Exploring steering effects using Bell tests

Exploring steering effects using Bell tests

Hsin-Pin Lo,1Atsushi Yabushita,1and Li-Yi Hsu2

1_{Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan, Republic of China} 2_{Department of Physics, Chung Yuan Christain University, Chungli 320, Taiwan, Republic of China}

(Received 22 February 2013; published 18 February 2014)

We study the interconnection among steering, uncertainty, and quantum correlation. The practical steering criteria for bipartite quantum entanglement are proposed using the multiple applications of the Maassen-Uffink uncertainty relation. The results show that distant steering causes less uncertainty than that in any local-hidden-variable theories. The reduction of uncertainty in the quantum steering of entangled states leads to quantum correlation. The steering visibility for the mixed states is also discussed. The experimental results on the proposed steering criteria are reported.

DOI:10.1103/PhysRevA.89.022115 PACS number(s): 03.65.Ud, 03.67.Mn

I. INTRODUCTION

When two entangled qubits are spatially separated, the state of one qubit can be “steered” or piloted (as it was termed by Schr¨odinger) by the prior measurement of the other one. Historically, this effect has led to disputes on the completeness of quantum theory. To depict the Einstein, Podolsky, and Rosen (EPR) paradox in a nutshell, here we consider a composite system composed of two qubits, which are entangled and spatially separated. According to quantum theory, once the first qubit (system) is measured using one of two noncommuting observables, the second qubit (system) will be left in different wave functions. To preserve the locality, Einstein, Podolsky, and Rosen (EPR) [1] asserted “No real change can take place in the second system in consequence of anything that may be done to the first system.” Furthermore, EPR claimed that the same reality may be assigned to two different wave functions required by the consequences of measuring the remote systems. The above argument regarding locality and reality led EPR to the conclusion that quantum theory is incomplete, and implied the existence of local hidden variables (LHV).

Thanks to Schr¨odinger, local hidden variables as an explanation of steering was implicitly rejected then [2]. In addition, Cavalcanti et al. have verified that the notions of EPR paradox and steering are an equivalent depiction of nonlocality [3]. These previous works motivated us to tackle the steering effect using Bell tests on quantum nonlocality. In the following, we consider the following scenario. Spatially separated observers Alice and Bob each hold half of the entangled systems. To perform local measurements, each party randomly chooses one of two or more noncommuting operators corresponding to physical quantities. To ensure that Alice is to steer Bob’s qubit, they initially agree that, in each round, Alice performs her local measurement prior to Bob’s. Without classical communication, the no-signal theorem states that Bob has no information about Alice’s local operation, which, nevertheless, lets Bob’s qubit behave as a steerable state resulting in violation of some specific bipartite Bell-type inequality.

On the other hand, according to EPR’s criteria for reality, Bob may indeed predict the value of “a” physical quantity with certainty [4]. However, Bob never predicts the values

of two noncommuting physical quantities with certainty. It is well known that uncertainty relations among noncommuting observables play an essential role in quantum theory. A useful steering criterion as an extended version of EPR’s reality was proposed by Reid [5], which can be stated as follows. Without disturbing a system in any way, one can predict the value of a physical quantity with some specific uncertainty. Caval-canti et al. further developed Reid’s proposal as quantitative criteria. In detail, due to Alice’s measurement and hence the steering effect on Bob’s observables, she can estimate or infer Bob’s observables using her knowledge of local measurement outcomes. Such interference or estimation can indeed reduce the uncertainty, which has been experimentally validated in the continuous variable case [6]. As shown by Cavalcanti et al., the product uncertainty relations including the inference variances of noncommuting observables can be served as the steering criteria [3,7,8]. The other kind of steering criteria are based on additive convexity of uncertainty relations, which are sums of convex functions [3,7,8].

