Unifying decoherence and the Heisenberg Principle

Abstract

We exhibit three inequalities involving quantum measurement, all of which are sharp and state independent. The first inequality bounds the performance of joint measurement. The second quantifies the trade-off between the measurement quality and the disturbance caused on the measured system. Finally, the third inequality provides a sharp lower bound on the amount of decoherence in terms of the measurement quality. This gives a unified description of both the Heisenberg uncertainty principle and the collapse of the wave function.

Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off

“…Thus this can be regarded as an "operator-valued inner product". It satisfies a Cauchy-Schwarz inequality proven by Janssens [33], which we recapitulate here as we expect it to be useful for future steps in this line of investigation. Proof.…”

Measurement disturbance and conservation laws in quantum mechanics

“…The Cauchy-Schwarz inequality was proven by Janssens in Lemma 1 of Ref. [32], and we refer to Lemma 3 of Ref. [33] for an alternative proof.…”

A quantum reference frame size-accuracy trade-off for quantum channels

“…Proof. We begin by using a channel Λ to define an "operator-valued inner product" A|B := Λ(A * B) − Λ(A) * Λ(B), which satisfies a Cauchy-Schwarz type inequality (see [18] and Lemma 3 in [11]):…”

“…We now present proofs of Theorem 1 and Corollary 2. To prove Theorem 1, we need the following lemma [11,18]:…”