Quantum Steering Ellipsoids

week ending PRL 113, 020402 (2014) PHYSICAL REVIEW LETTERS 11 JULY 2014

Sania Jevtic,1,2 Matthew Pusey,3,2 David Jennings,2 and Terry Rudolph2 1Mathematical Sciences, John Crank 501, Brunel University, Uxbridge UB8 3PH, United Kingdom 2Controlled Quantum Dynamics Theory, Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom 3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2 L 2Y5, Canada (Received 13 May 2013; revised manuscript received 6 December 2013; published 8 July 2014) 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.

DOI: 10.1103/PhysRevLett.113.020402 PACS numbers: 03.65.Ta, 03.67.Mn

The Bloch sphere provides a simple representation for importantly, the representation reveals surprising structure the state space of the most primitive quantum unit—the in mixed state entanglement. We find that mixed state qubit—resulting in geometric intuitions that are invaluable entanglement decomposes into the simple geometric com- in countless fundamental information-processing scenarios. ponents of (a) the spatial orientation of the ellipsoid, (b) its The two-qubit system, likewise, constitutes the primitive distance from the origin, and (c) its size. We are also lead to unit for bipartite quantum correlations. However, the the surprising nested tetrahedron condition: a state is two-qubit state space is described by 15 real parameters separable if and only if its ellipsoid fits inside a tetrahedron with a surprising amount of structure and complexity. As that itself fits inside the Bloch sphere. such, it is challenging both to faithfully represent its states The representation also provides unity and insight for a and to acquire natural intuitions for their properties [1–3]. range of distinct features. The nested tetrahedron condition The phenomenon of steering was first uncovered by leads to a simple determination of the minimal number of Schrödinger [4] (and subsequently rediscovered by others product states in the ensemble of any separable state. We [5–7]), who realized that local measurements on Bob’s side note that the ellipsoid volume is an entanglement criterion, T of the pure state jψiAB could be used to “steer” Alice’s state and provide a formula for it in terms of detðρÞ and detðρ B Þ. into any convex decompositions of her reduced state ρA. Nonzero ellipsoid volume is a type of correlation inter- Hence, we say that for jψiAB, steering is “complete” within mediate between discord and entanglement. Alice’s Bloch sphere. For a two-qubit mixed state ρ,itis Beyond these new insights, we also feel that this method known [8] that the convex set of states that Alice can be of compactly depicting any two-qubit state in three steered to is an ellipsoid EA, 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 ρA within the steering ellipsoid EA are accessible. More

Published by the American Physical Society under the terms of FIG. 1 (color online). Ellipsoid representation of a two-qubit the Creative Commons Attribution 3.0 License. Further distri- state. For any two-qubit state ρ, the set of states to which Bob can bution of this work must maintain attribution to the author(s) and steer Alice forms an ellipsoid EA in Alice’s Bloch sphere, the published article’s title, journal citation, and DOI. containing her Bloch vector a.Bob’sBlochvectorb is also shown.

0031-9007=14=113(2)=020402(5) 020402-1 Published by the American Physical Society week ending PRL 113, 020402 (2014) PHYSICAL REVIEW LETTERS 11 JULY 2014

