Testing Bell inequalities with weak measurements

0.23 MB·English

by  Shmuel Marcovitch

Testing Bell inequalities with weak measurements
Shmuel Marcovitch and Benni Reznik
School of Physics and Astronomy, Raymond and Beverly Sackler
Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel.
Quantum theory is inconsistent with any local hidden variable model as was first shown by Bell.
To test Bell inequalities two separated observers extract correlations from a common ensemble of
identical systems. Since quantum theory does not allow simultaneous measurements of noncom-
muting observables, on each system every party measures a single randomly chosen observable out
of a given set. Here we suggest a different approach for testing Bell inequalities that is experimen-
tallyrealizable bycurrentmethods. WeshowthatBellinequalitiescanbemaximally violatedeven
whenallobservablesaremeasured oneachmemberoftheensemble. Thisispossiblebyusingweak
1 measurements that produce small disturbance, at the expense of accuracy. However, our approach
1 doesnotconstituteanindependenttestofquantumnonlocalitysincethelocalhiddenvariablesmay
0 correlate the noise of the measurement instruments. Nevertheless, by adding a randomly chosen
2 precise measurement at the end of every cycle of weak measurements, the parties can verify that

n thehidden variables were not interfering with the noise, and thusvalidate thesuggested test.
a
J
8 I. INTRODUCTION
1 tA tB
Quantum nonlocality [1] has been tested [2, 3] in dif-
] tB
ph fseimrepnltesmyastneimfesstaatniodniissgniovwenwbyidCellyauascecre-Hptoerdn.e-SIhtsimmonoyst- q2A (cid:86)z t2A 2 q2B (cid:86)34(cid:83)
- Holt (CHSH) inequality [4], in which every cycle A and
t
n B measure one out of two randomly chosen observables
a a1,a2 1 and b1,b2 1 . In a local hidden vari- tB
qu able m∈od{e±l B}≤2, where∈ {± } (cid:86)x t1A 1 (cid:86)(cid:83)4
[ q1A q1B
B = E(a1b1)+E(a1b2)+E(a2b1) E(a2b2). (1) (cid:92)
| − | AB
2
Above E(a b ) is the joint expectation value. The EPR
v i j A B
6 state
3
1
2 ψ = ( A B + A B ), (2)
3 | i √2 |↑z↑zi |↓z↓z i FIG. 1: Sequential CHSH setup. Parties A and B measure
. two observables at sequential times. The knobs control the
5 saturates the maximal bound 2√2 [5] with i.e. a1 = inaccuracyofthedevices. Intheaccuratemeasurementlimit
00 σx, a2 = σz, b1 = σπ/4, and b2 = σ3π/4, where σπ/4 = the pointer readings correspond to the possible values of the
1 1 observables( 1). Intheinaccurate(weak)limitthereadings
1 √2(σx+σz) and σ3π/4 = √2(σx−σz). are noisy,and±themicroscopic values cannot be inferred. We
: In the regular setting every cycle the parties measure
v requirethat thedistance between theparties d= XA XB
i a single observable. In our approach to be referred to satisfies d ctA2 and d ctB2. | − |
X as the sequential setting, for each member of the ensem- ≪ ≪
r ble A measures both a1 and a2 (one after another), and
a
similarly B measures both b1 and b2, as is schematically
describedinFig. 1. Obviously,asdiscussedabove,foror- decreasingthe accuracyof measurements,one in fact re-
dinary measurements the first measurement of a1 would gains the maximal violation of Bell inequalities. This
randomize the result in the subsequent measurement of result is stated in theorem 1 and corollary 1. The out-
a2, and no violation of Bell inequalities would be man- comes of inaccurate measurements may lie outside the
ifested. However, as is well known, there is a trade-off usual range of the measured observables. In this case
