Measurement Incompatibility vs EPR-steering

In the previous chapter we have dedicated ourselves to understanding which quantum states can and cannot lead to nonlocal correlations when arbitrary measurements are performed. But as mentioned before, quantum nonlocality is based on two central features of quantum theory, namely entanglement and incompatible measurements. In this chapter and the next one we explore the converse problem, that is, which sets of incompatible measurements lead to quantum nonlocality?

Incompatible Measurements EPR Steering

Quantum Entanglement

Figure3.1: The observation of EPR steering, a form of quantum nonlocality, implies the presence of both entanglement and incompatible measurements.

Whether the converse links hold is an interesting question. Here we solve the problem in this direction by showing that any set of incompatible measure-ments can be used to demonstrate EPR steering (green arrow).

In the particular case of projective measurements, it was shown that in-compatible measurements can always lead to Bell nonlocality [7,9]. Note that in this case, compatibility is uniquely captured by the notion of com-mutativity [67]. However, for general measurements,i.e., POVMs, no general result is known. In this case, there are several inequivalent notions of com-patibility. Here we focus on the notion of joint measurability, see e.g.[68], as this represents a natural choice in the context of quantum nonlocality. Some

46

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 47 previous works have addressed the connection between general incompatible measurements and nonlocality before [69,70]. One can show that every set of incompatibleprojectivemeasurements can lead to a CHSH Bell inequality violation [7,8] and the strongest known result is due to Wolf et al. [9], who showed that any set of two incompatible POVMs with binary outcomes can lead to violation of the CHSH Bell inequality. However, this result may not be extended to the general case (of an arbitrary number of POVMs with arbitrarily many outcomes), since pairwise joint measurability does not imply full joint measurability in general [71,72].

Here we explore the relation between compatibility of general quantum measurements and quantum nonlocality to show direct link between joint measurability and EPR steering. Specifically, we show that for any set of POVMs that is incompatible (i.e. not jointly measurable), one can find an entangled state, such that the resulting statistics violates a steering inequality.

Hence the use of incompatible is a necessary and sufficient ingredient for demonstrating EPR steering.

We also remark that the main result of this chapter was independently discovered by Uolaet al. [73].

3.1 Measurement Incompatibility in Quantum Mechanics

In quantum mechanics, we say that a set of measurements {M_a|x}^x=N_x=1 is jointly measurable when there exists a joint measurementM_~awith outcomes labelled as~a = (ax=1,ax=2, . . . ,ax=N) that can recover all measurement as marginals, that is

M_~a ≥0,

∑

~a

M_~a= I, M_a|x=

∑

~a\a_x

M_~a. (3.1) For instance, two dichotomic measurements{M_a1|1}and{M_a2|2}are jointly measurable if there exits a joint measurement{Ma₁,a₂}such that

M_a1|1= Ma₁,0+M_a1,1, M_a2|2= M0,a₂+M_1,a2. (3.2) The concept of joint measurability also admits an equivalent definition inspired by Fine’s theorem (see section1.2.2and reference [74]).

Lemma4. A set of measurements{Ma|x}, is jointly measurable if and only if there exist probability distributions p(.|x,λ)and a single POVM{M_λ}such that

M_a|x=

∑

λ

p(a|x,λ)M_λ (3.3) Proof. If the set of measurements{M_a|x}, is jointly measurable, there exits a joint measurement{M_~a}with~a= (a0,a1, . . . ,aN)such thatM_a|x=_∑~a\axM_~a. In order to show that we can write jointly measurable operators in the form of eq3.3, we just need to set our hidden variable asλ:=~aand the response functions asp(a|x,λ):=δa,a_x.

For the “only if” condition, we explore the existence of p(a|x,λ) and M_λ to defineM_~a :=_∑λ∏xp(a|x,λ)M_λ, with ∏xbeing the product over all

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 48 possible response functions. It is easy to verify that M_λ constitutes a valid quantum measurement, and by settingλ=~awe recover the definition of joint measurability presented in equation (3.1).

Although in this thesis we only explore joint measurability to classify measurement incompatibility, we remark the existence of some other notions of measurement incompatibility in quantum mechanics that have applications in different contexts. In particular, one can prove the strict hierarchy between commutation, non-disturbance [75], co-existence [76], and joint measurability, with joint measurability being the weaker notion of compatibility. Also, all these notions coincide for the particular case of projective measurements [67].

3.2 Equivalence Between Measurement Compatibility and EPR-Steering

We now show that a set of measurements can lead to EPR-steering correlations if and only if it is not jointly measurable. A quick look at the alternative formulation of joint measurability presented in lemma4and the definition of an unsteerable assemblage (definition4) may already suggest a connection between these concepts. We now present the main result of this chapter.

