Quantum steering ellipsoids, extremal physical states and monogamy

Any 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 Euler's inequality for the circumradius and inradius of a triangle.


Introduction
The Bloch vector representation of a single qubit is an invaluable visualisation tool for the complete state of a two-level quantum system.Properties of the system such as mixedness, coherence and even dynamics are readily encoded into geometric properties of the Bloch vector.The extraordinary effort expended in the last 20 years on better understanding quantum correlations has led to several proposals for an analogous geometric picture of the state of two qubits [1,2,3].One such means is provided by the quantum steering ellipsoid [4,5,6,7], which is the set of all Bloch vectors to which one party's qubit could be 'steered' (remotely collapsed) if another party were able to perform all possible measurements on the other qubit.
It was shown recently [6] that the steering ellipsoid formalism provides a faithful representation of all two-qubit states and that many much-studied properties, such as entanglement and discord, could be obtained directly from the ellipsoid.Moreover steering ellipsoids revealed entirely new features of two-qubit systems, namely the notions of complete and incomplete steering, and a purely geometric condition for entanglement in terms of nested convex solids within the Bloch sphere.
However, one may well wonder if there is much more to be said about twoqubit states and whether the intuitions obtained from yet another representation could be useful beyond the simplest bipartite case.We emphatically answer this in the affirmative.Consider a scenario with three parties, Alice, Bob and Charlie, each possessing a qubit.Bob performs measurements on his system to steer Alice and Charlie.We show that the volumes V A|B and V C|B of the two resulting steering ellipsoids obey a tight inequality that we call the monogamy of steering (Theorem 6): We also prove an upper bound for the concurrence of a state in terms of the volume of its steering ellipsoid (Theorem 4).Using this we show that the well-known CKW inequality for the monogamy of concurrence [8] can be derived from the monogamy of steering.The monogamy of steering is therefore strictly stronger than the CKW result, as well as being more beautiful and geometrically intuitive.
The picture that emerges, which was hinted at in Ref. [6] by the nested tetrahedron condition for separability, is that the volume of a steering ellipsoid is a fundamental property capturing much of the non-trivial quantum correlations.But how large can a steering ellipsoid be?Clearly the steering ellipsoid cannot puncture the Bloch sphere.However, not even all ellipsoids contained in the Bloch sphere correspond to physical states.We begin our analysis by using known constraints on the set of single-qubit completely positive maps [9].The Choi isomorphism between linear maps and operators then gives us the necessary and sufficient conditions for a steering ellipsoid to represent a valid quantum state (Theorem 1).The conditions relate the ellipsoid's centre, semiaxes and orientation in a highly non-trivial manner.
We subsequently clarify these geometric constraints on physical states by considering the limits they impose on steering ellipsoid volume for a fixed ellipsoid centre.This gives rise to a family of extremal volume states (Figure 3) which, in Theorem 3, allows us to place bounds on how large an ellipsoid may be before it becomes first entangled and then unphysical.The maximal volume states that we give in equation ( 10) are found to be very special.In addition to being Choi-isomorphic to the amplitudedamping channel, these states maximise concurrence over the set of all states that have steering ellipsoids with a given centre (Theorem 5).This endows steering ellipsoid volume with a clear operational meaning and suggests a definition for a new measure of quantum correlation called obesity.
A curious aside of the steering ellipsoid formalism is its connection with classical Euclidean geometry.By investigating the geometry of separable steering ellipsoids, in Section 4.4 we arrive at a novel derivation of a famous inequality of Euler's in two and three dimensions.On a plane, it relates a triangle's circumradius and inradius; in three dimensions, the result extends to tetrahedra and spheres.Furthermore, we give a generalisation of Euler's result to ellipsoids, a full discussion of which appears in Ref. [10].
The term 'steering' was originally used by Schrödinger [11] in the context of his study into the complete set of states/ensembles that a remote system could be collapsed to, given some (pure) initial entangled state.The steering ellipsoid we study is the natural extension of that work to mixed states (of qubits).Schrödinger was motivated to perform such a characterisation by the EPR paper [12].The question of whether the ensembles one steers to are consistent with a local quantum model has been recently formalised [13] into a criterion for 'EPR steerability' that provides a distinct notion of nonlocality to that of entanglement: the EPR-steerable states are a strict subset of the entangled states.We note that the existence of a steering ellipsoid with nonzero volume is necessary, but not sufficient, for a demonstration of EPR-steering.It is an open question whether the quantum steering ellipsoid can provide a geometric intuition for EPR-steerable states as it can for separable, entangled and discordant states, although progress has recently been made [14].

