PROBABILISTIC THEORIES: WHAT IS SPECIAL ABOUT QUANTUM MECHANICS?
PROBABILISTIC THEORIES: WHAT IS SPECIAL ABOUT QUANTUM MECHANICS?
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概率理论：量子力学有什么特别之处？
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发表时间：
2009
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通讯作者：
G. Mauro
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作者：
G. Mauro
量子力学（QM）是一种非常特殊的概率理论，但我们不知道是什么操作原理使它如此。所有公理化的尝试都至少有一个数学性质的假设。在这里，我将分析导出QM作为公平操作框架的数学表示的可能性，即一组规则，允许实验者在合适的测试的基础上对未来事件进行预测，例如，没有来自不可控来源的干扰。任何公平的操作框架都需要满足两个假设：NSF：没有来自未来的信号--可以根据过去的测试进行预测; PFAITH：存在一个准备忠实的状态--可以准备任何状态并校准任何测试。我将证明，所有满足NSF的理论都承认事件的C-代数表示为效应的线性变换。基于动态独立性的非常一般的概念，很容易看出所有这样的概率理论都是无交互的非信令（简称非信令）-公平操作框架的另一个要求。假设PFAITH然后意味着当地的可观测性原则，张量积结构的线性空间的状态和效果，不可能的位承诺和额外的功能，这样一个操作定义的转置，标量产品的影响，弱自对偶的理论，等等。PFAITH假设的对偶假设一个类似的效应假设会给出额外的量子特征，比如隐形传态。然而，所有可能的后果，这些假设仍然需要调查，这是不清楚的，如果我们可以推导出质量管理从目前的假设。QM的特殊之处在于，效应也构成了一个C-代数。更确切地说，这对所有量子-经典混合理论都是正确的，对应于QM加超选择规则。然而，尽管效果的总和可以在操作上加以定义，但效果的概念却排斥任何种类的组合。基于演化的原子性（AE）的自然假设，通过Choi-Jamiolkowski同构（CJ）将效应与原子事件相鉴别，可以定义效应的合成。以这种方式，量子-经典混合体在非信号概率理论的大竞技场中被选择，包括Popescu-Rohrlich盒。CJ同构在操作上下文中看起来很自然，并且希望它将很快转换为操作原则。
Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. All axiomatization attempts suffer at least one postulate of a mathematical nature. Here I will analyze the possibility of deriving QM as the mathematical representation of a fair operational framework, i.e. a set of rules which allows the experimenter to make predictions on future events on the basis of suitable tests, e.g. without interference from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: no-signaling from the future—for the possibility of making predictions on the basis of past tests; PFAITH: ex- istence of a preparationally faithful state—for the possibility of preparing any state and calibrating any test. I will show that all theories satisfying NSF ad- mit a C ∗ -algebra representation of events as linear transformations of effects. Based on a very general notion of dynamical independence, it is easy to see that all such probabilistic theories are non-signaling without interaction (non- signaling for short)—another requirement for a fair operational framework. Postulate PFAITH then implies the local observability principle, the tensor- product structure for the linear spaces of states and effects, the impossibility of bit commitment and additional features, such an operational definition of transpose, a scalar product for effects, weak-selfduality of the theory, and more. Dual to Postulate PFAITH an analogous postulate for effects would give additional quantum features, such as teleportation. However, all possible consequences of these postulates still need to be investigated, and it is not clear yet if we can derive QM from the present postulates only. What is special about QM is that also effects make a C ∗ -algebra. More precisely, this is true for all hybrid quantum-classical theories, corresponding to QM plus super-selection rules. However, whereas the sum of effects can be operationally defined, the notion of effect abhors any kind of composition. Based on the natural postulate of atomicity of evolution (AE) one can de- fine composition of effects by identifying them with atomic events through the Choi-Jamiolkowski isomorphism (CJ). In this way the quantum-classical hy- brid is selected within the large arena of non-signaling probabilistic theories, including the Popescu-Rohrlich boxes. The CJ isomorphism looks natural in an operational context, and it is hoped that it will soon be converted into an operational principle.