Super-Activation of Quantum Steering

The whole is other than the sum of the parts Kurt Koffka

“Canρ⊗ρ be nonlocal whenρ is local?” This question led physicists to a phenomenon calledsuper activation of quantum nonlocality. We start this chapter by discussing known results on Bell nonlocality in the regime where the parties can perform collective measurements on many copies of a given quantum state. We then introduce the concept of super activation of quantum steering and explore some particularities of EPR-steering to significantly enlarge the set of states that can have their (EPR) nonlocality super activated.

In particular, we show that all entangled two-qubit states are steerable when sufficiently many copies are considered.

5.1 The Many Copies Paradigm On Bell Nonlocality

Letρbe a bipartite state acting onC^d⊗C^dshared by Alice and Bob. Alice’s local measurements are then described by POVM elements, positive semi-definite operatorsAaacting onC^d. If we now consider that the parties share two copies ofρ, Alice can perform measurements acting on the larger space C^d⊗_C^d. Some positive operators acting in C^d⊗_C^d cannot be written in the separable form ofA1⊗A2nor as convex combinations of operators that factorise. Measurements including POVM elements that cannot be described by convex combinations of positive operators that do not factorise are refereed asentangling measurementsor collective measurements. Collective measure-ments play an important role in some quantum information tasks such as quantum teleportation [90] and in this chapter we exploit the power of such measurements in nonlocality scenarios.

In a standard quantum Bell/EPR-steering scenario, the parties perform measurements on a single-copy of an entangled quantum stateρduring many rounds and later analyses the final statistical data. In order to have enough statistics for deciding if the state is Bell/EPR nonlocal, the parties should have

64

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 65 access to many copies ofρusing a single copy in each round. Since Alice and Bob share many copies ofρ, one can imagine a scenario where in each round of the experiment, instead of the parties performing measurements on a single copy of the state, they perform collective measurements onkcopies ofρ.

In a seminal paper, Peres considered a Bell scenario where the parties have access to probabilistic pre-processing and many copies of some specific bipartite state [91] to show that some states that are local in the standard Bell regime can become nonlocal when probabilistic local pre-processing and many copies are considered. Later, without considering any pre-processing, Liang and Doherty [92] showed that the maximal violation of the CHSH inequality [16] of a particular state ρcan be increased if the parties perform measurements on multiple copies of it. A next step was made by Navascués and Vértesi [93] who have shown that there exists bipartite statesρthat do not violate the CHSH inequality butρ⊗ρdoes.

The first example of super activation of quantum nonlocality appeared in 2012when Palazuelos exploited a phenomenon named asunbounded violations of Bell inequalities[94] to find a family of entangled states that have a LHV model for general POVMs but do violate a Bell inequality when multiple copies shared by the parties are jointly measured [95]. Exploring the methods proposed by Palazuelos, Cavalcantiet alpresented a general criterion that is sufficient to show that a stateρABviolates a Bell inequality when sufficiently many copies are considered, establishing a strong connection with quantum teleportation [96,44].

In this chapter we consider steering scenarios where Alice and Bob share many copies of a particular bipartite state to present a simple sufficient criterion for obtaining steering in this many copies regime. We remark that the set of states captured by our criteria is considerably larger than the set detected by the criterion of Cavalcantiet aland we also prove that all two-qubit states and qubit-qutrit states are EPR-steerable for some number of copiesk.

5.2 Super Activation of Bell Nonlocality

In this section we will reproduce the proof presented by Cavalcantiet al[96] that explores some methods of Palazuelos [95]. An important step of the proof relies on the existence ofunbounded violations of Bell inequalities[94], in particular, on the Khot-Visnoi (KV) game [97]. The KV game is a bipartite Bell inequality for a scenario where the parties haveN= ²_d^d inputs anddoutputs per input. In the standard notation presented at equation (1.4), the KV game takes the form

B_KV =

∑

abxy

c_ab|xyp(ab|xy)≤ ^C

d, (5.1)

with positive coefficientscab|xy≥0 and a universal constantCthat does not depend on the number of inputs or outputs.

