Quantum Steering Ellipsoids

Abstract

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...

Geometric representation of two-qubit entanglement witnesses

“…Since |ν| = 1, this describes the linear transformation of a unit sphere and in fact gives the ellipsoid E as defined in (2) and (3) [13]. r (ν) is therefore a point on the surface of E, parametrised by ν.…”

“…[11,13,15,16] we perform a reversible, trace-preserving local filtering operation that transforms R to a canonical operator R:…”

“…When R is a state shared between Alice and Bob, we can represent it using Alice's steering ellipsoid E. This gives the set of Bloch vectors to which Alice's qubit can be collapsed given all possible local measurements by Bob [12,13]. E necessarily lies inside the Bloch sphere [19].…”

“…E necessarily lies inside the Bloch sphere [19]. E is defined by its centre vector c and a real 3 × 3 ellipsoid matrix Q [13]:…”

“…The steering ellipsoid formalism [11][12][13][14] gives a faithful representation of two-qubit states analogous to the Bloch vector picture for a single qubit. Any two-qubit state may be represented by an ellipsoid inside the Bloch sphere, but not all ellipsoids inside the Bloch sphere represent a two-qubit state.…”