Corrigendum: Quantum steering ellipsoids, extremal physical states and monogamy (2014New J. Phys. 16 083017)

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Corrigendum: Quantum steering ellipsoids, extremal physical states and monogamy (2014 New J. Phys. 16 083017)

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New J. Phys.17(2015) 019501 doi:10.1088/1367-2630/17/1/019501

CORRIGENDUM

Corrigendum: Quantum steering ellipsoids, extremal physical states and monogamy (2014 New J. Phys. 16 083017)

Antony Milne¹, Sania Jevtic², David Jennings¹, Howard Wiseman³and Terry Rudolph¹

1 Controlled Quantum Dynamics Theory, Department of Physics, Imperial College London, London SW7 2AZ, UK

2 Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK

3 Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia

E-mail:antony.milne@gmail.com

The proof of theorem 6(a) is incorrect, although the monogamy of steering result VA B∣ + VC B∣ ⩽ 4^π 3

holds as stated. Here we give a corrected proof, which reveals a remarkable new result relating the volume of Aliceʼs steering ellipsoid to the centre of Charlieʼs:VA B∣ = ^πcC B

4 3

2 . We are very grateful to Michael Hall for identifying the mistake, independently verifying our numerical tests and assisting with the corrected proof.

Proof.The pure three-qubit state held by Alice, Bob and Charlie is∣ϕ_ABC〉. The canonical transformation

 

ϕ ϕ ϕ

∣ 〉 → ∣∼ 〉 = ⊗ ⊗ ∣ 〉

( ρ )

ABC ABC ABC

1

2_B leavesA B^∣ andC B∣ invariant. To prove the monogamy of steering we therefore need consider only canonical states for which∼=

b 0. (Whenρ_Bis singular and the canonical transformation cannot be performed, no steering by Bob is possible; we then haveVA B∣ =VC B∣ =0so that the bound holds trivially.)

We begin by showing thatV_{A B}∣ = ^4πc_{C B}

3

2 . Denote the eigenvalues ofρ∼ = ∣ϕ∼ 〉〈ϕ∼ ∣

AB trC ABC ABC as{ }λi . For a canonical state Charlieʼs Bloch vector coincides withcC B∣ , and soρ∼ = ∣ϕ∼ 〉〈ϕ∼ ∣ = + σ

c ∣

tr ( · )

C AB ABC ABC C B

1

2 .

By Schmidt decomposition we therefore have{ }λ_i ={ (1¹ + c_{C B}∣ ), (1−c_{C B}∣ ), 0, 0}

2

1

2 .

From the expression forV_{A B}∣ given in [1] we obtainV_{A B}∣ =^64π ∣detρ∼_AB∣

3

TA sincedetρ∼_AB= 0. Define the reduction map [2,3] asΛ( )X =trX− X. Following [4] we note thatdetρ∼_AB^TA = det ((σ_y ⊗)ρ∼_AB^TA(σ_y ⊗)) and that(σy ⊗)ρ∼ABT (σy ⊗)=(Λ⊗)(ρ∼AB)= 1⊗−ρ∼AB

2

A , where we have used the fact that Bobʼs local

state is maximally mixed. Since the eigenvalues of¹⊗− ∼ρ_AB

2 are{1 −λi}

2 we obtaindetρ∼_AB^TA=∏_i ⁽1 ^{− λ}i⁾ 2

=(−1cC B∣ )( cC B∣ )( )( )

2 1 2

1 2

1

2 =−¹c_{C B}

16

2 , which givesV_{A B}∣ =^4πc_{C B}

3 2 . From theorem 3 we haveVC B∣ ⩽ Vcmax

C B =^4π(1−c_{C B}∣ )

3

2. Hence V_{A B}∣ + V_{C B}∣ ⩽ ^4πc_{C B}∣

3 +

π − c ∣

(1 _{C B})

4

3 = ^4π

3 .

□

References

[1] Jevtic S, Pusey M, Jennings D and Rudolph T 2014Phys. Rev. Lett.113020402 [2] Cerf N J and Adami C 1999Phys. Rev.A60893

[3] Horodecki M and Horodecki P 1999Phys. Rev.A594206

[4] Augusiak R, Demianowicz M and Horodecki P 2008Phys. Rev.A77030301 OPEN ACCESS

RECEIVED

22 December 2014

ACCEPTED FOR PUBLICATION

22 December 2014

PUBLISHED

27 January 2015

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