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arXiv:1806.10048 · Physical Review A

Discrimination between quantum common causes and quantum causality

Mingdi Hu, Yuexian Hou

Published Jun 26, 2018• 10 citations
arXivHTMLPDF

Abstract

In classic cases, Reichenbach's principle implies that discriminating between common causes and causality is unprincipled since the discriminative results essentially depend on the selection of possible conditional variables. For some typical quantum cases, K. Reid et al. [Nat. Phys. 11, 414 (2015)] presented the statistic $C$ which can effectively discriminate quantum common causes and quantum causality over two quantum random variables (i.e., qubits) and which only uses measurement information about these two variables. In this paper, we formalize general quantum common causes and general quantum causality. Based on the formal representation, we further investigate their decidability via the statistic $C$ in general quantum cases. We demonstrate that (i) $C\ensuremath{\in}\left[\ensuremath{-}1,\frac{1}{27}\right]$ if two qubits are influenced by quantum common causes; (ii) $C\ensuremath{\in}\left[\ensuremath{-}\frac{1}{27},1\right]$ if the relation between two qubits is quantum causality; (iii) a geometric picture can illuminate the geometric interpretation of the probabilistic mixture of quantum common causes and quantum causality. This geometric picture also provides a basic heuristic to develop more complete methods for discriminating the cases corresponding to $C\ensuremath{\in}\left[\ensuremath{-}\frac{1}{27},\frac{1}{27}\right]$. Our results demonstrate that quantum common causes and quantum causality can be discriminated in a considerable scope.

DOI
10.1103/physreva.97.062125

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In classic cases, Reichenbach's principle implies that discriminating between common causes and causality is unprincipled since the discriminative results essentially depend on the selection of possible conditional variables. For some typical quantum cases, K. Reid et al. [Nat. Phys. 11, 414 (2015)] presented the statistic CCC which can effectively discriminate quantum common causes and quantum causality over two quantum random variables (i.e., qubits) and which only uses measurement information about these two variables. In this paper, we formalize general quantum common causes and general quantum causality. Based on the formal representation, we further investigate their decidability via the statistic CCC in general quantum cases. We demonstrate that (i) C\ensuremath∈[\ensuremath−1,127]C\ensuremath{\in}\left[\ensuremath{-}1,\frac{1}{27}\right]C\ensuremath∈[\ensuremath−1,271​] if two qubits are influenced by quantum common causes; (ii) C\ensuremath∈[\ensuremath−127,1]C\ensuremath{\in}\left[\ensuremath{-}\frac{1}{27},1\right]C\ensuremath∈[\ensuremath−271​,1] if the relation between two qubits is quantum causality; (iii) a geometric picture can illuminate the geometric interpretation of the probabilistic mixture of quantum common causes and quantum causality. This geometric picture also provides a basic heuristic to develop more complete methods for discriminating the cases corresponding to C\ensuremath∈[\ensuremath−127,127]C\ensuremath{\in}\left[\ensuremath{-}\frac{1}{27},\frac{1}{27}\right]C\ensuremath∈[\ensuremath−271​,271​]. Our results demonstrate that quantum common causes and quantum causality can be discriminated in a considerable scope.
Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations.Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize.The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution.In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand.Motivated by this, we introduce a quantum counterpart to the principle.We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way.Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models and provide examples of how the formalism works.
It has long been recognized as a difficult problem to determine whether the observed statistical correlations between two classical variables arise from causality or from common causes. Recent research has shown that, in a quantum theoretical framework, the mechanisms of entanglement and quantum coherence provide advantages in tackling this problem. In some particular cases, quantum common causes and quantum causality can be effectively distinguished by using observations alone. However, these solutions do not apply to all cases. There still exist a large class of cases in which quantum common causes and quantum causality cannot be distinguished. In this paper, along the line of considering unitary transformation as causality in the quantum world, we formally show that quantum common causes and quantum causality are universally separable. Based on the analysis, we further provide a general method to discriminate the two.
Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations.Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize.The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution.In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand.Motivated by this, we introduce a quantum counterpart to the principle.We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way.Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models and provide examples of how the formalism works.
Quantum mechanics challenges our intuition on the cause-effect relations in nature. Some fundamental concepts, including Reichenbach's common cause principle or the notion of local realism, have to be reconsidered. Traditionally, this is witnessed by the violation of a Bell inequality. But are Bell inequalities the only signature of the incompatibility between quantum correlations and causality theory? Motivated by this question, we introduce a general framework able to estimate causal influences between two variables, without the need of interventions and irrespectively of the classical, quantum, or even postquantum nature of a common cause. In particular, by considering the simplest instrumental scenario-for which violation of Bell inequalities is not possible-we show that every pure bipartite entangled state violates the classical bounds on causal influence, thus, answering in negative to the posed question and opening a new venue to explore the role of causality within quantum theory.
We study quantum causal inference in a setup proposed by Ried et al (2015 Nat. Phys. 11 414) in which a common cause scenario can be mixed with a cause–effect scenario, and for which it was found that quantum mechanics can bring an advantage in distinguishing the two scenarios: whereas in classical statistics, interventions such as randomized trials are needed, a quantum observational scheme can be enough to detect the causal structure if the common cause results from a maximally entangled state. We analyze this setup in terms of the geometry of unital positive but not completely positive qubit-maps, arising from the mixture of qubit channels and steering maps. We find the range of mixing parameters that can generate given correlations, and prove a quantum advantage in a more general setup, allowing arbitrary unital channels and initial states with fully mixed reduced states. This is achieved by establishing new bounds on signed singular values of sums of matrices. Based on the geometry, we quantify and identify the origin of the quantum advantage depending on the observed correlations, and discuss how additional constraints can lead to a unique solution of the problem.
Abstract Reichenbach’s common cause principle (RCCP) is a metaphysical claim about the causal structure of the world. It entails that all correlations can be explained causally either by pointing at the causal connection between the correlated entities or by displaying a common cause of the correlation. We contribute to its mathematical side. Firstly, after adopting the RCCP axioms, we indicate the importance of positive covariance of events in connection with RCCP. In fact, RCCP is meaningful only for positively correlated events. Secondly, we find explicit requirements for a state to have a common cause in a quantum logic. Afterwards, we compare RCCP in the standard (Boolean) case and in the quantum setup that admits new properties of correlation, impossible in the classical case. We show that the notion of maximal correlation differs considerably in these two cases. We answer an open question by providing a counterexample based on a quantum logic given by a free orthomodular lattice. Subsequently, we find a relation between the Darboux property and RCCP. We invent a new technique for obtaining the central result that atomless σ-complete quantum logics are common cause complete. A variety of new examples is presented. Finally, we contribute to a fundamental question whether a common cause incomplete quantum logic can be embedded into one that is common cause complete. We contribute by an advanced construction that allows for a positive answer to this question for quantum logics with finitely many atoms and even for some quantum logics with countably many atoms.