Consider now that Alice and Bob share ad-dimensional maximally en-tangled stateφ⁺_d and use it to play the above described KV game withN= ²_d^d inputs and d outputs per input. For this scenario, one can explicitly find

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 66

projective measurementsA_a|xandB_b|yand guarantee that B_KV^Q =

∑

abxy

cab|xytr(φ⁺_dAa|x⊗Bb|y)≥ ^C

0

ln²d (5.2) for some universal constant C⁰ [98]. With this quantum strategy we can verify that, regardless the values of the universal constants C and C⁰, the ratio between attainable quantum value and maximal local one ^d

ln²d C C⁰ grows arbitrarily as a function of the dimension of the quantum systemd. We see that by increasing the dimensiondof their local systems, Alice and Bob can have an unbounded violation of a Bell inequality.

Observe now that that there is a formal mathematical equivalence between two parties sharing kcopies of ad-dimensional maximally entangled state φ_d⁺

= _∑^d−1_i=0|iii and two parties sharing a single copy of a d^k maximally entangled one. That is,

φ⁺_d⊗k

=

d−1

∑

i=0

|iii

!⊗k

=

d^k−1 i=0

∑

|iii=φ⁺_dk

E. (5.3)

We now analyse kcopies of ad-dimensional isotropic state ISO_d =Vφ⁺_d + (1−V)_d^I₂, which can also be parametrised by its entanglement fraction¹

F=tr(φ⁺_dISO_d) =V+(1−V)

d² , (5.4)

to assume the form

ISO=Fφ⁺_d + (1−F)^I−φ⁺_d

d²−1. (5.5) The advantage of parametrising the isotropic state in terms its entanglement fractionF and not in terms of its visibilityVis that ^I−φ

+ d

d²−1 lies in a subspace that is orthogonal toφ_d⁺, hence roughly speaking,Fquantifies the “amount of maximally entangled state” in the isotropic state. Exploring the entanglement fraction parametrisation, we see that when two parties sharek-copies ofISO_d their final state can be described by

ISO^⊗k_d =F^kφ⁺_kn+ (₁−F^k)ρ_N, (5.6) whereρ_Nis a quantum state that is orthogonal toφ⁺_dk,i.e., tr(φ⁺_dkρ_N) =0.

With the above equation, we can lower bound the minimal d^k-output KV inequality violation ofkcopies of a d-dimensional isotropic state with

1The entanglement fractionFof a bipartite stateρis defined as its trace with the maximally entangled stateF:=tr(ρφ⁺_d).

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 67

entanglement fractionFas follows tr

BKVISO^⊗k_d

=

∑

abxy

cab|xytr

Aa|x⊗Bb|yISO^⊗k_d

=

∑

abxy

c_ab|xytr

A_a|x⊗B_b|yF^kφ_d⁺_k+ (1−F^k)ρN

≥F^k

∑

abxy

c_ab|xytr

A_a|x⊗B_b|yφ_d⁺_k

≥ ^F

kC⁰

ln²(d^k)^{. (}

5.7) (5.8) Since the local bound of thed^k-output KV inequality is given by _d^C_k we have a Bell violation whenever the inequality

F^kC⁰ ln²(d^k) > ^C

d^k ⇐⇒ (Fd)^k>kln²(d)^C

C⁰ (5.9) is satisfied. It follows then that ifF>1/d, there will always exist a number of copiesksuch thatISO_d^⊗kis nonlocal.

In order to tackle non-isotropic states, we remark that any bipartite quantum state ρ can be mapped into an isotropic state with local unitary operations and shared randomness by preserving its entanglement fraction tr(ρφ_d⁺)via anisotropic twirling operation[44]. That is, letρbe a bipartite state shared by Alice and Bob, we always have the identity

Z

UU⊗UρU^†⊗U^†dU =tr(ρφ⁺_d)φ⁺_d + (1−tr(ρφ⁺_d))^I−φ⁺_d

d²−1 (5.10) where U is the complex conjugate ofU and the integral is taken over all unitary operators acting onC^duniformly distributed by the Haar measure [44].

We now observe that since local unitaries cannot transform a Bell-local state into a nonlocal one, Alice and Bob can perform local unitariesUAand UBon their states before applying the twirling operation. That motivates the definition ofmaximal entanglement fraction

Fmax = max

U_A,U_Btr(UA⊗UBρU_A^† ⊗U^†_Bφ_d⁺), (5.11) and we now see that any quantum state acting onC^d⊗C^dthat has a maximal entanglement fraction is greater than 1/dviolates a Bell inequality for some sufficiently large number of copiesk.