The canonical, aligned state
A Hermitian operator with unit trace acting on the Hilbert space C 2 ⊗ C 2 may be expanded in the Pauli basis {1, σ} ⊗2 as For a two-qubit state, ρ is positive semidefinite, ρ ≥ 0. The local Bloch vectors are given by a = tr(ρ σ ⊗ 1) and b = tr(ρ 1 ⊗ σ), whilst bipartite correlations are contained in the matrix T ij = tr(ρ σ i ⊗ σ j ) [3].Requiring that ρ ≥ 0 places non-trivial constraints on a, b and T .Alice's steering ellipsoid E A is described by its centre where the Lorentz factor , and a real, symmetric 3 × 3 matrix The eigenvalues of Q A are the squares of the ellipsoid semiaxes s i and the eigenvectors give the orientation of these axes.Together with a specification of Bob's local basis, the geometric data (E A , a, b) provide a faithful representation of two-qubit states (Figure 1) [6].When Bob is steered by Alice, we can consider his ellipsoid E B , described by c B and Q B .This amounts to swapping a ↔ b and T ↔ T T in the expressions for c A and Q A .
Bob's steering of Alice is said to be complete when, for any convex decomposition of a into states in E A or on its surface, there exists a POVM for Bob that steers to it [6].All nonzero volume E A correspond to states that are completely steerable by Bob.When Bob's steering is complete, a lies on an ellipsoid E A scaled down by a factor b = |b|; for incomplete steering of Alice, a lies strictly inside this scaled-down ellipsoid.Aside from these straightforward, necessary restrictions on a and b, finding whether any two-qubit operator ρ describes a physical state usually involves obscure functions of the components of the matrix T , resulting from the requirement that ρ ≥ 0. However, these functions become much clearer in the context of the steering ellipsoid.
It will prove very useful to perform a reversible, trace-preserving local filtering operation that transforms ρ to a canonical state ρ.Crucially, Alice's steering ellipsoid is invariant under Bob's local filtering operation, so the same E A describes both ρ and ρ.We may perform the transformation [5] ρ provided that Bob's reduced state ρ B = tr A ρ is invertible (the only exception occurs when ρ B is pure, in which case ρ is a product state for which no steering is possible).In this canonical frame, Bob's state is maximally mixed ( b = 0) and Alice's Bloch vector coincides with the centre of E A ( a = c A ).The ellipsoid matrix is given by Q A = T T T , and so the semiaxes are s i = |t i |, where t i are the signed singular values of T .The local filtering operation preserves positivity: ρ ≥ 0 if and only if ρ ≥ 0. It also maintains the separability of a state: ρ is entangled if and only if ρ is [15].We may therefore determine the positivity and separability of ρ by studying its canonical state ρ.
Applying state-dependent local unitary operations on ρ, we can achieve the transformations a . We can always find O A and O B that perform a signed singular value decomposition on T , i.e.O A T O T B = diag(t).Bob's rotation O B has no effect on E A , but O A rotates E A about the origin (treating c A as a rigid rod) to align the semiaxes of E A parallel with the coordinate axes.Note there is some freedom in performing this rotation: the elements of t can be permuted and two signs can be flipped, but the product t 1 t 2 t 3 is fixed.
Both the positivity and entanglement of ρ are invariant under such local unitary operations.We therefore need only consider states that have E A aligned with the coordinates axes in this way.The question of physicality of any general operator of the form (2) therefore reduces to considering canonical, aligned states In the steering ellipsoid picture, this restricts our analysis to looking only at steering ellipsoids whose semiaxes are aligned with the coordinate axes: 3 ).In the following, unless stated otherwise, we will only refer to Alice's steering ellipsoid; we therefore drop the label A so that E ≡ E A , Q ≡ Q A and c ≡ c A .