betweentheaccuracyofthemeasurementandthedistur- local hidden variables may interfere with the noise to
bance caused to the system [6]. The limit in which indi- violate Bell inequalities. Therefore our approach does
vidualmeasurementsprovidevanishing informationgain notconstitute anindependent test ofquantumnonlocal-
was firstanalyzedby Aharonovet. al. (in the contextof ity. However,byaddingasinglerandomlychosenprecise
post-selection) [7] and was termed weak measurements. measurementattheendofeachcycle,thepartiescande-
In this limit the measurements become inaccurate and termine whether the hidden variables interfere with the
onecouldexpectthatnonlocalcorrelationswouldnotbe noise. The expected negative result implies that the in-
observed. accurate measurements still provide a valid test. This is
The main result of this paper is that as one gradu- manifested in corollary 2. We then proceed by quantita-
ally reduces the disturbance caused to the system, by tivelyillustratingthetransitionfromstrongtoweakmea-
2
surements in the sequential setting in realistic finite en- After the measurementsthe state ofthe systemsandthe
semblesandshowthatsimultaneouslydifferentBelltests pointers is given by
can be attained using the same ensemble. We conclude
m m
by discussing implications of our approach,in particular
maximal violation of the Leggett and Garg inequalities |Ψi=U(tm) φ(ql)− φ′(qk) φ(ql)Ok(tk)
[8] and possible realizations of the suggested method. hYl Xk lY6=k
+ φ′(qk)φ′(ql) φ(qs)Ol(tl)Ok(tk) (9)
k<l s=k,l
X Y6
II. VIOLATION OF BELL INEQUALITIES m
1
2 3
WITH WEAK MEASUREMENTS + 2 φ′′(qk) φ(ql)Ok(tk)+o(ǫ )|ψi.
Xk lY6=k i
Theorem1. Letasystem ψ bemeasuredsequentially
by an arbitrary number of ob|seirvables O (i = 1,...,m) Let us compute EQM(qiqj), where without loss of gener-
i ality i<j. Define U(t )ψ = δ ψ , U(t )O (t )ψ
awteatikmmeseatsiuwrehmereent0s,<t1 <t2 <···<tm. In the limit of = αkr|ψri, and U(jtj|)Oil(tl)POk(rt|k)|rψii = j kηrklk|ψ|rii,
where ψ is any basis and k,l j. Expanding Eq.
r
P | i ≤ P
(9) in the ψ basis up to the jth measurement yields
EQM(qi)= ψ Oi(ti)ψ , (3) | ri
h | | i
EQM(qiqj)=Re ψ Oi(ti)Oj(tj)ψ , (4) j j
h | | i
Ψ = δ φ(q ) αkφ(q ) φ(q )
| i r k − r ′ k l
pwohienrteerq.iTishtehoebrseeardvianbgleosfOtheariethwmriettaesnurienmtheentHaepispeanrabteurgs Xr h Yk Xk lY6=k
i + φ(q )φ(q ) φ(q )ηkl
representation, O (t ) U (t )O U(t ), U(t) = e iH0t, ′ k ′ l s r (10)
i i † i i i −
and H0 is the free Ham≡iltonian. Xk<l sY6=k,l
j
1
Proof. The full Hamiltonian is + 2 φ′′(qk) φ(ql)ηrkk+o(ǫ3)|ψri.
m Xk lY6=k i
H =H0+ δ(t−ti)piOi, (5) To compute EQM(qiqj) we trace out the system and qk,
Xi k 6= i 6= j. Using qφ2(q)dq = 0, φ(q)φ′(q)dq = 0, it
can be shown that to second order operations of O do
where [q ,p ] = i (we take ~ = 1). The second term in R R k
i i not contribute. Therefore,
Eq. (5) corresponds to the von-Neumann measurement
interaction, where for simplicity instantaneous measure- 1
E (q q )= (δ ηij +δ η ij +α iαj +αiα j)+o(ǫ4)
ments are assumed. In addition, we assume identical QM i j 4 r∗ r r r∗ ∗r r r ∗r
initial Gaussian wavepackets for the pointers: Xr
4
=Re ψ O (t )O (t )ψ +o(ǫ ).
i i j j
h | | i
φ(qi,t=0)= ǫ 1/4e−ǫqi2/4, (6) (11)
2π
(cid:16) (cid:17) This completes the proof of theorem 1.
where ǫ 1 = [∆q (t = 0)]2 σ2. The initial state of the
− i
≡
system and the apparatuses Using the same considerations it can be shown that
Eqs. (3,4) are valid for simultaneous measurements too.
