The quantum steering ellipsoid of a two-qubit state is the set of Bloch vectors that Bob can collapse Alice's qubit to, considering all possible measurements on his qubit. We provide an elementary construction of the ellipsoid for arbitrary states, calculate its volume, and explain how this geometric representation can be made faithful. The representation provides a range of new results, and uncovers new features, such as the existence of "incomplete steering" in separable states. We show that entanglement can be analyzed in terms of three geometric features of the ellipsoid and prove that a state is separable if and only if it obeys a "nested tetrahedron" condition. The Bloch sphere provides a simple representation for the state space of the most primitive quantum unit-the qubit-resulting in geometric intuitions that are invaluable in countless fundamental information-processing scenarios. The two-qubit system, likewise, constitutes the primitive unit for bipartite quantum correlations. However, the two-qubit state space is described by 15 real parameters with a surprising amount of structure and complexity. As such, it is challenging both to faithfully represent its states and to acquire natural intuitions for their properties [1][2][3].The phenomenon of steering was first uncovered by Schrödinger [4] (and subsequently rediscovered by others [5][6][7]), who realized that local measurements on Bob's side of the pure state jψi AB could be used to "steer" Alice's state into any convex decompositions of her reduced state ρ A . Hence, we say that for jψi AB , steering is "complete" within Alice's Bloch sphere. For a two-qubit mixed state ρ, it is known [8] that the convex set of states that Alice can be steered to is an ellipsoid E A , see Fig. 1.The purpose of this Letter is to show that this steering ellipsoid is the natural generalization of the Bloch sphere picture, in that it can be used to give a faithful representation of an arbitrary two-qubit state in three dimensions, and moreover, that the core properties of the state and its correlations are made manifest in simple geometric terms.By adopting this representation, we are led to a range of novel results for both separable and entangled states.First, it reveals a new feature of separable quantum states, called incomplete steering, where not all decompositions of ρ A within the steering ellipsoid E A are accessible. More importantly, the representation reveals surprising structure in mixed state entanglement. We find that mixed state entanglement decomposes into the simple geometric components of (a) the spatial orientation of the ellipsoid, (b) its distance from the origin, and (c) its size. We are also lead to the surprising nested tetrahedron condition: a state is separable if and only if its ellipsoid fits inside a tetrahedron that itself fits inside the Bloch sphere.The representation also provides unity and insight for a range of distinct features. The nested tetrahedron condition leads to a simple determination of the minimal number of product states...

The question of which two-qubit states are steerable (i.e. permit a demonstration of EPR-steering) remains open. Here, a strong necessary condition is obtained for the steerability of two-qubit states having maximally-mixed reduced states, via the construction of local hidden state models. It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions are also obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of n/(n An) 2 over arbitrary unit hemispheres for any positive matrix A.

The proof of theorem 6(a) is incorrect, although the monogamy of steering result AbstractAny two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman-Kundu-Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.

Distinguishing hot from cold is the most primitive form of thermometry. Here we consider how well this task can be performed using a single qubit to distinguish between two different temperatures of a bosonic bath. In this simple setting, we find that letting the qubit equilibrate with the bath is not optimal, and depending on the interaction time it may be advantageous for the qubit to start in a state with some quantum coherence. We also briefly consider the case that the qubit is initially entangled with a second qubit that is not put into contact with the bath, and show that entanglement allows for even better thermometry.Comment: Minor changes to match published version, references updated. 5 pages, 2 figure