5.3 Super Activation of EPR-Steering

We now address the question of super activating EPR-steering. How can Alice exploit collective measurements to steer Bob when they share several copies

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 68 of a quantum state? We first point out that, since all Bell-nonlocal states are steerable it follows that allk-copy Bell-nonlocal states are alsok-copy (two-way) steerable. In this section we exploit some particularities of EPR-steering to greatly increase the set ofk-copy steerable states. In particular, we show that all quantum states violating thereduction criterion[99], a useful criterion for entanglement detection, arek-copy steerable.

Main Result7. All quantum statesρ_ABviolating the reduction criterion,i.e., for which the operator

I⊗ρB−ρAB (5.12)

is not positive definite, are k-copy steerable from Alice to Bob. In particular, all entangled two-qubit and qubit-qutrit states are two-way steerable.

The first step to get our main result is to notice a simple corollary of lemma 2of chapter2.

Corollary 3. Letρ be a bipartite entangled state, Λ^⊗k, k ∈ _Nbe a positive map acting on Bob’s system, such that

ρ_F = (I⊗_Λ)(ρ)

tr[(I⊗Λ)(ρ)]^{. (5.13)} If Alice can steer Bob withρ^⊗n_F , Alice can steer Bob with the stateρ^⊗n.

We remark here that the positive mapsΛfor whichΛ^⊗k is also positive are calledk-tensor stable positive maps. If a map isk-tensor stable positive for all k∈ _Nwe simply say that a map istensor stable. The only known tensor stable positive maps up to now are completely positive maps and completely positive maps composed with transposition²[100].

It can be shown that states violating the reduction criterion,i.e.,I⊗ρ_B−ρ_AB is not positive, are entangled [99]. Moreover, by applying local filtering op-erations on Bob’s side, a state violating this criteria can be transformed into another state ρ⁰_AB that satisfies tr(φ_d⁺ρ⁰_AB) > 1/d. This was first shown in reference [99] and since the proof can bring some intuition, we also present it here.

Lemma7. Letρ_ABbe a bipartite state acting onC^d⊗C^din which I⊗ρ_B−ρ_ABis not positive definite. There exists a local filter operation F_Bsuch that the filtered state

ρ⁰_AB= ^I⊗F_B^†ρ_ABI⊗FB

tr(I⊗F_B^†ρ_ABI⊗F_B) ⁽ 5.14) satisfiestr(ρ⁰_ABφ_d⁺)>1/d.

Proof. If the operator I⊗_Λ(ρ_AB)is not positive definite, there exists a pure quantum stateψsuch that

2Actually, the existence of any tensor stable positive map that is not completely positive and not a composition of a completely positive map with transposition would imply the existence of an NPT-bound entangled state, solving a big open problem on entanglement distillation.

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 69

tr((I⊗ρB−ρ_AB)ψ)<0 (5.15) or equivalently, tr(ρABψ)>tr(ρBψB). If|ψi=_∑ijαij|iji, define

FB:=_∑ijα_ij

√

d|iihj|whereα_ij is the complex conjugate ofα_ij. so that|ψi= I⊗F_B^†

ψ_d⁺

, and construct the filtered state is

ρ⁰_AB:= ^I⊗F_B^†ρ_ABI⊗F_B

tr(I⊗F_B^†ρ_ABI⊗FB)^{. (} 5.16) Now, straightforward calculation shows thatF_B^†FB=dId, and

tr(I⊗F_B^†ρABI⊗FB) =dtr(ρBψB),

tr(φ_d⁺I⊗F_B^†ρ_ABI⊗FB) =tr(ψρ_AB) (5.17) and since we have tr(ρ_ABψ)>_tr(ρ_Bψ_B)by hypothesis, it follows that

tr(ρ⁰_ABφ⁺_d)>1/d. (5.18)

Since Bell nonlocal states are necessarily (two way) steerable. Any state ρ_AB that violates the reduction criterion can be transformed into another one that has an entanglement fraction tr(ρABφ_d⁺)greater then 1/d. Hence, it follows from the result by Cavalcantiet al[96] that all quantum states violating the reduction criteria are (two way) steerable (see also section 5.2). In the subsection5.3.3we present an alternative proof of our main result that entirely focuses on EPR-steering and do not rely on the super-activation of nonlocality.

Our alternative proofs are also “more economical” in terms of the number of copies and dimension required for having the super-activation of EPR-steering phenomenon.