Physical state conditions and chirality
To obtain conditions for the physicality of a two-qubit state, we employ the Choi isomorphism between linear maps and operators [16].The single-qubit map isomorphic to a two-qubit operator ρ is defined by Φ[ρ](η) = 2 tr B ((1 ⊗ η T )ρ), where η gives the argument of the map.A canonical state ρ has b = 0, which is equivalent to the isomorphic map Φ[ ρ] being trace-preserving (TP) [17].Similarly, the condition of positivity of an operator corresponds to complete positivity (CP) of the isomorphic map.We can therefore use results for when a TP single-qubit map is CP to give conditions for the physicality of ρ.Note that we are interested in the general case that c = 0, corresponding to the isomorphic map being non-unital.
Ruskai et al. [17] give a thorough characterisation of such conditions, but the expressions found by Braun et al. [9] will be more convenient for our purposes.For an operator of the form (6), ρ ≥ 0 if and only if We find that det ρ = 1 256 (c 4 − 2uc 2 + q), where u = 1 To obtain geometric conditions for the physicality of ρ, we express these conditions in terms of rotational invariants.Some care is needed with the term t 1 t 2 t 3 , which could be positive or negative.Since describes the chirality of E.
Let us say that Bob performs Pauli measurements on ρ and obtains the +1 eigenstates as outcomes, corresponding to Bloch vectors x, ŷ and ẑ.These vectors form a right-handed set.These outcomes steer Alice to the Bloch vectors c + t 1 x, c + t 2 ŷ and c + t 3 ẑ respectively.When Bob's outcomes and Alice's steered vectors are related by an affine transformation involving a proper (improper) rotation, Alice's steered vectors form a right-handed (left-handed) set and χ = +1 (χ = −1).We therefore refer to χ = +1 ellipsoids as right-handed and χ = −1 ellipsoids as left-handed.Note that a degenerate ellipsoid corresponds to χ = 0, since at least one t i = 0 (equivalently s i = 0).Theorem 1.Let ρ be an operator of the form (6), described by steering ellipsoid E with centre c, matrix Q and chirality χ.This corresponds to a two-qubit state, ρ ≥ 0, if and only if Proof.Rewrite the conditions in (7) using the ellipsoid parameters Q, c and χ.
As with the criteria for entanglement given in equation ( 4) of Ref. [6], we can identify three geometric contributions influencing whether or not a given steering ellipsoid describes a physical state: the distance of its centre from the origin, the size of the ellipsoid and the skew ĉT Qĉ.In addition, the physicality conditions also depend on the chirality of the ellipsoid, which relates to the separability of a state.
Theorem 2. Let ρ be a canonical two-qubit state of the form (6), described by steering ellipsoid E.
(a) E for an entangled state ρ must be left-handed.
(b) E for a separable state ρ may be right-handed, left-handed or degenerate.For a separable left-handed E, the corresponding right-handed E is also a separable state and vice-versa. Proof.
(a) An entangled state ρ must have det ρ T B < 0 [6] (following from the Peres-Horodecki criterion) and a non-degenerate ellipsoid, hence χ = ±1 a priori.Partial transposition ρ → ρ T B is equivalent to t 2 → −t 2 and hence to χ → −χ.All quantum states achieve det ρ ≥ 0, so for an entangled ρ, we have det ρ > det ρ T B .Using the form for det ρ given in Theorem 1, an entangled canonical state must have (b) The ellipsoid for a separable state may be degenerate or non-degenerate and so χ = 0 or χ = ±1 a priori.For a two-qubit separable state ρ, the operator ρ T B is also a separable state [18].Since partial transposition is equivalent to χ → −χ, this means that both the χ and the −χ ellipsoids are separable states.For the degenerate case, χ = 0.For a non-degenerate ellipsoid, both the χ = +1 and χ = −1 ellipsoids are separable states.
Recall that a local filtering transformation maintains the separability of a state.Although the chirality of an ellipsoid is a characteristic of canonical states only, we can extend Theorem 2 to apply to any general state of the form (2) by defining the chirality of a general ellipsoid as that of its canonical state.
As an example, consider the set of Werner states given by ρ [19].Although Werner's original definition does not impose the restriction p ≥ 0, states with p ≤ 0 can be obtained from the partial transposition of states with p ≥ 0. We will see in Section 4.2 that ρ W (p) is described by a spherical E of radius p centred on the origin.Figure 2 illustrates Theorem 2 for ρ W (p).