1 dimensions should be of interest to a range of researchers in which occurs with probability 2, and where Alice’s Bloch both the theoretical and experimental quantum sciences. vector is now a þ Tx. The set of all states Alice can end up The Pauli basis.—Let σμ ¼f1; σx; σy; σzg, μ ¼ 0; 1; 2; 3 with is simply the unit sphere of possible x, shrunk and “ ” denote the homogeneous Pauli basis. AnyP single-qubit rotated by T and translated by a, i.e., an ellipsoid centered Eˆ Eˆ 1 3 X σ a Hermitian operator can be written ¼ 2 μ¼0 μ μ, at with orientation and semiaxes given by the eigenvec- ˆ TTT where the Xμ ¼ trðEσμÞ are components of the real vector tors and eigenvalues of . The ellipsoid dimension is ˆ rankðTÞ¼rankðΘÞ − 1. Points inside the ellipsoid can be X. Demanding that E ≥ 0 is equivalent to X0 ≥ 0 and P reached via convex combinations of projective measure- X2 ≥ 3 X2, and we can identify Eˆ as a positive-operator 0 i¼1 i ments, and conversely, a POVM element is a positive valued measure (POVM) element. operator and so can be spectrally decomposed into a In a similar way, any two-qubitP state ρ can be written 1 3 mixture of projectors, thus, giving a point within the in the Pauli basis as ρ Θμνσμ ⊗ σν, where ¼ 4 μ;ν¼0 ellipsoid. Θ ρσ ⊗ σ μ; ν μν ¼ trð μ νÞ is real for all . As a block matrix Now, consider a general state with b ≠ 0.Ifb ¼ 1, then ρ 1 bT Θ a; b is a product state, in which case there is no steering and the we have ¼ða T Þ, where are the Bloch vectors of steering ellipsoid is the single point a. For the case b<1, −ð1=2Þ the reduced states ρA and ρB of ρ, respectively, and T is a we find that the SLOCC operator 1 ⊗ ð2ρBÞ corre- 3 × 3 matrix encoding the correlations [2]. If Bob does a sponds to a Lorentz boost Lb by a “velocity” b that ˆ POVM and obtains outcome E, he steers Alice to the state transforms ρB to the maximally mixed state (which has ˆ b 0 ρ~ proportional to trB½ρð1 ⊗ EÞ, which, in the Pauli basis, is ¼ ). We refer to this special filtered state as the 1 1 “canonical state” on the SLOCC orbit SðρÞ. Since SLOCC given by the four-vector 2 ΘX, with probability 2 ð1 þ b · xÞ T operations on Bob do not affect Alice’s steering ellipsoid, where x X1;X2;X3 . ¼ð Þ ’ The four-vector formalism is related to the idea of the parameters of an arbitrary state s steering ellipsoid Θ L stochastic local operations and classical communication are obtained by simply boosting by b and reading off the ellipsoid parameters. This gives a steering ellipsoid (SLOCC) [1], which are operations of the form 2 0 † centered at cA ¼ða − TbÞ=ð1 − b Þ, with orientation and ρ → ρ SA ⊗ SBρ SA ⊗ SB , where SA;SB are invert- ffiffiffiffi ¼ ð Þ s pq ible complex matrices. The set of states attainable from ρ semiaxes lengths i ¼ i given [9] by the eigenvectors q under SLOCC is called the SLOCC orbit of ρ, and denoted and eigenvalues i of the ellipsoid matrix S ρ Θ Θ0 ð Þ. Under this action, the matrix transforms as ¼ 1 bbT Λ ΘΛT Λ Q T − abT 1 TT − baT : A B where AðBÞ are proper orthochronous Lorentz A ¼ 2 ð Þ þ 2 ð Þ ð2Þ transformations [9]. Significant in what follows, for a 1 − b 1 − b 0 SLOCC operation affecting only Bob (Θ ΘΛB), the ¼ E set of states Alice is steered to is unaffected, since: X is To obtain B, the ellipsoid at B, we simply perform a 0 swap of A and B, which corresponds to transposing Θ and in the forward light cone if and only if X ΛBX is, ¼ b → a a → b T → TT E E and Θ0X ΘX0. sends , , . Hence, A and B always ¼ Θ − 1 Previously, in [8], a range of SLOCC techniques were have the same dimensionality, rankð Þ . This completes E a b employed to study entanglement and steering for two-qubit the construction of the geometric data ( A, , ) for a given ρ ρ mixed states; however, this approach encounters problems state . Next, we describe the reverse direction: obtaining E a b when applied to certain separable states and, moreover, is from an ellipsoid A and the vectors and . ρ — not suited to addressing the geometric features of interest. Reconstruction of from geometric data. Given a; b; E Q ; c ρ T The techniques developed here follow a different line, and A ¼ð A AÞ, to recover , we need .In[9],we circumvent both of these issues. prove that this matrix is given by — Construction of the quantum steering ellipsoid. We 1 pffiffiffiffiffiffiffi γ − 1 pffiffiffiffiffiffiffi now provide an alternative construction of the steering T γc bT Q O Q ObbT ; ¼ γ A þ A þ b2 A ð3Þ ellipsoid EA to that in [15], which applies even when EA is degenerate. pffiffiffiffiffiffiffi where O ∈ Oð3Þ satisfies a ¼ cA þ QAOb. This specifies Our construction of EA is easiest to understand in the ρ b 0 O up to a rotation O0 ∈ Oð3Þ such that O0b ¼ b. The action case when the state has ¼ . For such a state, suppose 0 1 of O can be encoded, for example, by a coloring of EA as in X Bob projects his qubit onto the pure state ¼ðx Þ with [16]. In this way, the steering ellipsoid can be used as a faithful representation of ρ. O0 corresponds to a local unitary x ¼ 1. Given this outcome, Alice is steered to and/or partial transpose on Bob’s system, and so is irrelevant for any correlation properties such as entanglement. “Complete” and “incomplete” steering.—The steering 1 0T 1 1 Y ΘX ; ellipsoid specifies which states Bob can steer Alice to. ¼ ¼ a T x ¼ a Tx ð1Þ þ A more subtle question is which decompositions of