In this paper, we attempt to explore the steering effect in the Bell tests. The basic idea is that the steering effect in the Bell tests can be revealed with the knowledge of the time order of local measurements. Here we propose the uncertainty relations as the steering criteria, which can be regarded as alternative expressions of Clauser-Horne-Shimony-Holt (CHSH) inequality [9,10] and chained Bell inequalities [11] in terms of conditional marginal probabilities. On this basis, the experimental setup of the proposed criteria is almost the same as the corresponding Bell tests, except that the time ordering of the local measurements must be recorded or previously settled. Our results show that the steering effect can reduce less uncertainty than LHV theory in the Bell tests, which is the reason why quantum theory can violate Bell inequalities.

Before proceeding further, here we compare the proposed criteria with Cavalcanti et al.’s steering criteria and Bell-type inequalities. First, researches on quantum entanglement are unfolded in two different perspectives, the nonlocality and steering effect. The former focuses on correlation relations, whereas the latter focuses on interference ability. The joint probabilities revealing correlations are usually exploited in Bell-type inequalities, whereas, based on Alice’s estimation, the steering criteria usually include conditional probabilities

or inference variances of local observables [3]. Second, the conditional probabilities are multiplied in the proposed criteria, whereas the joint probabilities are summed in Bell-type inequalities. Cavalcanti et al. constructed the steering criteria using either the product or the sums of inference invariance [3,7,8].

The paper is organized as follows. Section II introduces Maassen and Uffink uncertainty relation as the building blocks of the proposed criteria. Then we manipulate them to construct the proposed criteria in Sec.III. The two-setting and infinite-setting cases are studied respectively. Experimental verification of two-setting case is shown in Sec.IV. In Sec.V, we make discussions on local hidden states and visibility of the steering effect, followed by the conclusions.

II. MAASSEN AND UFFINK UNCERTAINTY RELATION

In 1961, Pollack et al. [12,13] proposed the uncertainty relation for continuous time-limiting and band-limiting signal processing. In 1988, Maassen and Uffink (MU) [14] revised the relation as quantum uncertainty relation as follows. Given a qubit|ψ, one performs the measurement using the orthonor-mal basis B_yb, where y is the index of the basis, and b= {0,1} is the measurement outcome. The maximum probability of a certain outcome given the basis is py = max{p(b =

0|ψ,y), p(b = 1|ψ,y)}. The MU uncertainty relation can be formulated as p₀p₁  1+ c 2 2 , (1) where c= max b,b |B b 0|B b

1|. Here, |ψ is the maximally proba-ble state, if the equality of Eq. (1) holds.

In this paper, we exploit Eq. (1) as the building block of the proposed uncertainty relations, which can serve as the steering criteria. We demonstrate that the steering effect in quantum theory can result in less uncertainty than that in theories with no steering (e.g., local hidden variables). Wehner and Oppenheimer [15] first considered the connection between uncertainty relations and Bell-type inequalities. The relations to be proposed in the present work are much simpler than theirs from the operational viewpoint. In the following, |ψψ| = 1 2[I + (−1) b_ψ_{· −}→_σ ], |BybB b y| = 12[I+ (−1) b_n y· − →_{σ], and|A}a xA a x| = 1 2[I + (−1) a_χ

x· −→σ], where the vector

− →_{σ = (σ}

x,σy,σz) and therein σx, σy, and σz are the Pauli

matrices. The vectorsχxandnyare depicted in Fig.1.

Before proceeding further, we assign two operational meanings to Eq. (1). In the one-qubit case, we consider a single photon impinging on a series of three polarizers. The first and third polarizer axes are fixed atn0 andn1, respectively, and n0· n1 = c. Without loss of generality, let the measurement outcome b= 0 denote that the state of polarization is |B0

0 (|B0

1) when the photon passes the first (third) polarizer. Then designate the second polarizer axis as the direction vector 

ψ. After a photon passes through this polarizer, the state of polarization then becomes |ψ. Once the photon passes through the first polarizer, the probability passing through the second and third polarizer is P = p(B0

1|ψ)p(ψ|B00). On the

θ

FIG. 1. (Color online) The four coplanar state vectors are on the Bloch sphere. The angle θ isπ

4 and 0 for the best-case and worst-case scenarios, respectively.