Extremal ellipsoid states
We will now use Theorems 1 and 2 to investigate ellipsoids lying on the entangledseparable and physical-unphysical boundaries by finding the largest area ellipses and largest volume ellipsoids with a given centre.The ellipsoid centre c is a natural parameter to use in the steering ellipsoid representation, and the physical and geometric results retrospectively confirm the relevance of this maximisation.In particular, we will see in Section 5 that the largest volume physical ellipsoids describe a set of states that maximise concurrence.
The methods used to find extremal ellipsoids are given in full in the Appendix, but the importance of Theorem 2 should be highlighted.The ellipsoid of an entangled state must be left-handed.For non-degenerate E we can therefore probe the separableentangled boundary by finding the set of extremal physical E with χ = +1; these must correspond to extremal separable states.The physical-unphysical boundary is found by studying the set of extremal physical E with χ = −1.Clearly the separable-entangled boundary must lie inside the physical-unphysical boundary since separable states are a subset of physical states.For the case of a degenerate E with χ = 0, any physical ellipsoid must be separable and so there is only the physical-unphysical boundary to find.

Two dimensions: largest area circles and ellipses in the equatorial plane
We begin by finding the physical-unphysical boundary for E lying in the equatorial plane.For a circle of radius r, centre c, we find that E represents a physical (and necessarily separable) state if and only if r ≤ 1 2 (1 − c 2 ).The physical ellipse with the largest area in the equatorial plane is not a circle for c > 0. For E with centre c = (c, 0, 0), the maximal area physical ellipse has minor semiaxis Noting that that both s 1 and s 2 are monotonically decreasing functions of c with s 1 ≤ s 2 ≤ 1  2 , we see that the overall largest ellipse is the radius 1  2 circle centred on the origin.Our results describe how a physical ellipse must shrink from this maximum as its centre is displaced towards the edge of the Bloch sphere.
Note that the unit disk does not represent a physical state; this corresponds to the well-known result that its Choi-isomorphic map is not CP (the 'no pancake' theorem).In fact, Ref. [9] gives a generalisation of the no pancake theorem that immediately rules out such a steering ellipsoid: a physical steering ellipsoid can touch the Bloch sphere at a maximum of two points unless it is the whole Bloch sphere (as will be the case for a pure entangled two-qubit state).

