One-to-One Mapping between Steering and Joint Measurability Problems

Abstract

Quantum steering refers to the possibility for Alice to remotely steer Bob's state by performing local measurements on her half of a bipartite system. Two necessary ingredients for steering are entanglement and incompatibility of Alice's measurements. In particular, it is known that for the case of pure states of maximal Schmidt rank the problem of steerability for Bob's assemblage is equivalent to the problem of joint measurability for Alice's observables. We show that such an equivalence holds in general; namely, the steerability of any assemblage can always be formulated as a joint measurability problem, and vice versa. We use this connection to introduce steering inequalities from joint measurability criteria and develop quantifiers for the incompatibility of measurements.

Sufficient criterion for guaranteeing that a two-qubit state is unsteerable

“…This can be seen as follows. A set of measurements {M ±|x } is jointly measurable if and only the assemblage given by σ ±|x = ρ , where ρ is a full-rank quantum state [33], is unsteerable. Choosing ρ = 1 1/2 we get the corresponding assemblage σ ±|x = ρ 1/2 M ±|x ρ 1/2 = 1 2 M ±|x .…”

N-term pairwise-correlation inequalities, steering, and joint measurability

“…In recent years there is a surge of research activity dedicated to explore the notion of measurement incompatibility and its connection with the counter-intuitive quantum notions like non-locality, contextuality and nonmacrorealism [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In particular, it is known that measurement incompatibility plays a key role in bringing to surface the violations of the so-called no-go theorems in the quantum world.…”

“…However, this result may not hold, in general, for Bell non-locality tests where more than two incompatible POVMs with any number of outcomes are employed i.e., it is possible to identify a set of non-jointly measurable POVMs, which fail to reveal Bell-type non-locality, in general [24]. Interestingly, there exists a one-to-one equivalence [16][17][18] between measurement incompatibility and quantum steerability (i.e., Alice's ability to non-locally alter Bob's states by performing local measurements on her part of the quantum state [26]). More specifically, a set of fuzzy POVMs is said to be incompatible if and only if it can be used to show steering in a quantum state.…”

Dynamics of incompatibility of quantum measurements in open systems

“…Accordingly, we take an ancillary qubit (in addition to our open qubit system described above), and set the initial state of the combined system to be the maximally entangled state Ψ 0 = 1 √ 2 (|00 + |11 ). We stress that this specific choice is not crucial, since the connection between incompatibility and entanglement holds for any bipartite state [9]. Analogous to eq.…”

“…Concerning our specific choice of the incompatibility measure [19], there are alternatives, such as "incompatibility weight" [18] based on the steering quantification in [7]. The resulting incompatibility measure has recently been compared with ours in [9] (see especially Fig. 2 therein); while there are strong quantitative differences, the qualitative behaviour is relatively similar except for special cases.…”