A variant of the quantum marginal problem was known from early sixties as N-representability problem. In 1995 it was designated by National Research Council of USA as one of ten most prominent research challenges in quantum chemistry. In spite of this recognition the progress was very slow, until a couple of years ago the problem came into focus again, now in framework of quantum information theory. In the paper I give an account of the recent development.
A universal formulation of the quantum uncertainty regarding quantum indeterminacy, quantum measurement, and its inevitable observer effect is presented with additional focus on the representability of quantum observables over a given state. The operational tangibility of the framework assures that the resultant general relations admit natural operational interpretations and characterisations, and thereby perhaps most importantly, their experimental verifiability. In view of the universal formulation, Heisenberg's original philosophy of the uncertainty principle, most typically exemplified in …
Consider a projector matrix P, representing the first order reduced density matrix in a basis of orthonormal atom-centric basis functions. A mathematical question arises, and that is, how to break P into its natural component kernel projector matrices, while preserving N-representability of P. The answer relies upon 2- projector triple products, P'jPP'j. The triple product solutions, applicable within the quantum crystallography of large molecules, are determined by a new form of the Clinton equations, which - in their original form - have long been used to ensure N-representability of density…
Classical optimisation of reduced density matrix estimations with classical shadows using N-representability conditions under shot noise considerations
A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational tangibility of the framework, the resultant general relations admit natural operational interpretations and characterisations, and are thus also experimentally verifiable. In view of the universal formulation, Heisenberg's philosophy of the uncertainty principle is also revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in…
Linus Pauling contributions span structural biology, chemistry in its broadest definition, quantum mechanical theory, valence bond theory, and even nuclear physics. A principal tool developed and used by Pauling is Xray, and electron, diffraction. One possible extension of the Pauling oeuvre could be the marriage of crystallography and quantum mechanics. Such an effort dates back to the sixties and has now flourished into an entire subfield termed quantum crystallography. Quantum crystallography could be achieved through the application of Clinton equations to yield N-representable density mat…
Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability
我们最近提出了一个构造性的解决方案，以解决两电子约化密度矩阵（2-RDM）的N-可表示性问题--一个系统的方法来构造完备条件，以确保2-RDM表示一个现实的N-电子量子系统[D. A. Mazziotti，Phys. Rev. Lett. 108，263002（2012）]。在本文中，我们提供了更多的细节，并推导出进一步的N-可表示性条件的2-RDM，从建设性的解决方案。由此产生的条件可以分为一个层次的约束，被称为（2，q）-正性条件，其中q表示它们从q体算子的非负性导出。除了已知的T_1和T_2条件外，我们还得到了一类新的（2，3）-正性条件.我们还得到了3类（2，4）-正性条件，6类（2，5）-正性条件和24类（2，6）-正性条件.所得到的约束可以分为两种一般类型：（i）提升条件，即从较低的（2，q）-正性条件提升到较高的（2，q+1）-正性条件而产生的条件;（ii）纯条件，即不能从较低条件的简单提升导出的条件。所有的提升条件和q>3的纯（2，q）-正性条件都需要模型哈密顿量中系数的张量分解。新的N-可表示性条件的子集可以与先前已知的条件一起使用，以实现多电子量子系统的基态能量和2-RDM的多项式标度计算，即使在存在强电子相关的情况下。
It is virtually impossible to directly solve the Schrödinger equation for a many-electron wave function due to the exponential growth in degrees of freedom with increasing particle number. The two-body reduced density matrix (2-RDM) formalism reduces this coordinate dependence to that of four particles irrespective of the wave function's dimensionality, providing a promising path to solve the many-body problem. Unfortunately, errors arise in this approach because the 2-RDM cannot practically be constrained to guarantee that it corresponds to a valid wave function. Here we approach this so-call…
The quantum marginal problem is concerned with characterizing which collections of quantum states on different subsystems are compatible in the sense that they are the marginals of some multipartite quantum state. Presented here is a countable family of inequalities, each of which is necessarily satisfied by any compatible collection of quantum states. Additionally, this family of inequalities is shown to be sufficient: every incompatible collection of quantum states will violate at least one inequality belonging to the family.
Clarifying the relation between the whole and its parts is crucial for many problems in science. In quantum mechanics, this question manifests itself in the quantum marginal problem, which asks whether there is a global pure quantum state for some given marginals. This problem arises in many contexts, ranging from quantum chemistry to entanglement theory and quantum error correcting codes. In this paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are …
One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the…
Quantum machine learning offers a promising advantage in extracting information about quantum states, e.g. phase diagram. However, access to training labels is a major bottleneck for any supervised approach, preventing getting insights about new physics. In this Letter, using quantum convolutional neural networks, we overcome this limit by determining the phase diagram of a model where analytical solutions are lacking, by training only on marginal points of the phase diagram, where integrable models are represented. More specifically, we consider the axial next-nearest-neighbor Ising (ANNNI) H…
With the recent claim of a quantum advantage demonstration in photonics by Zhong et al, the question of the computation of lower-order approximations of boson sampling with arbitrary quantum states at arbitrary distinguishability has come to the fore. In this work, we present results in this direction, building on the results of Clifford and Clifford. In particular, we show: 1) How to compute marginal detection probabilities (i.e. probabilities of the detection of some but not all photons) for arbitrary quantum states. 2) Using the first result, how to generalize the sampling algorithm of Clif…
We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find …
Mixed States Uniquely Determined by Marginals and Additivity
LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University（教育部实验室（北京航空航天大学）及北京航空航天大学数学科学学院）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