Three dimensions: largest volume spheres
In three dimensions we find distinct separable-entangled and physical-unphysical boundaries.Inept states [20]  (The name 'inept' was introduced because such states arise from the inept delivery of entangled qubits to pairs of customers: the supplier has a supply of pure entangled states |φ and always delivers a qubit to each customer but only has probability r of sending the correct pair of qubits to any given pair of customers.)The two parameters r and that describe an inept state can easily be translated into a description of the steering ellipsoid: E has c = (0, 0, (2 − 1)(1 − r)) and Q = diag(r 2 , r 2 , r 2 ).Thus an inept state gives a spherical E of radius r.Note that inept states with = 1  2 have null Bloch vectors for Alice and Bob and are equivalent to Werner states.The corresponding E are centred on the origin.
The separable-entangled boundary for a spherical E with centre c corresponds to . Any left-or right-handed sphere smaller than this bound describes a separable state.The physical-unphysical boundary is r = 1 − c.A spherical E on this boundary touches the edge of the Bloch sphere, and so this is just the constraint that E should lie inside the Bloch sphere.All left-handed spherical E inside the Bloch sphere therefore represent inept states.Right-handed spheres whose r exceeds the separableentangled bound cannot describe physical states since an entangled E must be lefthanded.Note how simple the physical state criteria are for spherical E: subject to these conditions on chirality, all spheres inside the Bloch sphere are physical.The same is not true for ellipsoids in general; there are some ellipsoids inside the Bloch sphere for which both the left-and right-handed forms are unphysical.

Three dimensions: largest volume ellipsoids
The largest volume separable E centred at c is an oblate spheroid with its minor axis oriented radially.For an ellipsoid with c = (0, 0, c), the major semiaxes are ).The largest volume physical E centred at c is also an oblate spheroid with its minor axis oriented radially.For an ellipsoid with c = (0, 0, c), the major semiaxes are s 1 = s 2 = √ 1 − c and the minor semiaxis is s 3 = 1 − c.These extremal ellipsoids are in fact the largest volume ellipsoids with centre c that fit inside the Bloch sphere.
The volume V of these maximal ellipsoids can be used as an indicator for entanglement and unphysicality.Our calculations have been carried out for a canonical ρ, but since steering ellipsoids are invariant under the canonical transformation 5, the results are directly applicable to any general ρ.Proof.Find the volume of the ellipsoids on the separable-entangled and physicalunphysical boundaries using V = 4π 3 s 1 s 2 s 3 .
This result extends the notion of using volume as an indicator for entanglement, as was introduced in Ref. [6].We see that the largest volume separable ellipsoid is the Werner state on the separable-entangled boundary, which has a spherical E of radius 1 3 and c = 0. We have tightened the bound by introducing the dependence on c.In fact, Theorem 3 gives the tightest possible such bounds, since we have identified the extremal E that lie on the boundaries.Note that for all c we have , with equality achieved only for c = 1 when E is a point with V = 0 and ρ is a product state.This confirms that the two boundaries are indeed distinct and that the separable E are a subset of physical E.

Applications to classical Euclidean geometry using the nested tetrahedron condition
Recall the nested tetrahedron condition [6]: a two-qubit state is separable if and only if E fits inside a tetrahedron that fits inside the Bloch sphere.We used Theorem 1 and ellipsoid chirality to algebraically find the separable-entangled boundary for the cases that E is a circle, ellipse, sphere or ellipsoid.The nested tetrahedron condition then allows us to derive several interesting results in classical Euclidean geometry.We give a very brief summary of the work here; a full discussion is given in Ref. [10].
Euler's inequality r ≤ R 2 is a classic result relating a triangle's circumradius R and inradius r [21].In Section 4.1 we investigated the largest circular E in the equatorial plane, finding that E represented a physical (and necessarily separable) state if and only if r ≤ 1 2 (1 − c 2 ).By the degenerate version of the nested tetrahedron condition, this gives the condition for when E fits inside a triangle inside the unit disk (R = 1).We therefore see that our result implies Euler's inequality, since 0 ≤ c ≤ 1.
We can pose the analogous question in 3 dimensions.Let S r be a sphere of radius r contained inside another sphere S R of radius R. If the distance between the sphere centres is c, what are the necessary and sufficient conditions for the existence of a tetrahedron circumscribed about S r and inscribed in S R ?This question was answered by Daniellson using some intricate projective geometry [22], but there is no known proof using only methods belonging to classical Euclidean elementary geometry [23].By considering the steering ellipsoids of inept states (Section 4.2) we have answered precisely this question, finding the necessary and sufficient condition for the existence a nested tetrahedron.Our result is found to reproduce Daniellson's result that the sole condition is c 2 ≤ (R + r)(R − 3r).In fact, our work extends these results to give conditions for the existence of a nested tetrahedron for the more general case of an ellipsoid E contained inside a sphere.These very non-trivial geometric results can be straightforwardly derived from Theorem 1 by understanding the separability of two-qubit states in the steering ellipsoid formalism.

