Incompatible observables underlie pillars of quantum physics such as contextuality and entanglement. The Heisenberg uncertainty principle is a fundamental limitation on the measurement of the product of incompatible observables, a 'joint' measurement. However, recently a method using weak measurement has experimentally demonstrated joint measurement. This method [Lundeen, J. S., and Bamber, C. Phys. Rev. Lett. 108, 070402, 2012] delivers the standard expectation value of the product of observables, even if they are incompatible. A drawback of this method is that it requires coupling each observable to a distinct degree of freedom (DOF), i.e., a disjoint Hilbert space. Typically, this 'read-out' system is an unused internal DOF of the measured particle. Unfortunately, one quickly runs out of internal DOFs, which limits the number of observables and types of measurements one can make. To address this limitation, we propose and experimentally demonstrate a technique to perform a joint weak-measurement of two incompatible observables using only one DOF as a read-out system. We apply our scheme to directly measure the density matrix of photon polarization states.

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资源高效联合弱测量的理论与实验

不相容的可观察量是量子物理学支柱的基础，例如上下文和纠缠。海森堡测不准原理是对不相容的可观测量的乘积测量的基本限制，即“联合”测量。然而，最近一种使用弱测量的方法已经通过实验证明了联合测量。这种方法 [Lundeen, JS, and Bamber, C. Phys. 牧师莱特。108, 070402, 2012] 提供可观察的乘积的标准期望值，即使它们不兼容。这种方法的一个缺点是它需要将每个可观测量耦合到不同的自由度 (DOF)，即不相交的希尔伯特空间。通常，这个“读出”系统是被测粒子未使用的内部自由度。不幸的是，很快就会耗尽内部自由度，这限制了一个人可以进行的可观察的数量和测量类型。为了解决这一限制，我们提出并通过实验证明了一种技术，该技术仅使用一个 DOF 作为读出系统来对两个不兼容的可观测对象进行联合弱测量。我们应用我们的方案来直接测量光子偏振态的密度矩阵。