other hand, in quantum theory, the probability

p(m|n) = |m|n|2= p(n|m)

representing the state |n (|m) is measured using the orthonormal basis {|m,|m⊥} ({|n,|n⊥}), and the posts-elected state is |m (|n). Here, m and n can represent

B0

y and ψ or vice versa. Therefore, p(ψ|B00)= p(B00|ψ) and p(B0

1|ψ)p(ψ|B00)= p(B10|ψ)p(B00|ψ), which was the probability of photon ψ passing both polarizers,n0 andn1. As a result, let py = p(By0|ψ); then the maximal probability

Pmaxis the MU uncertainty relation. Finally, the equality holds whenny· ψ=

 1+c

2 . From the perspective of local hidden variables, the measurement result can always be set to b= 0, and the photon always passes through the second polarizer. In the LHV case, p0= p1= 1, there is no uncertainty.

In the two-qubit case, Alice prepares a two-qubit system and then sends Bob one of these two qubits. Even though Reid showed that there is unavoidable uncertainty inherent to the steering effect [5], Alice and Bob still can collaborate to reduce the uncertainty to the minimum from the perspective of the MU relation. As for Alice’s operation, she attempts to pilot the state of Bob’s qubit into a certain maximally probable state. On the other hand, Bob also adjusts his measurement observables to maximize the probability product of Eq. (1). Notably, even though the equality in Eq. (1) holds, it is not enough to guarantee that Alice can steer Bob’s state. One of the examples is the case when the qubit at Bob’s hand is initially in the maximally probable state, which is unentangled with Alice’s qubit.

Here we sketch the proposed quantum steering criteria as follows. Alice can convince Bob of her steering ability with piloting his qubit into a set of nonorthogonal states. Math-ematically speaking, we exploit the MU relation repeatedly with different conditional states. In this case, if the uncertainty is too large, such that it can be explained using local hidden variables, Bob can reject her claim of steerability. In contrast, if the uncertainty were too small (as in the finite multisetting case), it cannot be explained using quantum theory.

III. STEERING CRITERIA

Wiseman et al. [16] revisited Schr¨odinger’s concept of steerability in terms of tasks. From this foundation, we propose the following steering-convincing task. In the general scheme of this task, Alice first prepares two bipartite steerable boxes, and then sends one of the boxes to remote Bob. Then, in the steering phase, Alice inputs the random bit x into her accessible box, which outputs a bit α. Finally, in the checking phase, Bob also inputs the random bit y into his accessible box, which outputs a bit β. If the nonlocal input-output relation were physically realized as

α+ β = xy + cax+ cby mod 2,

where x, y, α, β, ca, and cb ∈ {0,1}, given Alice’s steering

information (x,α), Bob can predict with certainty the output

β= α + xy + cax+ cbymod 2. According to Reid’s criteria,

Bob can predict with some specific uncertainty. Here we evaluate and quantify the uncertainty due to the steering effect as follows. We define the value

Mαβ=

1 

x,y₌₀

p(β = α + yx + cax+ cby|α,x,y). (2)

In the proposed protocol, Bob and Alice are to minimize the uncertainty. That is, they aim to maximize both Mαβand Mαβ.

If Bob were to predict with certainty, Mαβ= Mαβ = 1.

In the quantum version, the boxes are replaced by two bipartite qubits. The two-setting scenario is described in detail as follows. In the preparation phase, Alice initially prepares the two-qubit Bell state

| +_{ = √}1 2(|0|0 + |1|1) = 1 √ 2 χ 0 xχx0  +χ_x1χ_x1,

where |χx0 = cos ϕ|0 + sin ϕ|1, and χx0|χx1 = 0. In the

steering phase, Alice randomly chooses the orthonormal basis {|A0

x,|A1x}, x ∈ {0,1}, as the measurement basis. As a result,

the states of both qubits are simultaneously collapsed into the same state|χxj = |A