020402-2 week ending PRL 113, 020402 (2014) PHYSICAL REVIEW LETTERS 11 JULY 2014 P ’ n Eˆ β = Eˆ ρ p α Alice s reduced state he can steer to. Clearly, a necessary to i¼1ðtrð iÞ trð BÞÞ i i. Hence, her steered Bloch condition is that all of the states in the decomposition must vector will be a convex combination of the Bloch vectors be in EA, surprisingly however, it turns out that this is not for the αi—in other words her steering ellipsoid is con- sufficient. tained in T . Consider some nonproduct two-qubit state with We prove, in [9], that the nontrivial converse holds: any ellipsoids EA and EB. The following are equivalent [9]: ellipsoid that fits inside a tetrahedron that itself fits inside (1) (Complete steering of A) For any convex decomposition the Bloch sphere must arise from a separable state, and of a into vectors in EA or on its surface, there exists a thus, the nested tetrahedron condition is both necessary POVM for Bob that steers Alice to it. (2) The affine span and sufficient for separability of the state. of EB contains the origin. This key geometric insight leads to some nontrivial These conditions hold for all nondegenerate ellipsoids corollaries. For example, for any separable state ρ, the (which includes all entangled states) as well as all states minimal number ofP product states in an ensemble b 0 ρ n p α ⊗ β n Θ where ¼ . However, complete steering is not symmetric: decomposition ¼ i¼1 i i i is ¼ rankð Þ.If 1 Θ 1 n 1 the state ρ ¼ 2 ðj00ih00jþjþ1ihþ1jÞ has complete steer- rankð Þ¼ , we have a product state, and so ¼ , while ing of Alice by Bob, but not vice versa. if rankðΘÞ¼2, we have that EA is a line segment and we The three geometric contributions to entanglement.— form a decomposition of ρ using the endpoints of this The Peres-Horodecki criterion [17,18] asserts that a two- segment, giving n ¼ 2. The case rankðΘÞ¼3 is slightly TB qubit state ρe is entangled if and only if ρe has a negative more involved, but follows from the fact any ellipse inside a eigenvalue. Furthermore, it can be shown [15] that, at most, tetrahedron inside the unit sphere also lies inside a triangle TB TB in the unit sphere [9,21]. Finally, it is known [20] that any one eigenvalue of ρe can be negative, and that ρe is full TB separable state can be written using four product states, rank for all entangled states [19]. Hence, det ρe < 0 is a which covers the case rankðΘÞ¼4. Combining this with necessary and sufficient condition for entanglement. ρ the above results on complete steering provides a natural Suppose is entangled, then any state in its SLOCC orbit geometrical classification of two qubit states, as in Fig. 2. SðρÞ is also entangled [15], including the canonical state — ρ~ ∈ S ρ ρ Quantum discord and ellipsoid orientation. Quantum ð Þ. It follows that is entangled if and only if discord has received much attention as a measure of the ρ~TB < 0 ρ ρ~ E detð Þ . However, the states and share the same A, quantumness of correlations (see [22] for details) in which and so, expanding detðρ~TB Þ < 0 in the geometric represen- ρ zero discord