quantum tomography, for states, POVMs, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes.
Quantifying nonclassicality in qubit systems via positive operator-valued measures
该研究提出基于POVM不锐利度参数的量子比特非经典性度量，推导其显式表达式，最大值为1/4，关联Kolmogorov一致性等，为量子技术认证提供实验工具。
We report an optical detector with tunable positive operator-valued measures (POVMs). The device is based on a combination of weak-field homodyne techniques and photon-number-resolving detection. The resulting POVMs can be continuously tuned from Fock-state projectors to a variety of phase-dependent quantum-state measurements by adjusting different system parameters such as local oscillator coupling, amplitude and phase, allowing thus not only detection but also preparation of exotic quantum states. Experimental tomographic reconstructions of classical benchmark states are presented as a demon…
The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutati…
Computable lower bound of the parameterized entanglement monotone
本文研究了两种纠缠单调子（q-纠缠度和α-纠缠度）的下界，利用信息完备的(N,M)-正算子值测量方法，为双量子位态和双量子位态提供了下界计算，并展示了基于(N,M)-POVM的下界优于GSIC-POVM和SIC-POVM，且优于文献中基于正部分转置和重排准则的下界。
We study dynamical correlations of two coupled large spins depending on the time and on the spin quantum numbers. In the high-temperature approximation, we obtain analytical expressions for the mutual informations, quantum and classical parts of correlations. The latter was obtained performing the non-orthogonal projective (POVM) measurements onto the spin coherent states of two spins or one spin, as well as by means of the orthogonal projective measurement of von Neumann. Contribution from quantum correlations is much less than that from classical ones and decreasing with the increase in spin…
Intersubjectivity as a principle determining physical observables and non-classicality
本文通过量化主体间性条件，证明投影值测度（PVM）是唯一满足所有粗粒化操作下主体间性的正算子值测度（POVM），并刻画了经典理论，从而确立了物理可观测量和经典性的完整特征。
Suppose we want to distinguish two quantum pure states. We consider the case in which no classical knowledge on the two states is given and only a pair of samples of the two states is available. This problem is called quantum pure-state identification problem. Our task is to optimize the mean identification success probability, which is averaged over an independent unitary invariant distribution of the two reference states. In this paper, the two states are assumed bipartite states which are generally entangled. The question is whether the maximum mean identification success probability can be…
The results of space-like separated measurements are independent of distant measurement settings, a property one might call two-way no-signalling. In contrast, time-like separated measurements are only one-way no-signalling since the past is independent of the future but not vice-versa. For this reason temporal correlations that are formally identical to non-classical spatial correlations can still be modelled classically. We define non-classical temporal correlations as the ones which cannot be simulated by propagating in time a classical information content of a quantum system. We first show…
Generalized measurement schemes on one part of bipartite states, which would leave the set of all separable states insensitive are explored here to understand quantumness of correlations in a more general perspecitve. This is done by employing linear maps associated with generalized projective measurements. A generalized measurement corresponds to a quantum operation mapping a density matrix to another density matrix, preserving its positivity, hermiticity and traceclass. The Positive Operator Valued Measure (POVM) -- employed earlier in the literature to optimize the measures of classical/qua…
Recently foundational issues of applicability of the formalism of quantum mechanics (QM) to cognitive psychology, decision making, and psychophysics attracted a lot of interest. In particular, in \cite{DKBB} the possibility to use of the projection postulate and representation of "mental observables" by Hermitian operators was discussed in very detail. The main conclusion of the recent discussions on the foundations of "quantum(-like) cognitive psychology" is that one has to be careful in determination of conditions of applicability of the projection postulate as a mathematical tool for descri…
Optimization of the mean efficiency for unambiguous (or error free)discrimination among $N$ given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit $N$-sphere for equiprobable states and on an $N$-ellipsoid if the states are given with different probabil…
We study the problem of separating the data produced by a given quantum measurement (on states from a memoryless source which is unknown except for its average state), described by a positive operator valued measure (POVM), into a "meaningful" (intrinsic) and a "not meaningful" (extrinsic) part. We are able to give an asymptotically tight separation of this form, with the "intrinsic" data quantfied by the Holevo mutual information of a certain state ensemble associated to the POVM and the source, in a model that can be viewed as the asymptotic version of the convex decomposition of POVMs into …
Enhanced observable estimation through classical optimization of informationally over-complete measurement data -- beyond classical shadows