j

x, j ∈ {0,1}. In the checking phase, Bob

randomly chooses{|B0

y,|By1}, y ∈ {0,1}, as the measurement

basis. Now we consider the ideal case: they agree beforehand that these vectors, (χ₀,χ₁,n₀, andn₁), are always coplanar in the Bloch sphere. Without loss of generality, we set |A0 0 = |0, |A10 = |1, |A a 1 = √1₂(|0 + (−1) a_{|1), and n} y = 1 √ 2[z+ (−1)

y_{x] so that the value of c in the MU relation is}

1 √

2. Consequently,|χ 0

0, |χ01, |χ10, and |χ11 are all maximally probable states. Hereafter, the probability with which they obtain the results a and b from the x and y measurement directions, respectively, is p(Bb

y|A a

x)= p(b|a,x,y). In the

following, we state the steering criteria as the main result. And then, Bob’s qubit is then steerable so long as either

1 4 < M00, M11  1 2  1+√1 2 4 , (3) or 1 4 < M01, M10  1 2  1+√1 2 4 (4)

holds. Without loss of generality, hereafter we set ca=

cb= 0, and hence we have p(β = yx|0,x,y) = p(β = 1 +

yx|1,x,y) = 1₂(1+√1 2), and therefore M00= M11= [ 1 2(1+ 1 √ 2)]

4_{. In this way, Alice can maximally steer Bob’s state.} In contrast, we consider two worst-case scenarios that result in M00 = M11 =1₄. First, the measurement basis is set as 

χ0= n0 and χ1 = −n1, and the measurement probabilities are p(z|z,0,0) = p(z|z,1,1) = 1 and p(z|z,x = y) = 1

2 for ∀z ∈ {0,1}. In the perspective of local hidden variables, take a predetermined output that can be represented as binary vector (a|x=0, a|x=1, b|y=0, b|y=1); M00and M11can then be obtained with the equal mixing vectors (0, 0, 0, 0), (0, 0, 0, 1), (1, 1, 1, 0), and (1, 1, 1, 1). Second, letχ0= n0 = n1andχ1· χ0=√1₂, the probabilities are then p(z|z,0,0) = p(z|z,0,1) = 1 and

p(0|z,1,0) = p(0|z,1,1) = 1₂. Alice cannot convince Bob that his qubit is steerable in such cases.

As the end of this section, we interpret M00 and M11 as a polarized single photonn₁ that passes through a series of five polarizers. The polarizations of the first through fourth polarizers are set to beχ0,n0,χ1, andn⊥1, respectively. Here,

M00 is the probability of a photon passing through the fifth polarizer once it has passed the first one. If we simultaneously rotate these polarizers by 90◦, the corresponding probability becomes M11. However, in the single-photon case for local hidden variables, this detection probability is zero, as the photon’s original polarization and the final polarizers are orthogonal to one another. Thus, from the perspective of local hidden variables, the single-qubit case is not completely equivalent to the two-qubit one.

IV. EXPERIMENTAL SETUP AND RESULTS

We consider the polarization-entangled photon pair to demonstrate the steering process, and the experimental setup is shown in Fig. 2. Here, we use second harmonic (390 nm) of a Ti:sapphire pulse laser (780 nm). The second harmonic pumps a type-II BBO (β-barium borate) crystal to generate the polarization-entangled photon pair. Half wave plates (HWPs) and BBO crystals are used to compensate for group delay and walkoff in the photon pair. The local measurement was performed using a quarter wave plate and an HWP. The filters were used to suppress noise, passing only photons with a wavelength of 780± 3 nm. The polarizers were set at 0◦ (90◦) for the horizontal (vertical) polarization measurement. Each of the entangled photons was coupled to a single-mode fiber connected to a single-photon-counting module (SPCM). The photon-counting signal of Alice (SPCM A) was used as the stop trigger, introducing electronic delay. Meanwhile, the nondelayed signal of Bob (SPCM B) was used as the start trigger in the coincidence measurement. Note that Alice always measured the photon earlier than Bob. It is an obvious advantage that it is possible to use the same experimental setup to test the CHSH inequality (S = 2.76 ± 0.06). The only difference is that we must ensure the measurement order of the two local measurements. Experimentally, the results satisfied our steering criteria of Eq. (3), as M₀₀exp= 0.45 ± 0.02, and

M₁₁exp= 0.51 ± 0.03. Consequently, Alice can exactly steer

Bob’s qubit into the maximally probable states in the entangled photon pair.