Applications to mixed state entanglement: ellipsoid volume and concurrence
The volume of a state provides a measure of the quantum correlations between Alice and Bob, distinct from both entanglement and discord [6].We will now study the states corresponding to the maximal volume physical ellipsoids.By deriving a bound for concurrence in terms of ellipsoid volume, we see that maximal volume states also maximise concurrence for a given ellipsoid centre.

Maximal volume states
Recall that the largest volume ellipsoid with c = (0, 0, c) has major semiaxes s 1 = s 2 = √ 1 − c and minor semiaxis s 3 = 1 − c.We will call this E max c . With the exception of c = 1, which describes a product state, these correspond to entangled states and so are described by left-handed steering ellipsoids.Using (6), the canonical state for where ).This describes a family of rank-2 'X states' parametrised by 0 ≤ c ≤ 1.Some examples are shown in Figure 3.
The density matrix of an X state in the computational basis has non-zero elements only on the diagonal and anti-diagonal, giving it a characteristic X shape.X states were introduced in Ref. [24] as they comprise a large class of two-qubit states for which certain correlation properties can be found analytically.In fact, steering ellipsoids have already been used to study the quantum discord of X states [5].In the steering ellipsoid formalism, E for an X state will be radially aligned, having a semiaxis collinear with c. when we reparametrise c = 2(1 − p) and also make the change |ψ + → |ψ c .The Horodecki state is a rank-2 maximally entangled mixed state [25].ρ H may be extended (see, for example, Refs.[26,27]) to the generalised Horodecki state Note that this has two free parameters, α and p. Setting α = 1/2p and reparametrising c = 2(1 − p), we see that our ρ max c states form a special class of the generalised Horodecki states described by the single parameter c.
The maximal volume states have a clear physical interpretation when we consider the Choi-isomorphic channel: ρ max c is isomorphic to the TP single qubit amplitudedamping (AD) channel with decay probability c [26].For a single qubit state η, this channel is , where [28] If Alice and Bob share the Bell state |ψ ) and Alice passes her qubit through this channel, we obtain a maximal volume state centred at c = (0, 0, c), i.e. ρ max c = (Φ AD ⊗ 1)(|ψ + ψ + |).

Bounding concurrence using ellipsoid volume
Physically motivated by its connection to the entanglement of formation [29], concurrence is an entanglement monotone that may be easily calculated for a two-qubit state ρ.Define the spin-flipped state as ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) and let λ 1 , ..., λ 4 be the square roots of the eigenvalues of ρρ in non-increasing order.The concurrence is then given by Concurrence ranges from 0 for a separable state to 1 for a maximally entangled state.In principle one may find C(ρ) in terms of the parameters describing the corresponding steering ellipsoid E, but the resulting expressions are very complicated.It is however possible to derive a simple bound for C(ρ) in terms of steering ellipsoid volume.
For a general state ρ, the volume V = 4π 3 γ 4 b | det Θ(ρ)| [6].From the local filtering transformation, and using det L A = det L B = 1, we have For a Bell-diagonal state | det Θ(τ )| = |t 1 t 2 t 3 | and so we obtain , from which the result then follows.
This bound will be of central importance in the derivation of the CKW inequality in Section 6. Theorem 4 also suggests how ellipsoid volume might be interpreted as a quantum correlation feature called obesity.If we define the obesity of a two-qubit state as Ω(ρ) = | det Θ(ρ)| 1/4 then Theorem 4 shows that concurrence is bounded for any two-qubit state as C(ρ) ≤ Ω(ρ).Note that this definition also suggests an obvious generalisation to a d-dimensional Hilbert space, Ω(ρ) = | det Θ(ρ)| 1/d .