Main Result5. A set of measurements{M_a|x}acting onC^dis not jointly measur-able if and only there exits an entangled stateρ_ABacting inC^d⊗C^dsuch that the resulting assemblageσ_a|x=trA(ρ_ABMa|x⊗I)is steerable.

Proof. Since a set of compatible measurements can be seen as a single POVM with many outcomes, the ’if’ part is straightforward. We will show that{σ_a|x} admits a decomposition of the formσawhen{M_a|x}is jointly measurable, for any stateρAB. ConsiderM_~a, the joint observable for{M_a|x}, and define Alice’s local variable to beλ=~a, distributed according toΠ(~a) =tr(M_~aρ_A), where ρ_A =trB(ρ_AB). Next Alice sends the local stateσ_~a =trA(M_~a⊗IρAB)/Π(~a). When asked by Bob to perform measurementx, Alice announces an outcome

aaccording to p(a|x,~a) =δa,a_x.

We now move to the ’only if’ part. Since it suffices to show that there exists a quantum stateρABsuch that the assemblage trA(M_a|x⊗IρAB)is steerable, we set the stateρ_ABas the maximally entangled stateφ⁺=|φ⁺ihφ⁺|acting on dimensionC^d⊗_C^d.The identity M_a|y⊗I|φ⁺i=I⊗M^T_a|y|φ⁺i, withM_a|y^T being transposition ofMa|y, allows us to write the resulting assemblage as

σ_a|x=_trA(M_a|x⊗Iφ⁺) = ^M

T a|x

d . (3.4)

Our goal is now to show thatσ_a|xis steerable whenever{Ma|x}is not jointly measurable. For that we will look for a logical contradiction by supposing that {M_a|x} is not jointly measurable and σ_a|x is unsteerable. Since σ_a|x is unsteerable, there exist distributionsπ(.)andpA(a|x,λ)and quantum states ρ_λsuch that

σ_a|x=

∑

λ

π(λ)p(a|x,λ)ρ_λ, (3.5)

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 49 and identity3.4allows us to write

A_a|x=d

∑

λ

π(λ)p(ax|x,λ)ρ^T_λ. (3.6) We will now set M_λ := π(λ)ρ_λ^Td and show that M_λ is a valid quantum measurement, that is, positive and the sum of all elements is the identity operator. Clearly,M_λ ≥0, and in order to show that∑λM_λ=I, we sum on all possible valuesaon both sides of equality (3.6)

∑

a

A_a|x=

∑

a

d

∑

λ

π(λ)p(ax|x,λ)ρ^T_λ (3.7) I=d

∑

λ

π(λ)ρ^T_λ (3.8)

=

∑

λ

M_λ, (3.9)

and it follows from lemma 4 that {M_a|x} is jointly measurable, what is contradiction.

From a more practical perspective, the proof ot the above theorem shows that, if one part performs incompatible measurements{M_a|x}on a maximally entangled stateφ⁺, the resulting assemblageσ_a|x=trA(M_a|x⊗Iφ⁺)is guar-anteed to be steerable. Moreover, since reversible local filtering operations do not change the steerabiliy of an assemblage (corollary2), it follows that ifψis a pure entangled state¹of Schmidt rank²equal to the rank of the measurements {M_a|x}, the assemblage tr_A(M_a|x⊗Iψ)is also guaranteed to be steerable.

3.3 Applications Of The Correspondence Between Joint Measurability And EPR-Steering

Besides providing novel physical interpretations and a method to construct steering assemblages, the deep connection between steering and joint meas-urability can be used as a bridge to connect two different fields of quantum mechanics. In this section we present some of these applications.

3.3.1 Applying Results From EPR-steering to Measurement Incompatibility

Imagine that a physicist wants for performallprojective measurements jointly.

Clearly, that cannot be done perfectly, however that can become possible by working with noisy POVMs. In order to address this question, we define anη white noise POVM of a projective measurement{Pa}via

Pa^η:=ηPa+ (1−η)^I

dtrPa (3.10)

1Notice that all pure entangled states can written as|ψi= ^UA⊗FB|φ⁺i

kUA⊗FB|φ⁺ik, that is, a local unitary UAon one side, and a local filterFBon the other side.

2The Schmidt decomposition theorem [10] states that all pure entangled states|ψican be written as∑iαi|iiiwithαi≥0 for some orthonormal basis{|ii}. The Schmidt rank of|ψiis then the minimal number of non-zero coefficientsα_i.