FIG. 2. (Color online) The experimental setup [9]. Tsunami: the laser system; QWP: quarter wave plate; HWP: half wave plate.

V. DISCUSSIONS AND CONCLUSIONS A. Local hidden states

Wiseman et al. pointed out the fundamental differences among quantum nonseparability, Bell nonlocality, and steering in between [16]. In brief, quantum states and local hidden variables are involved and considered in quantum nonsepara-bility and Bell nonlocality, respectively. To test the steering criteria, the local hidden states were proposed [16]. Once the other remote quantum system is measured, the steerable states in the quantum system cannot be described as hidden local states [16], which can be stated as follows. Given the preparation procedure c, the joint probability for a local hidden state can be written as

p(a,b|x,y,c) =

λp(λ|c)p(a|x,c,λ)pQ(b|y,c,λ),

where pQ(b|y,c,λ) represent probability distributions, which

are compatible with a quantum state. However, the measure-ment outcomes of the one-qubit system can be simulated using local hidden variables, which was pointed out by Bell [17]. In this sense, the local hidden states can be completely simulated by local hidden variables, and hence cannot be exploited for the violation of either Eq. (3) or (4).

B. Multisetting case

Now we consider the bipartite multisetting case. The following proposed uncertainty relation can be regarded as an alternative description of chained Bell inequalities. In this process, Alice and Bob each randomly input Axand By, where

x, y∈ {1,2, . . . ,N}, and the outputs are a and b ∈ {0,1}, respectively. We define MN αβas M_αβN = N  y₌₁ p(b= β|a = α,x = y + 1) × p(b = β + δy,N|a = α,x = y), (5)

where N+ 1 ≡ 1, and the parameter δy,N is unity when y=

N and zero otherwise. Using a similar approach for N = 2, to reach the upper bound in local hidden variables, we set

a(x)= α and b(y) = β, and furthermore, b(N) can be set deterministically to 0 or 1 with equal probability. As a result, the maximal MαβN is 14 for local hidden variables.

For the quantum version, we can exploit the same setup for chained Bell inequalities. Without loss of generality, Alice performs her measurement Ax = χx· −→σ with the

measurement outcome α= 0. This measurement steers Bob’s qubit state into|χxχx| = 1₂(I+ Ax). As for Bob, he performs

the measurement By with the basis{|ny,|n⊥y}. The outcome

b is denoted by 0 or 1 if the postselected state is |ny or

|n⊥

y, respectively. Let the angle between χyandnybe _2Nπ . As

a result, c= |χi|ni| = |χi₊₁|ni| = cos_2Nπ >|χi|n⊥i | =

|χi+1|n⊥i |,∀i ∈ {1,2, . . . ,N}, with χN+1= χ1. According to the MU relation, max p(ny−1|χy)p(ny|χy)= (1+c₂ )2, M₀₀N 

(1+c₂ )2N_{, and M}N 11  ( 1+c 2 ) 2N_{, replacing χ} i and ni with χi⊥

and n⊥_i , respectively. Then the steering criteria can be stated as follows. The box systems are steerable if, for some α,

β ∈ {0,1}, 1 4  M N αβ, M N αβ   1+ c 2 2N . (6)

Here, MαβN=2= Mαβ in Eq. (2). In particular, note that

limN→∞(1+c₂ )2N = 1, and therefore, Mαβ∞= 1.