Maximal volume states maximise concurrence
We now demonstrate the physical significance of the maximum volume steering ellipsoids by finding that the corresponding states ρ max c maximise concurrence for a given ellipsoid centre.This will also demonstrate the tightness of the bound given in Theorem 4.
The state ρ max c given in ( 10) is a canonical state with b = 0. Let us invert the transformation (5) to convert ρ max

Monogamy of steering
The maximal volume states ρ max c have particular significance when studying a monogamy scenario involving three qubits.Monogamy scenarios and steering ellipsoids have been used before to study the Koashi-Winter relation [5].Here we show that ellipsoid volume obeys a monogamy relation that is strictly stronger than the CKW inequality for concurrence monogamy, giving us a new derivation of the CKW result.Subscripts labelling the qubits A, B and C are reintroduced so that Alice's ellipsoid E is now called E A , the maximal volume state ρ max c is now called ρ max c A , and so on.We begin by considering a maximal volume two-qubit state shared between Alice and Bob. Proof.
Since concurrence is a symmetric function with respect to swapping Alice and Bob we must also have For any two-qubit state, the volumes of E A and E B are related by γ and so E B is also maximal volume for the centre c B , i.e.E B = E max c B .Now consider Scenario (a) shown in Figure 4, in which Alice, Bob and Charlie share a pure three-qubit state and Bob can perform a measurement to steer Alice and Charlie.Let E A|B , with volume V A|B and centre c A|B , be the ellipsoid for Bob steering Alice, and similarly for the ellipsoid E C|B with Bob steering Charlie.Lemma 3. Alice, Bob and Charlie share a pure three-qubit state for which the joint state ρ AB = ρ max c A|B given by (13), corresponding to E A|B being maximal volume.The ellipsoid E C|B is then also maximal volume, and the centres obey all of E A|B , E B|A , E C|B and E B|C are simultaneously maximal volume for their respective centres.
The monogamy of steering can easily be rephrased in terms of obesity.Although Alice and Bob's steering ellipsoid volumes are in general different, obesity is a partyindependent measure.When expressed using obesity, the two steering scenarios therefore give the same bound Ω 2 (ρ AB ) + Ω 2 (ρ BC ) ≤ γ −2 b .We now use the monogamy of steering to derive the Coffman-Kundu-Wootters (CKW) inequality for monogamy of concurrence [8].
Theorem 7. When Alice, Bob and Charlie share a pure three-qubit state the squared concurrences must obey the bound Proof.The result can be derived using either bound presented in Theorem 6; we will use Scenario (a).Theorem 4 tells us that The monogamy of steering is strictly stronger than the monogamy of concurrence since Theorem 6 implies Theorem 7 but not vice versa.Our derivation of the CKW inequality again demonstrates the significance of maximising steering ellipsoid volume for a given ellipsoid centre.
Finally, we note that the tangle of a three-qubit state may be written in the form . When there is maximal steering, so that the bounds in Theorems 6 and 7 are saturated, we have τ ABC = 0.The corresponding three-qubit state belongs to the class of W states [30] (assuming that we have genuine tripartite entanglement).The W state itself, |W = 1 √ 3 (|001 + |010 + |001 ), corresponds to the case that c A|B = c B|A = c C|B = c B|C = (0, 0, 1  2 ).