Related results for correlations of two-level system with
Ψ = ψ φ(q ), (7)
i
| i | i⊗ continuous weak measurements have been discussed
i
Y in [9], the case of post-selection in [10–12] and two
evolves in time according to sequential measurements in [13].
= U (t )e ipiOiU(t ). (8) Corollary 1. In the sequential setting we define
i † i − i
U U ≡
Yi Yi B = E(qAqB)+E(qAqB)+E(qAqB) E(qAqB). (12)
S 1 1 1 2 2 1 2 2
| − |
Each operation of p yields an order of ǫ. In the limit of
In the limit of weak measurements B B, coinciding
weak measurements, the inaccuracy σ (ǫ 0). S
→
→∞ → withthatdefinedinEq. (1). Inparticular,maximalvio-
Byexpanding tofirstorderinǫ,whilekeepingterms
i
to first order inU and tracing out the system and q , lationof2√2isobtainedwiththesameobservablesasin
j
j 6=i, one obtainsUEq. (3). OthBe r=egIuAlar sσeBttinagnwdiOthBO=1AI=AσxAσ⊗BIB., O2A =σzA⊗IB,
To derive Eq. (4) we expand to second order 1 ⊗ π/4 2 ⊗ 3π/4
1 Corollary 2. Assume that at the end of every
e−ipiOi =1−ipiOi− 2p2iOi2+o(ǫ3). cycleofweakmeasurementsthe partiesrandomlychoose
3
a single observable to measure strongly OA, OB respec-
s s
tively. Then E (qA,qB) = ψ OA OB ψ . This can
QM s s h | s ⊗ s | i 2 2
be straightforwardly seen from the weak disturbance of
2.5
thepreviouslymeasuredobservables. Inquantumtheory 106
the weakly measured correlations equal the strongly 2
measured ones.
1.5 105
1
III. TESTING LOCAL HIDDEN VARIABLE 104
MODELS IN THE SEQUENTIAL SETTING
0.5
103
In the sequential setting the outcomes of weak mea- (cid:16)1 0 1 2 3
e e e e e
surements may lie outside the usual range of the mea-
sured observables. Then the hidden variables can inter-
ferewiththenoisetoviolateCHSHinequality. Therefore
thesequentialsettingdoesnotconstituteanindependent FIG.2: ViolationofthesequentialCHSHtest. Solidline: BS
testofquantumnonlocality. The requirementofrandom asafunctionofthemeasurementsinaccuracyσ ∆q(t=0).
≡
choice seems inevitable in any independent test of quan- BS > 2 for σ > p[ 2ln(23/4 1)]−1 1.1425. Note
− − ∼
tum nonlocality. that precise sequential measurements satisfy CHSH inequal-
Nevertheless,physicallyonewouldexpectadditiveran- ity. Brokenline: ensemblesizeN3(σ)thatmanifestsviolation
withthreestandarddeviations. B 2.43correspondstothe
dom noise n: qiA = ai +nAi , qjB = bj +nBj such that minimal ensemblesize N3 =3088.SF∼orcomparison, theregu-
E(ainj)=0 and E(ninj)i6=j =0. Then lar non-sequentialprecise setting requires N3 =105.