Again, one should consider the operational interpretation of MαβN. Notably, M N 00can be modified to M₀₀N = N  y₌₁ |ny|χy|2|χy+1|ny|2 = N  y=1 p(χy+1|ny)p(ny|χy),

FIG. 3. (Color online) The 2N coplanar state vectors for the chained Bell inequalities are on the Bloch sphere.

where we set χy₊₁= χ₁⊥.In the proposed scenario, a single

photon impinges on 2N polarizers, as shown in Fig.3, which represents the quantum Zeno effect [18]. Let the polarization of the incoming photon beχ₁. The kth polarizer is oriented along the direction of vectorχk/2+1(n(k+1)/2) if k is even (odd) with

respect to Alice and Bob. In this case, M₀₀Ncan be interpreted as the probability of the photon passing through all 2N polarizers. Once the polarization of the photon and the orientation of the polarizers are rotated by 90◦, the transmission probability becomes M₁₁N.

C. Nonideal cases

First, suppose that a nonmaximal entanglement is prepared, e.g.,|φ = s|0|0 + t|1|1({s,t} ∈ R and s2+ t2= 1). Alice can steer Bob’s state into either |0 or |1 with certainty. However, from the unitary operation, |φ can be rewritten as √1

2(|+|ϕ+ + |−|ϕ−), where |ϕ± = s|0 ± t|1 is a nonorthogonal basisϕ₊|ϕ₋ = 0. Here, Alice can exploit the unambiguous quantum measurement of nonorthogonal states by a selective filter F = √ 1

1+|s2_−t2_|(|+ϕ ⊥

−| + |−ϕ+⊥|). Probabilistically, Alice’s qubit can pass through the filter, ensuring that Bob’s state will be steered into the same state as hers. Second, in the biased two-base local measurements, the upper and lower bounds of Mαβ in Eqs. (3) and (4) are

unchanged. Third, consider the Werner isotropic state [19],

W = (1 − η)1

4I⊗ I + η| + +|,

where 0 η  1, I is the identity. Such states can be regarded as depolarizing disturbances from the environment. Isotropic states are inseparable if η 1₃[20] and steerable if η1₂ [16] in the two-level case.

As for the proposed N -setting scheme, we consider the critical value of ηNsuch that M₀₀N = M₁₁N =1₄. That is,

1 4 =  1 2(1− ηN)+ ηN  1+ cos_2Nπ 2 2N , and hence ηN = (21−1/N − 1) sec π 2N.

That is, the steering effect becomes “visible” if η > ηN as

shown in Fig. 4. Obviously, η2= 2 − √ 2 > 1₂ and lim N→∞ηN = 1. 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 N

FIG. 4. The visibility of the steering effect.

The numerical result is shown in Fig.4. As a result, the two-setting setup is optimal for witnessing quantum steering.

D. Min-entropy representation

Finally, we express Eq. (2) in terms of the min-entropy H_∞(X|Y ) = − log₂max P (X|Y ), with a logarithm of base 2; for a random variable X∈ {x1, . . . ,xN} and

Y = y, let the arbitrary conditional probability distribution

be P (X|Y ) = {p(x1|y), . . . ,p(xN|y)}. According to Eq. (2),

we have1_x,y=0H_∞(b|a,x,y)  − log₂M, and equivalently, 4[2− log₂(2+√2)] − log₂M 2.

In conclusion, a practical criterion for quantum steering is proposed that allows us to relate the steering, the uncertainty relation, and Bell-type inequalities. Indeed, quantum steering can reduce less uncertainty than LHV. From the operational perspective, the proposed criteria can be physically realized using the same set of Bell tests. In particular, the two-setting case can be performed using the experimental setup of CHSH inequality. Characteristically, our proposed criteria exploit the MU relation to revise the joint probability terms of Bell-type inequalities as the product form of conditional probabilities. It can be easily verified that, once the measurement outcomes result in the maximal values of both M00 and M11 ([1₂(1+

1 √

2)]

4_{), CHSH inequality is maximally violated. In addition,} the two-setting case is the most robust against depolarized noise. Finally, we believe that the Bell tests and the steering criteria can be physically performed simultaneously.

ACKNOWLEDGMENTS

We acknowledge the financial support from the National Science Council of the Republic of China under Contracts No. NSC.102-2112-M-033-006-MY3, and No. NSC.101-2112-M-009-002-MY2.

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