5.3.1 All Entangled Two-Qubit States arek Copy Steerable

As announced before, all entangled two-qubit states arek-copy steerable for some number of copiesk. This result follows from fact that all entangled two-qubit states violate the reduction criterion. Actually, all entangledqubit-qutrit states violate the reduction criterion and in the next subsection we will discuss how our super activation results applies to quantum in which Alice and Bob have different local dimension.

The original proof that all entangled qubit-qutrit states violate the reduc-tion criterion be found in reference [99] and we reproduce it here for the sake of completeness.

Lemma8. A quantum stateρABacting onC²⊗_C³(qubit-qutrit) is separable iff I⊗ρ_B−ρ_AB≥0.

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 70 Proof. Before starting with the proof we define the positive (but not completely positive) operator

Λ(A):=Itr(A)−A, A:L(H_d)→L(H_d) (5.19) and notice that for any quantum stateρ_ABwe have: I⊗Λ(ρ_AB) =I⊗ρ_B−ρ_AB. Parametrising a two-by-two matrix as

A=a₀₀|0ih0|+a₀₁|0ih1|+a₁₀|1ih0|+a₁₁|1ih1|,

andσy=i|0ih1| −i|1ih0|. One can check by straightforward calculation that Λ(A):=Itr(A)−A

=Itr(A)−a₀₀|0ih0| −a₀₁|0ih1| −a₁₀|1ih0| −a₁₁|1ih1|

=a11|0ih0| −a01|0ih1| −a10|1ih0|+a00|1ih1|

= (σyAσy)^T. (5.20)

In other words, when applied to qubits, the mapΛcan be seen as a unitary transformation followed by the transposition map. Since unitary transform-ations do not change the spectrum of linear operators, we finish the proof by recalling that a qutrit-qubit stateρ_ABis separable if and only ifρ^T_AB^b ≥0 [55,56].

5.3.2 The Case of Asymmetric Local Dimension

In previous sections we have assumed that Alice and Bob have the same local dimensiond, but in principle, one can also consider states with different local dimensions. For instance, let ρ_AB be a bipartite state acting onC^dA ⊗_C^dB. Before proceeding, observe that any vector lying onC^dA can be embedded into the larger spaceC^dB and physically we can always understand qubit a qutrit in which its “qutrit component” is zero. This simple observation allows one to understand any stateρacting onC^dA⊗C^dB ans an stateρ acting on C^d⊗C^dwhered=_max(d_A,d_B). Adopting this convention we can apply our criterion presented in main result7to any bipartite state with asymmetric local dimensions.

For the particular case of entangled qubit-qutrit states, we know by lemma 8that they always violate the reduction criterion. Hence by combining lemma 3and lemma7it follows than that all qubit-qutrit state arek-copy steerable from the qubit part to the qutrit one.

5.3.3 An Alternative Proof For Super-Activation of Quantum Steering

We start this section by pointing that when we have lower-bounded the minimal Bell inequality violation ofkcopies of ad-dimensional isotropic state ISO^⊗k_d =F^kφ⁺_dk+ (1−F^k)ρN in equation (5.7). We have completely neglected the possible contribution by the termρN when we have used the inequality tr(BKVρ_N)≥0. Since we are neglecting the potential contribution ofρ_N, Alice

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 71 and Bob could have also exploited the isotropic twirling operation [99] (see section5.2) to perform the transformation

F^kφ⁺_dk+ (₁−F^k)ρ_N 7→F^kφ_d⁺_k+ (₁−F^k)^I−φ⁺_dk

d^2k−1 (5.21) before playing the KV game and obtain the precisely same result.

Equation5.21shows that we can transformkcopies of ad-dimensional iso-tropic state with entanglement fractionFinto a single copy of ad^k-dimensional isotropic state with entanglement fractionF^kvia local unitary transformations and shared randomness. In order to obtain our new proof, we just explore the simple criterion to decide whether the isotropic stateF^kφ_d⁺_k+ (1−F^k)^I−φ

+ dk

d^2k−1

is steerable presented in section 1.5.2. In section1.5.2we have shown that ad-dimensional isotropic stateVφ⁺_d + (1−V)_d^I₂ is steerable with projective measurements if and only if [5] (see section1.5.2)

V> ^∑

di=11 i −1

d−1 . (5.22)

Since the entanglement fraction of the isotropic state Vφ_d⁺+ (1−V)_d^I₂ is F = tr(ISO_dφ⁺_d) = V+ ^1−V_d2 . The necessary and sufficient criterion for projective steerability of the isotropic state can be equivalently rephrased as

F> (1+d)(_∑^d_i=1¹_i −1)−d

d² . (5.23)

Now, we consider that Alice and Bob sharekcopies of a d-dimensional state . We can easily find a sufficient criterion for steerability as a function of the dimension d, number of copies k, and entanglement fraction F, by substituting the correct values on the above equations. We can then state the following result as.