for one party roughly corresponds to them tation, we find that is entangled if and only if a physical possessing a nondisturbing projective measurement. Within c cnˆ Q steering ellipsoid with center ¼ and matrix satisfies the geometric representation, it is readily seen that ρ has A E c4 − 2c2 1 − Q 2nˆ TQnˆ h Q < 0; zero discord for if and only if A degenerates to a radial ð tr þ Þþ ð Þ ð4Þ line segment, while ρ has zero discord for B if and only if ffiffiffiffiffiffiffiffiffiffiffi p 2 2 EA is one dimensional and b ¼ 2jcA − aj=lA, where lA is the where hðQÞ≔1–8 det Q þ 2trðQ Þ − ½trðQÞ − 2trðQÞ, length of EA [9]. and we drop A; B labels as entanglement is a “symmetric” To illustrate the effect of the alignment of EA on the relation. This equation is manifestly invariant under global entanglement and discord of a state, we can consider a rotations, corresponding to local unitaries on the quantum ρ θ 1 1 state, and shows that correlations between the qubits one-parameterP family of states of the form ð Þ¼4 ð þ 1 σ ⊗ 1 T θ σ ⊗ σ manifest themselves in three geometric ways: (1) the dis- 2 z þ ij ijð Þ i jÞ, for which the ellipsoid tance c of the ellipsoid center from the origin, (2) the size of skew varies smoothly with θ while maintaining a constant “ ” nˆ TQnˆ E T θ the ellipsoid, and (3) its skew, captured by the term , volume for A.p Specifically,ffiffiffiffiffiffiffiffiffiffiffiffiffi we have that ð Þ¼ T which reflects the alignment of the ellipsoid relative to the RyðθÞKRy ðθÞ¼ QAðθÞ, and so RyðθÞ ∈ SOð3Þ gener- nˆ radial direction described by center unit vector . ates states with inequivalent correlations via rotation of The nested tetrahedron condition.—The condition for 1 T the steering ellipsoid around its own center cA ¼ð0; 0; Þ , entanglement given by equation (4) provides a compact 2 “ ” algebraic condition for nonseparability and uncovers con- note that this internal rotation is distinct from a global tributions from different geometric aspects. However, the rotation generated by a local unitary on the state. We K − 9=20 ; − 3=10 ; − 3=10 steering ellipsoid captures the distinction between sepa- choose ¼ diagð ð Þ ð Þ ð ÞÞ so that rable and nonseparable states in another elegant way: A ρðθÞ ≥ 0, for all θ ∈ ½0; πÞ. This family of states illustrates two-qubit state ρ is separable if and only if its steering opposing behavior of the discord and concurrence as a ellipsoid EA fits inside a tetrahedron that fits inside the function of θ, see Fig. 3. The entanglement favors an Bloch sphere. To prove necessity,P suppose Alice and Bob orientation in which the longest semiaxis is aligned (radial) ρ n p α ⊗ β c θ π=2 share a separable state ¼ i¼1 i i i. Since we can with A at the point ¼ , while discord is maximized always take n ≤ 4 [20], the Bloch vectors of the αi when the short semiaxis is radial, at θ ¼ 0; π [23]. define a (possibly degenerate) tetrahedron T within Volume of the ellipsoid.—The expression for the volume ˆ Alice’s Bloch sphere. Bob’s outcome E collapses Alice of EA provides a compact and nontrivial relation between