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 50 to ask “For which noise parameterηwill all projective measurements become jointly measurable?”. Clearly,η=0 implies global joint measurability, but is there anη>0 such that we can also do so? These question was raised in [77] where the authors have showed thatη ≤ 1/2 is sufficient for the existence of a joint observable for all white noisyqubitmeasurements. We will now completely answer the above question by exploiting our main result5and the LHS model for the isotropic state presented in [5]. Since the isotropic state ηφ⁺_d + (1−η)I/d²can lead to projective steerable assemblages if and only if η> ^∑dk=_d−1^11k−1, we can use the identity

σ_a|x^η =trA

P_a|x⊗I

ηφ_d⁺+ (1−η) ^I d²

=trA

ηPa|x+ (1−η)^I d

⊗Iφ⁺_d

=tr_A

P_a|x^η ⊗Iφ_d⁺

(3.11) to show that the set of allηwhite noise projective measurements is incompat-ible if and only ifη> ^∑dk=_d−1^11k−1.

Moreover, following the same method, one can exploit the POVM LHS model for the isotropic state presented in section2.1(based on the LHV model of [45]) to get a similar result, now considering general POVMs. For that, we define anηwhite noise POVM via the map on the POVM elements acting on C^das

A^η_a :=ηAa+ (1−η)^I

dtr(Aa). (3.12) If follows then from the same argument we have used above that if we consider theseη-noisy POVMs and

η≤ (3d−1)(d−1)^d−1

(d+1)d^d , (3.13) it is possible to perform all possibled-dimensional POVMs jointly.

3.3.2 Applying Results From Measurement Incompatibility To EPR-steering

There is a simple criterion based on Bloch vectors [10] to decide if two dicho-tomic qubit measurements can be jointly performed [78,79]. If follows from our main result5that we can explore this criterion to decide whenever qubit assemblages with two dichotomic measurements are steerable. We now apply an established result on joint measurability to evaluate the probability of a two-qubit Werner state to be steerable when Alice performs two (uniformly) random projective measurements. The criterion for deciding whenever two dichotomic qubit measurements are jointly measurable presented in [78] is the following

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 51 Lemma5. Two dichotomic qubit measurements A₁and A₂with POVM elements described by

A_0|x= ¹

2(I+~ax·~σ), A_1|x=I−A_0|x; ~σ= [σx,σy,σz] (3.14) are jointly measurable iff

k~a1+~a2k+k~a1−~a2k ≤2 (3.15) With the joint measurability criterion presented above we can find a simple criterion for joint measurability of two Bloch vectors with normηseparated by an angleφ∈[0,π/4].

Lemma6. Two dichotomic qubit measurements A₁and A₂with POVM elements described by

A_0|x= ¹

2(I+~ax·~σ), A_1|x=I−A_0|x; ~σ= [σ_x,σ_y,σ_z] (3.16) with Bloch same vector norms k~a1k = k~a2k = η and separated by an angleφ ∈ [0,π/4]are jointly measurable iff

η≤

s 1

1+sinφ or equivalently φ≤sin⁻¹ 1

η²−1

(3.17) Proof. Without loss of generality we assume that the vector~a1is pointing in thezdirection and that~_a2=η(cosφzˆ+sinφxˆ). The proof now follows from straightforward calculation. We start by substituting the particular choice of vectors in the left hand side of the inequality (3.15)

k~a1+~a2k+k~a1−~a2k=η(k(1+cosφ)zˆ+sinφxˆk+k(1−cosφ)zˆ−sinφxˆk)

=η

q(₁+_cosφ)²+_sin²φ+ q

(₁+_cosφ)²+_sin²φ

=η p

2+_{2 cos}φ+^p₂−2 cosφ

. (3.18)

Then we explore trigonometric relations to obtain h

η p

2+2 cosφ+^p2−2 cosφ i2

=η² 4+4 sinφ

. (3.19) With the above identity, we see that for our particular choice of vectors, the inequality (3.15) now reads as

η≤

s 1 1+_sinφ.

We now present one application of the above lemma, where we evaluate the probability of Alice steering Bob by performing two uniformly random projective measurements on a two-qubit Werner state.

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 52 Theorem4. The probability of Bob having a steering assemblage when Alice per-forms two uniformly random projective measurements on a two-qubit Werner state ηψ⁻+ (1−η)₄^I is given by

p(η) = s

2η²−1

η⁴ (3.20)

0.75 0.8 0.85 0.9 0.95 1

0 0.2 0.4 0.6 0.8 1

Visibility parameterη

ProbabilityofHavingSteering

Figure 3.2: Probability of having a steering assemblage when Alice per-forms two uniformly random measurements on a two-qubit Werner state W₂=ηψ⁻+ (1−η)₄^I.

