Experimental test of error-disturbance uncertainty relation with continuous variables

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The uncertainty relation is one of the fundamental principles in quantum mechanics and plays an important role in quantum information science. We experimentally test the error-disturbance uncertainty relation (EDR) with continuous variables for Gaussian states. Two incompatible continuous-variable observables, amplitude and phase quadratures of an optical mode, are measured simultaneously using a heterodyne measurement system. The EDR values with continuous variables for coherent, squeezed, and thermal states are verified experimentally. Our experimental results demonstrate that Heisenberg’s EDR with continuous variables is violated, while Ozawa’s and Branciard’s EDRs with continuous variables are validated.