020402-3 week ending PRL 113, 020402 (2014) PHYSICAL REVIEW LETTERS 11 JULY 2014

FIG. 3 (color online). Discord (solid curve) and concurrence (dotted curve) of the state ρðθÞ as a function of the orientation θ of the ellipsoid. Entanglement is maximized when the major axis is radial.

matrix in equation (2), which may be rewritten as 2 2 VA ¼ð4π=3Þðj det Θj=ð1 − b Þ Þ. However, it turns out [9] that det Θ ¼ 16ðdet ρTB − det ρÞ; therefore,

64π j det ρ − det ρTB j VA ¼ : ð5Þ 3 ð1 − b2Þ2

The EB volume follows from VA via the simple rela- 2 2 2 2 tion VB ¼ VAð1 − b Þ =ð1 − a Þ . The ellipsoid volume is a nonlinear entanglement criterion. Specifically, the Werner state on the separable- entangled boundary has EA being the maximal sphere volume V⋆ ¼ 4π=81 inscribed inside the largest possible tetrahedron that can be inscribed inside the unit sphere [25]. We immediately deduce that any state with EA that has volume V>V⋆ must be entangled. Note that entangled states can have V ≤ V⋆. V>V FIG. 2 (color online). The classes of steering ellipsoids. Here, Since ⋆ can only be attained by entangled states, “Separable” (Sep.) and “Entangled” (Ent.) label the type of corre- whilst zero discord states have one-dimensional (degener- lation, while“Incomplete”(Incomp.) and“Complete”(Comp.) label ate) ellipsoids, we see that nonzero volume, or “obesity,” the type of steering. The large (green) dot is the reduced state in the is strictly stronger than discord but strictly weaker than respective Bloch sphere. States with EA, being three dimensional, entanglement. have nonzero volume (or simply “obesity”), and these are either Conclusion.—The quantum steering ellipsoid provides a ρ E entangled or separable. The state is separable if and only if A fits faithful representation of any two-qubit state and a natural A insideatetrahedroninsidetheBlochsphereof .Forseparablestates, geometric classification of states. It yields clear and the set EA can also be two dimensional (a steering pancake), or one dimensional (a steering needle), or trivially zero dimensional (not intuitive understanding into the usual key aspects of shown).Forthese cases,steering iseither“complete,” if all ensemble two-qubit states, uncovers surprising new features (such decompositions of a in EA are attainable (when the span of EB as the nested tetrahedron condition, skew and obesity, and 1 contains 2 1), otherwise, the steering is “incomplete.” Zero discord incomplete steering) while prompting novel questions, occurs only for radial steering needles. such as: can we use (4) to define a class of “least-classical” separable states for fixed (a, b, c)? Can we use the nested the steering properties of ρ and the ranks of ρ and ρTB . tetrahedron condition to provide a simple construction for The volume of any ellipsoid is proportional to the product the best separable approximation [28] for a state ρ? What is of its semiaxes V 4π=3 s1s2s3. Therefore, EA ¼ðpffiffiffiffiffiffiffiffiffiffiffiffiffiÞ the geometric characterization of the possible steering has volume VA ¼ð4π=3Þ det QA. Using the ellipsoid ellipsoids for an arbitrary two-qubit state? This would