Conclusions
Any two-qubit state ρ can be represented by a steering ellipsoid E and the Bloch vectors a and b.We have found necessary and sufficient conditions for the geometric data to describe a physical two-qubit state ρ ≥ 0. Together with an understanding of steering ellipsoid chirality, this is used to find the separable-entangled and physicalunphysical boundaries as a function of ellipsoid centre c.These boundaries have geometric and physical significance.Geometrically, they can be used to find very nontrivial generalisations of Euler's inequality in classical Euclidean geometry.Physically, the maximal volume ellipsoids describe a family of states that are Choi-isomorphic to the amplitude-damping channel.
The concurrence of ρ is bounded as a function of ellipsoid volume; this is used to show that maximal volume states also maximise concurrence for a given c.By studying a system of three qubits we find relations describing the monogamy of steering.These bounds are strictly stronger than the monogamy of concurrence and provide a novel derivation of the CKW inequality.Thus the abstract, mathematical question of physicality and extremal ellipsoids naturally leads to an operational meaning for ellipsoid volume as a bound for concurrence and provides a new geometric perspective on entanglement monogamy.
These results may find applications in other notions of how 'steerable' a state is.In particular, it should be possible to use our work to answer questions about EPRsteerable states [13].For example, what are the necessary constraints on ellipsoid volume such that no local hidden state model can reproduce the steering statistics?Beyond this, the results on monogamy of steering pave the way for looking at steering in a manyqubit system by considering how bounds on many-body entanglement are encoded in the geometric data.
This system can in fact be simplified before solving.In particular, the skew term ĉT Qĉ is awkward to deal with in full generality.However, by symmetry, any maximal ellipsoid must have one of its axes aligned radially and the other two non-radial axes equal.Since we are looking at ellipsoids aligned with the coordinate axes, we may therefore take c = (0, 0, c) and s 1 = s 2 .Maximal solutions could then have s 1 = s 2 > s 3 (an oblate spheroid), s 1 = s 2 < s 3 (a prolate spheroid) or s 1 = s 2 = s 3 (a sphere).
Extremal ellipses in the equatorial plane are found similarly using the KKT conditions.To maximise the area, we use the Lagrangian L = πs 1 s 2 + λ 1 g 1 + λ 2 g 2 and set Q = diag(s 2 1 , s 2 2 , 0) and c = (c, 0, 0) in the expressions for g 1 and g 2 .

Figure 1 .
Figure 1.An example of the geometric data: Alice's steering ellipsoid E A and the two Bloch vectors a and b.Together with a specification of Bob's local basis, this is a faithful representation of two-qubit states.

Figure 2 .
Figure 2. The physicality and separability of a steering ellipsoid depend on its chirality χ.This dependence is illustrated for the set of Werner states ρ W (p) with 0 ≤ p ≤ 1.

Theorem 3 .
Let ρ be an operator of the form (2), described by steering ellipsoid E with centre c and volumeV .Let V sep c = 2π 81 1 − 9c 2 + (1 + 3c 2 ) 3/2 and V max c = 4π 3 (1 − c) 2 .(a)If ρ is a physical state and V > V sep c then ρ must be entangled.(b)If V > V max c then ρ must be unphysical.

Figure 3 .
Figure 3.The geometric data for three maximal volume states ρ max c with (a) c = 0, (b) c = 0.5 and (c) c = 0.8.Since these are canonical states we have b = 0 and c = a.Note that E = E max c

Figure 4 .
Figure 4.The two scenarios for studying monogamy, with arrows between parties indicating the direction of steering.(a) Bob performs a measurement to steer Alice and Charlie, with corresponding steering ellipsoids E A|B and E C|B respectively.(b) Alice and Charlie perform measurements to steer Bob, with corresponding steering ellipsoids E B|A and E B|C respectively.

Lemma 2 .
If Alice and Bob share a state ρ max c A given by (13) then both E A and E B are maximal volume for their respective centres c A and c B .The steering ellipsoid centres obey γ 2 b