Theorem 5. Let Alice and Bob share k copies of a d-dimensional isotropic state Fφ_d⁺+ (1−F)^I−φ

+ d

d²−1. If the inequality

F^k > (1+d^k)(_∑^d_i=1^{k 1}_k−1)−d^k

d^k2 (5.24)

is satisfied, the k-copy state is two-way steerable.

When compared to the previous proof, the main advantage of this one is that we can tackle not only the case where Alice and Bob share arbitrary many copies, but also the case where they havek copies. We also remark that for large number of copiesk, the criterion presented in equation (5.24) reduces toF>1/d. Hence, when the number of copies can be as large as we want, this alternative proof requires the same entanglement fraction to show steering than the proof presented in section5.2. Of course, this could not be different, since forF≤1/d, the isotropic state is separable. Also, as noticed before, both proofs are based on the steerability of the isotropic state. The main difference is that the one based on nonlocality (section5.2) focuses on a single Bell inequality, this new proof covers all possible steering inequalities for the isotropic state.

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 72

5.4 Requirements on Dimension and Number of Copies For Super Activation

In the previous sections we have discussed how to reveal steering of a quantum stateρ by performing measurements onkcopies of it. We now restrict our attention to states that admit a LHS model. We say that the steerability of this state can be super-activated whenρhas an LHS model butρ^⊗kis steerable for some number of copiesk. Before we assumed that the parties have access to a very large number of copies kand we did not care about how large k must be. Here we seek for minimal requirements on number of copies and the dimension of the initial system in order to have a super-activation phenomenon.

By exploring theorem5and the the LHS models discussed in section1.5.2 we can easily find some conditions for super-activation with small number of copies. The idea is to start with a single copy of the isotropic state that is guaranteed to have a LHS model, and then to use theorem5to show EPR-steering super-activation. Straightforward calculation lead to some minimal requirements in terms of dimension of the initial isotropic statedand number of copiesk. For instance, if Alice and Bob start with a two-qubit isototropic state, they can super-activate steering for projective measurements withk=7 copies, or activate steering for general POVMs withk=24 copies. If Alice and Bob only have two copies of ad-dimension isotropic state, they can activate steering for projective measurements with dimensiond=6 with the current methods, we do not know how to super-activate steering with general POVMs.

Before finishing this section we point a simple method to show super-activation of nonlocality with only two copies, that isρ has an LHS model (and by consequence, a LHV one) for general POVMs andρ⊗ρviolates a Bell inequality (and by consequence it also violates a steering one). This method is implicit in Palazuelos’ paper on super activation of nonlocality [95] and we presented it here for completeness.

1. Choose a bipartite stateρacting onC^d⊗_C^dthat has a LHS model for general POVMs and isk-copy Bell-nonlocal.

2. There is a critical number of copies kC > 1 such thatk ≤ kC implies thatρisk-copies Bell/EPR-local andk>kC implies thatρ iskcopies Bell/EPR-nonlocal.

3. Define a new bipartite stateρ⁰ as two parties sharingk_Ccopies ofρ 4. By construction, the stateρ⁰is Bell/EPR-local but two copies of it violate

a Bell/steering inequality.

We then see that we can transform any example ofk-copy super activation of nonlocality (Bell or EPR-steering) into another one that requires onlyk=₂ copies. The “price” to pay for this construction is that the dimension of the quantum state that shows super activation with two copies can become significantly larger than the initial one that can be super-activable withk>2 copies.

CHAPTER5. SUPER-ACTIVATION OF QUANTUM STEERING 73

5.5 Super Activation of Nonlocality And Entanglement

5.5 Super Activation of Nonlocality And Entanglement

Dans le document Quantum entanglement and measurement incompatibility as resources for nonlocality (Page 65-76)