020402-4 week ending PRL 113, 020402 (2014) PHYSICAL REVIEW LETTERS 11 JULY 2014 potentially be useful when using ellipsoids to visualize the [13] L. Chen and D. Ž. Đoković, Phys. Rev. A 86,062332 results of two-qubit state tomography. In [9],wehave (2012). provided a discussion of several extensions to the work [14] A. Milne, S. Jevtic, D. Jennings, H. Wiseman, and described in this Letter, beyond the two-qubit scenario to T. Rudolph, arXiv:1403.0418. higher dimensional systems. [15] F. Verstraete, J. Dehaene, and B. DeMoor, Phys. Rev. A 64, 010101 (2001). We wish to acknowledge Antony Milne for his useful [16] J. B. Altepeter, E. R. Jeffrey, M. Medic, and P. Kumar, in comments, and Zuzana Gavorova and Peter Lewis for their Conference on Lasers and Electro-Optics, 2009 and 2009 early contributions to this topic. M. P. and S. J. acknowl- Quantum Electronics and Laser Science Conference [i.e. edges support by the EPSRC; S. J. acknowledges funding International Quantum Electronics Conference]: CLEO/ QELS [i.e. CLEO/IQEC] 2009; June 2009, Baltimore, MD, by EPSRC Grant No. EP/K022512/1; D. J. acknowledges USA; [including] PhAST 2009 (IEEE/Optical Society of funding by the Royal Commission for the Exhibition of America, Piscataway, NJ, 2009), p. IWC1. 1851; T. R. acknowledges support by the Leverhulme [17] A. Peres, Phys. Rev. Lett. 77, 1413 (1996). Trust. Research at Perimeter Institute is supported in part [18] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. by the Government of Canada through NSERC and by the A 223, 1 (1996). Province of Ontario through MRI. [19] A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A 58, 826 (1998). [20] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [21] M. Vidrighin, “mathoverflow.net/questions/109153,” [1] J. E. Avron, G. Bisker, and O. Kenneth, J. Math. Phys. (2012). (N.Y.) 48, 102107 (2007). [22] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 [2] R. Horodecki and M. Horodecki, Phys. Rev. A 54, 1838 (2001). (1996). [23] Discord can be calculated analytically when θ ¼ 0, π=2, and [3] I. Bengtsson and K. Życzkowski, Geometry of Quantum at these points ρ is an X state [24] and this agrees with our States: An Introduction to Quantum Entanglement findings. (Cambridge University Press, Cambridge, England, 2006). [24] C. S. Mingjun Shi, Fengjian Jiang, and J. Du, New J. Phys. [4] E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 (1935). 13, 073016 (2011). [5] N. Gisin, Phys. Lett. A 210, 151 (1996). [25] More formally, it was shown [26,27] that every convex body [6] L. P. Hughston, R. Jozsa, and W. K. Wootters, Phys. Lett. A n in R Euclidean space contains a unique maximal volume 183, 14 (1993). ellipsoid, and that a Euclidean ball is that maximum if and [7] R. W. Spekkens and T. Rudolph, Quantum Inf. Comput. 2, only if there exist vectors fxkg in the set of contact points of 66 (2001). the ball withP the convex body,P and positive numbers fckg [8] F. Verstraete, Ph.D. thesis, Katholieke Universiteit Leuven T 2 such that ckxk ¼ 0 and ckx xk ¼ r 1,wherer is the (2002). k k k radius of the ball. Inspection of the insphere for the regular [9] See Supplemental Material at http://link.aps.org/ tetrahedron in the Bloch sphere shows that it is the case for supplemental/10.1103/PhysRevLett.113.020402 which in- fxkg being the four intersection points. cludes Refs. [10–14], for details. [26] F. John, in Studies and Essays Presented to R. Courant on [10] R. Frosch, Four-tensors, the Mother Tongue of Classical his 60th Birthday, Jan. 8, 1948 (Interscience, New York, Physics (VDF Hochschulverlag, Zurich, 2006). 1948), pp. 187–204. [11] B. Dakić, V. Vedral, and Č. Brukner, Phys. Rev. Lett. 105, [27] K. Ball, Geometriae Dedicata 41, 241 (1992). 190502 (2010). [28] M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 [12] S. K. Goyal, B. Neethi Simon, R. Singh, and S. Simon, (1998). arXiv:1111.4427.

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