研究了具有固定边缘态的所有二分量子态的凸集。最近，Parthasarathy [Ann. Henri Poincaré（即将出现），quant-ph/0307182，[1]]描述了这个集合中的极端状态。在这里，我们提出了一个替代的必要和充分条件，一个国家与给定的边缘是极值。我们的方法是基于一个典型的二分态和一定类的完全正映射之间的对偶，并具有的优点是，它更容易检查和构造显式的极端状态的例子。在2 × 2维中，我们给出了一个简单的新证明，证明了所有具有最大混合边缘的极端态都是最大纠缠波函数的投影。我们还证明，在更高的维度，这并不成立，并构建一个明确的例子，极端状态的最大混合边缘在3 × 3的维度，这是不是最大纠缠。这一结果的推广到高维进行了讨论。
英文摘要：
We study the convex set of all bipartite quantum states with fixed marginal states. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quant-ph/0307182, [1]]. Here we present an alternative necessary and sufficient condition for a state with given marginals to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension 2 x 2 we give a simple new proof for the fact that all extremal states with maximally mixed marginals are precisely the projectors onto maximally entangled wave functions. We also prove that in higher dimension this does not hold and construct an explicit example of an extremal state with maximally mixed marginals in dimension 3 x 3 that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.
An extremal element of the convex set of composite quantum states in $M_2\otimes M_3$, whose marginals are all normalised identities has been constructed. It is found to be a mixed state and is entangled as well.
We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extremal point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extremal, then the whole map is extremal. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also…
We characterize the optimal correlative capacity of entangled, separable, and classically correlated states. Introducing the notions of the infimum and supremum within majorization theory, we construct the least disordered separable state compatible with a set of marginals. The maximum separable correlation information supportable by the marginals of a multi-qubit pure state is shown to be an LOCC monotone. The least disordered composite of a pair of qubits is found for the above classes, with classically correlated states defined as diagonal in the product of marginal bases.
We report here on the results of numerical searches for PPT states with specified ranks for density matrices and their partial transpose. The study includes several bipartite quantum systems of low dimensions. For a series of ranks extremal PPT states are found. The results are listed in tables and charted in diagrams. Comparison of the results for systems of different dimensions reveal several regularities. We discuss lower and upper bounds on the ranks of extremal PPT states.
Consider a qubit-qutrit ($2 \times 3$) composite state space. Let $C(\{1}{2}I_2, \{1}{3}I_3)$ be a convex set of all possible states of composite system whose marginals are given by $\{1}{2}I_2$ and $\{1}{3}I_3$ in two and three dimensional spaces respectively. We prove that there exists no pure state in $C(\{1}{2}I_2, \{1}{3}I_3)$. Further we generalize this result to an arbitrary $m \times n$ bipartite systems. We prove that for $m < n$, no pure state exists in the convex set $C(ρ_A,ρ_B)$, for an arbitrary $ρ_A$ and rank of $ρ_B >m$.
Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic a…
Quantum Fisher information, as an intrinsic quantity for quantum states, is a central concept in quantum detection and estimation. When quantum measurements are performed on quantum states, classical probability distributions arise, which in turn lead to classical Fisher information. In this article, we exploit the classical Fisher information induced by quantum measurements, and reveal a rich hierarchical structure of such measurement-induced Fisher information. We establish a general framework for the distribution and transfer of the Fisher information. In particular, we illustrate three ext…
Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
We discuss the problem of characterizing upper bounds on entanglement in a bipartite quantum system when only the reduced density matrices (marginals) are known. In particular, starting from the known two-qubit case, we propose a family of candidates for maximally entangled mixed states with respect to fixed marginals for two qudits. Interestingly, it turns out such states are always quasidistillable. Moreover, they are extremal in the convex set of two qudit states with fixed marginals. Our observations are supported by numerical analysis.
We investigate the entanglement, CHSH nonlocality, fully entangled fraction and symmetric extendibility of two-qubit states that have a single maximally mixed marginal. Within this set of states, the steering ellipsoid formalism has recently highlighted an interesting family of so-called 'maximally obese' states. These are found to have extremal quantum correlation properties that are significant in the steering ellipsoid picture and for the study of two-qubit states in general.
We study state engineering through bilinear interactions between two remote qubits and two-mode Gaussian light fields. The attainable two-qubit states span the entire physically allowed region in the entanglement-versus-global-purity plane. Two-mode Gaussian states with maximal entanglement at fixed global and marginal entropies produce maximally entangled two-qubit states in the corresponding entropic diagram. We show that a small set of parameters characterizing extremally entangled two-mode Gaussian states is sufficient to control the engineering of extremally entangled two-qubit states, wh…
Working with multivariate probability distributions Sklar introduced the notion of copula in 1959, which turned out to be a key concept to understand the structure of distributions of composite systems. Roughly speaking Sklar proved that a joint distribution can be represented with its marginals and a copula. The main goal of this paper is to present a quantum analogue of the notion of copula. Our main theorem states that for any state of a composite quantum system there exists a unique copula such that they are connected to each other by invertible matrices, moreover, they are both separable …
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-state quantum mechanics, with the exception of the quaternionic "bit". Another is similar, except in that (i) it excludes the quaternionic bit, and (ii) the composite of two complex quantum systems comes wit…
In any probabilistic theory, we may say a bipartite state on a composite system AB steers its marginal state (on, say, system B) if, for any decomposition of the marginal as a mixture, with probabilities p_i, of states b_i of B, there exists an observable a_i on A such that the states of B conditional on getting outcome a_i on A, are exactly the states b_i, and the probabilities of outcomes a_i are p_i. This is always so for pure bipartite states in quantum mechanics, a fact first observed by Schroedinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but per…
Information-theoretic ideas have provided numerous insights in the progress of fundamental physics, especially in our pursuit of quantum gravity. In particular, the holographic entanglement entropy is a very useful tool in studying AdS/CFT, and its efficacy is manifested in the recent black hole page curve calculation. On the other hand, the one-shot information-theoretic entropies, such as the smooth min/max-entropies, are less discussed in AdS/CFT. They are however more fundamental entropy measures from the quantum information perspective and should also play pivotal roles in holography. We …
