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I Finite-dimensional quantum systems are represented by vectors in a Hilbert space |ψ〉 I Quantum states have continuous-time dynamics given by the Schrödinger equation i~∂t|ψ 〉 = H|ψ〉 where H is a Hamiltonian I General quantum measurements can disturb quantum states discontinuously and are defined by a set of measurement operators {Mk} with∑ kM kMk = I and |ψ 〉 prob. pk− → Mk|ψ〉 pk where pk = 〈ψ|M†kMk|ψ〉 Why study continuous quantum measurements? Theoretical consideration Experimental considerations Quantum dynamics are reversible, deterministic and continuous. However, quantum measurements are irreversible, non-deterministic, and discontinuous. Can we describe measurements continuously but allow for irreversible, non-deterministic outcomes? Many quantum mechanical systems either have naturally slow measurement times, or can only be probed weakly. For example: microwave cavities [BHL+90], homo- and heterodyne measurements in quantum optics [YIM86, SSH+87] Continuous decompositions as random walks In [OB05] it was shown that any quantum measurement {M1,M2} can be decomposed into a 1-dimensional continuous stochastic process In this scheme successive weak measurements (steps) M±(x) are applied at each time-step. These step operators are a function of the running total of measurement outcomes, the pointer variable x. This process can be seen as a 1-d random walk on a curve in operator space. The total walk operator M(x) describes the evolution which terminates at the desired operators M1 or M2. Total walk operator M(x) ∝ lim δ→0 b|x|/δc∏ j=1 M±(±jδ) Endpoint operators M1,2 = lim x→X1,2 M(x) Reversibility equation M∓(x ± δ)M±(x) ∝ 1 In [FB14] we’ve shown that any qubit-probe interacting in a fixed way with the system being measured can only decompose measurements of the form M1 = U1 (αΠS + βΠS⊥)