Proof. We can obtain the probability of the Werner state ηψ⁻+ (1−η)₄^I violating a steering inequality with two (uniformly) random projective meas-urements by integrating over all possible measmeas-urements by Alice. It follows from the identity

σ_a|x^η =tr_A

P_a|x⊗I

ηψ⁻+ (1−η)^I 4

=trA

ηP_a|x+ (1−η)^I 2

⊗Iψ⁻

=tr_A

P_a|x^η ⊗Iψ⁻

, (3.21)

that having anη-white noise singlet is equivalent than having aη-white noisy projective measurements. And the main result5 states that the resulting assemblage is steerable if and only if theη white noisy measurements are jointly measurable. We can now use the necessary and sufficient condition presented in lemma6to obtain our desired probability distribution. It is then enough to evaluate the integral over all possible measurements that satisfy

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 53

φ≤sin⁻¹

1 η² −1

. We then complete the proof by evaluating the integral p(η) =

Z

φ∈[0,π/2]H

sin⁻¹ 1

η²−1

sinφdφ

=

Z _φ=π/2 φ=sin⁻¹

1 η2−1

sinφdφ

=cos

sin⁻¹ 1

η² −1

−cos(π/2)

= s

1− 1

η²−1 2

= s

2η²−1

η⁴ , (3.22)

whereHhere stands for the Heaviside function (H(x) =1 ifx ≥0 and H(x) =0 ifx<0).

3.3.3 A direct correspondence Between Joint Measurability and EPR-Steering

The main difference between assemblages and quantum measurements is the fact that while the sum over outcomes of assemblages should be a quantum state, measurements should sum to the identity operator. More specifically, one can define valid assemblages and a valid set of measurements as

σ_a|x≥0,

∑

a

σ_a|x=ρ_B∀x, with trρ_B=_{1; (}³_.²³₎ M_a|x≥0,

∑

a

M_a|x= I∀x. (3.24)

These sets of operators are, respectively, unsteerable and jointly measurable whenever the exists distributionsπ(.),p(.|x,λ), statesρ_λand measurements M_λsuch that

σ_a|x=

∑

λ

π(λ)pA(a|x,λ)ρ_λ; M_a|x=

∑

λ

pA(a|x,λ)M_λ. (3.25) We now recall that ifΛis a positive operator that has an inverseΛ⁻¹that is also positive, the assemblageσ_a|xis steerable if and only if _tr^Λ(σ_Λ(ρ^a|x)

B) is steerable (corollary2). Notice that by setting

Λ(σ_a|x):=ρ^−1/2_B σ_a|xρ^−1/2_B (3.26)

CHAPTER3. MEASUREMENT INCOMPATIBILITY VS EPR-STEERING 54 we can transform essentially³ any assemblage satisfying∑aσ_a|x = ρ_B into another one that satisfies∑aσ_a|x⁰ = _d^I without loosing its steering property.

Hence, the transformation Λ(σ_a|x) = ρ^−1/2_B σ_a|xρ^−1/2_B allows us to directly translate the question “Is this assemblage steerable?” to “Is the set of meas-urements jointly measurable?”.

This observation was also made by Uolaet al, who exploited this connection to translate results from one field to the other [80].

3.4 Final Remarks

This deep connection between EPR-steerability of assemblages and joint meas-urability of quantum measurements provides new physical interpretations for both concepts. From a practical perceptive, it allows non-trivial results from one field to be easily exported to the other. We have presented some of these applications here and we believe that more interesting results can be also extracted.

We also mention that EPR-steering has a natural definition on multipartite systems. In this case, one define some parties as “black boxes”, cases where we do not input any restriction on the response functions, and some other parties as “white boxes”, where the parties response functions are always the trace with a quantum (hidden) state [81]. We have also studied the connection between joint measurability and genuine multipartite steering, and although no strong results were found, we have strong numerical evidences that when two parties perform quantum tomography and a single party performs a set of measurements{A_a|x}acting onC². We can have genuine multipartite steering if and only if the set of measurements{A_a|x}is not compatible.

In the next chapter we will analyse how joint measurability relates to Bell nonlocality and show that this strong connection does not hold true any more.

3If the stateρB is not full rank, its inverseρ⁻_B1 is not defined. In order to overcome this technical detail, we first restrict the assemblageσ_a|xto a subspace where all operators are full-rank (also note that, sinceσ_a|x≥0 the range ofσ_a|xis contained in the range ofρB). We also remark that ifρBis an unidimensional projector (rank one operator), the assemblage is trivially unsteerable.

Chapter 4

Measurement Incompatibility vs

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