本文提出在信息过完备测量完成后优化对偶POVM算子，以在经典阴影需指数级开销时实现零方差可观测量估计，并在随机泡利测量量子模拟中显著降低统计误差。
Dual frame optimization for informationally complete quantum measurements
本文利用测量对偶框架推广经典影子后处理，提出参数化框架超算子和基于经验频率的免优化对偶框架，以几乎零额外代价改进信息完备测量的可观测量估计。
The free-induction decay (FID) of the transverse magnetization in a dipolar-coupled rigid lattice is a fundamental problem in magnetic resonance and in the theory of many-body systems. As it was shown earlier the FID shapes for the systems of classical magnetic moments and for quantum nuclear spins ones coincide if there are many quite equivalent nearest neighbors V in a solid lattice. In this paper, we reduce a multispin density matrix of above system to a two-spin matrix. Then we obtain analytic expressions for the mutual information and the quantum and classical parts of correlations at the…
This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator $ρ_0$. It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator $Π_0$, and maximizing $\textrm{Tr}(ρ_0 Π_0)$. In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a sem…
Quantum One-Time Memories from Stateless Hardware, Random Access Codes, and Simple Nonconvex Optimization
本文利用可经典访问的无状态硬件与 QRAC 构建 OTM，并通过 $\mathbb{C}^2$ 上 POVM 的非凸优化界定比特区分概率，证明其可抵抗多项式次经典查询。
The space of positive operator-valued measures on the Borel sets of a compact (or even locally compact) Hausdorff space with values in the algebra of linear operators acting on a d-dimensional Hilbert space is studied from the perspectives of classical and non-classical convexity through a transform $Γ$ that associates any positive operator-valued measure with a certain completely positive linear map of the homogeneous C*-algebra $C(X)\otimes B(H)$ into $B(H)$. This association is achieved by using an operator-valued integral in which non-classical random variables (that is, operator-valued fu…
Following initial work by Gregoratti and Werner [J. Mod. Optics 50, 913-933, 2003 and quant-ph/0403092] and Hayden and King [quant-ph/0409026], we study the problem of the capacity of a quantum channel assisted by a "friendly (channel) environment" that can locally measure and communicate classical messages to the receiver. Previous work [quant-ph/0505038] has yielded a capacity formula for the quantum capacity under this kind of help from the environment. Here we study the problem of the environment-assisted classical capacity, which exhibits a somewhat richer structure (at least, it seems to…
Hilbert space-valued Gaussian processes, and quantum states
本文研究了算子值核及其因子分解，提出了更明确的实现与新应用，包括 Hilbert 空间值高斯过程的协方差分析、量子门优化、POVMs 新结果及量子测量反问题的新方法。
Feedback-based quantum optimization and its classical counterpart: quantum advantage and the power of classical algorithms
本文比较了量子反馈优化算法及其经典对应，发现量子算法在解的质量上占优，而经典算法收敛更快，并展示了经典算法在高阶无约束二元优化中的可扩展性。
When Classical Correlations Certify Entanglement Recovery
本文通过引入比互补性更宽松的不可约性条件，建立了纠缠与态可区分性之间的定量联系，并应用于纠缠蒸馏、量子纠错及测量权衡，提出了统一框架。
Simultaneous Decoding of Classical Coset Codes over 3-User Quantum Interference Channel : New Achievable Rate Regions
研究三用户经典-量子干扰信道上的比特流通信，提出基于余集码和POVM解码的新策略，推导出新的可行速率区域上界，涵盖现有所有上界并证明在特定例子中严格更大。
From Classical to Quantum: Polymorphisms in Non-Commutative Probability
本文建立经典（标量值）与非交换（算子值）多态性之间的平行关系，重点应用于随机过程的条件分布，并借助正算子值测度（POVM）与量子态，阐明量子信息中的纠缠，揭示两种框架的相互作用。
We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group $G$ with its dual $\hat G$. The method is based upon the Pontryagin duality allowing to establish an isometrical isomorphism between the space of Hilbert-Schmidt operators in $L^2(G)$ and the Hilbert space $L^2(\hat G\times G)$. Any such a measure determines a pair of hybrid (containing classical and quantum parts) quantum channels consisting of the measurement channel and the channel transmitting an initial quantum state to the ensemble o…
Grover's algorithm can be employed in global optimization methods providing, in some cases, a quadratic speedup over classical algorithms. This paper describes a new method for continuous global optimization problems that uses a classical algorithm for finding a local minimum and Grover's algorithm to escape from this local minimum. Simulations with testbed functions and comparisons with algorithms from the literature are presented.
Non-classical optimization of entangled photons through complex media
提出复杂介质中的光学非经典优化概念，通过直接优化输出符合率控制纠缠光子对穿过散射介质，并证明其解不同于经典优化且可用于模拟多自旋相互作用自旋玻璃模型。
The recent literature on near-term applications for quantum computers contains several examples of the applications of hybrid quantum/classical variational approaches. This methodology can be applied to a variety of optimization problems, but its practical performance is not well studied yet. This paper moves some steps in the direction of characterizing the practical performance of the methodology, in the context of finding solutions to classical combinatorial optimization problems. Our study is based on numerical results obtained applying several classical nonlinear optimization algorithms t…
Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving
本文提出受量子启发的经典优化算法CACAO，利用局部反绝热驱动的量子Lyapunov控制与经典自旋动力学启发式求解组合优化，并与量子退火、FALQON和CD-FQA比较，验证其在万自旋系统上的表现。
Linear-time classical approximate optimization of cubic-lattice classical spin glasses
本文研究了立方晶格经典自旋玻璃的近似优化问题，证明线性时间经典元启发式方法可作为量子加速的上界，并鼓励开发更高效的启发式算法。
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