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 or both entangled. So considering copulas instead of states is a separability preserving transformation and efficiently decreases the dimension of the state space of composite systems. The method how we prove these results draws attention to the fact that theorem for states can be achieved by considering the states as quantum channels and using theorems for channels and different kind of positive maps.
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 or both entangled. So considering copulas instead of states is a separability preserving transformation and efficiently decreases the dimension of the state space of composite systems. The method how we prove these results draws attention to the fact that theorem for states can be achieved by considering the states as quantum channels and using theorems for channels and different kind of positive maps.
Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.
Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this paper, we show that any copula can be naturally mapped to a multipartite maximally entangled state. Thus, the task of learning joint probability distributions becomes the task of learning maximally entangled states. We prove that a variational ansatz we christen as a ``qopula'' based on this insight leads to an exponential advantage over classical methods of learning some joint distributions. As an application, we train a quantum generative adversarial network (QGAN) and a quantum circuit Born machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables in historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to eight qubits. Our experimental results show interesting findings such as the resilience against noise, outperformance against equivalent classical models and 20--1000 times less iterations required to converge as compared to equivalent classical models.
Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.
Abstract We study emerging notions of quantum correlations in compound systems. Based on different definitions of quantumness in individual subsystems, we investigate how they extend to the joint description of a composite system. Especially, we study the bipartite case and the connection of two typically applied and distinctively different concepts of nonclassicality in quantum optics and quantum information. Our investigation includes the representation of correlated states in terms of quasiprobability matrices, a comparative study of joint and conditional quantum correlations, and an entanglement characterization. It is, for example, shown that our composition approach always includes entanglement as one form of quantum correlations. Yet, other forms of quantum correlations can also occur without entanglement. Finally, we give an outlook towards multimode systems and temporal correlations.
Multivariate distribution modeling is a crucial process employed to capture dependencies in experimental data in areas like physics, chemistry, finance, and climate science. Copula distributions are a special class of multivariate distributions that have uniform marginals and have been studied extensively to model non-trivial dependencies in high-dimensional data. In the last decade, various ways of applying machine learning techniques to model copulas have been proposed to improve the pre-existing parametric models. Recently, with the advent of uniting machine learning and quantum computing, quantum generative models have been used to generate various data objects like images, probability distributions, natural language data, etc, using hybrid models that have both classical and quantum components. In this paper, we build on the recent work on utilizing quantum circuits to generate copulas and aim to provide a near-holistic study of how different quantum generative models can be implemented and tuned to capture correlations in real-world finance data.