Abstract
The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.
Highlights
One of the most fundamental features of quantum mechanics is the fact that it is impossible to prepare states that have sufficiently precise simultaneous values of incompatible observables
In [12] for the more general case of binary-valued measurements in arbitrary dimensional Hilbert spaces. The characterisation they give, which is formulated in terms of the expectation values of anticommutators, leads to the results we present (Lemmas 1–3 below, albeit in a different mathematical framework), Kaniewski et al use it to derive state-dependent entropic uncertainty relations for this generalised scenario, which bound the sum of the entropies considered and are not tight in the sense we consider
We give the bounds on all of these relations in terms of the norm |r| of the Bloch vector representing the state ρ = 21 (1 + r · σ ), which is directly linked to the purity of the state, so that, if a bound on this is known, tighter uncertainty relations can be obtained; for pure states |r| = 1, and the most general form is recovered
Summary
One of the most fundamental features of quantum mechanics is the fact that it is impossible to prepare states that have sufficiently precise simultaneous values of incompatible observables. The first issue can, in some cases, be avoided by considering more complicated expressions or different measures of incompatibility [12,13] This approach can be used to give non-trivial state-dependent uncertainty relations, which have the property that they can be experimentally verified without knowing the observables A and B and, are of interest for device-independent cryptography [12]. One ideally wants an uncertainty relation that depends on the state of the system only via the (operationally defined) measures of uncertainty, which ensures that the relation is an operational statement constraining the uncertainties directly and can be evaluated without prior knowledge of the system’s state This is the case, for example, with the position-momentum uncertainty relation described earlier. It makes sense to look for tighter, state-independent relations capable of addressing these issues, and it is this situation we tackle in this paper
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