E(qAqB)=E(a b ), (13)
i j i j
and no violation of CHSH inequality can be shown. By
corollary 2 we can use the regular test of nonlocality as
a certificate for the sequential setting. That is, let each N
3
partypreciselymeasurearandomlychosenobservableat
theendofeverycycle. Bell’stheoremstatesthatthecor-
relations of the precise measurements would not violate
CHSH inequality. If the parties observe that the weakly
measuredcorrelationsdiffer fromthe preciselymeasured
ones,thentheyknowthatthesequentialtestisnotvalid.
However, then they also know that the hidden variables
(”maliciously”) interfere with the noise.
IV. TRANSITION FROM STRONG TO WEAK
MEASUREMENTS FIG. 3: Simultaneous violation of several CHSH tests. The
partiesperformnCHSHexperimentsbeforestartingthecur-
The strict limit of weak measurements requires an in- rent CHSH sequence. BS > 2 for σ > σmin (dotted line),
finite ensemble, which is not practical. However, viola- where σmin = 23/4√n (for large enough n). Also shown is
theminimal valueof N3 (brokenline) and thecorresponding
tion can be observed using a finite ensemble, and finite
σ3(solidline)whichmanifestsviolationwith3standarddevi-
measurement apparatus inaccuracy σ ∆q(t = 0), as
≡ ations. N3scalesquadraticallywithn. Inset: Expectedviola-
illustrated in fig. 2. In fact, we find that BS > 2 for tiongivenσ3. BS(n)decreasesslowlywithnwhereBS 2.4.
relatively mild value of σ 1.15. As the inaccuracy in- ∼
∼
creases B 2√2. Also shown is the required ensemble
S
→
sizeN3 bywhichviolationismanifestedwith threestan- and the corresponding inaccuracy σ3 by which violation
dard deviations. The minimal ensemble is about thirty
is manifested.
times larger the one required to manifest B > 2 in the
We proceed by quantifying the required inaccuracy
usual setting.
of the measurements as depicted in figures 2 and 3.
Furthermore, simultaneous violation of (possibly dif-
By explicitly expanding the joint state of the system
ferent) Bell inequalities can be observed using the same
and the measurement apparatuses it can be shown that
ensemble. Forexample,assumethatthetwopartiesper- E(qAqB) = 1 , E(qAqB) = E(qAqB) = y and
formnCHSHsequencesoneachpairoftheensemblewith 1 1 √2 2 1 1 2 √2
a fixed inaccuracy σ before starting the current CHSH E(q2Aq2B) = √y22, where y ≡ e−ǫ/2 = e−2σ12. For exam-
test. In fig. 3 we show the minimal ensemble size N3 ple, the mutual state of the system and the measuring
4
devices after party A measures both σA and then σA Leggett and Garg (LG) [8]. LG discuss measurements
x z
and party B measures σB is onalocalsystemwhichareperformedatdifferenttimes.
π/4
They analyze the scenario in which at each cycle the ex-
z π/4 φ(q2A 1)φ(q1B 1) αφ(q1A 1) β(q1A+1) perimentalist chooses two out of m > 2 measurements
|↑ ↑ i − − − − and compute their mutual expectation value. Given a
+|↓z↑π/4iφ(q2A+1)φ(q1B−1)(cid:0)αφ(q1A−1)+β(q1A+1)(cid:1) realistic model and non-demolition measurements, cor-
+ z π/4 φ(q2A 1)φ(q1B+1)(cid:0)βφ(q1A 1)+α(q1A+1)(cid:1) relations satisfy the LG inequalities, which have a sim-
|↑ ↓ i − −
ilar structure to Bell inequalities. Interestingly, it was
+|↓z↓π/4iφ(q2A+1)φ(q1B+1)(cid:0)βφ(q1A−1)−α(q1A+1)(cid:1), shown that quantum theory violates the LG inequali-
(cid:0) (cid:1) ties using weak measurements [14–16] even if all observ-
1 1 ables are measured on each member of the ensemble. In
where α= [cos(π/8)+sin(π/8)], β = [sin(π/8)
2√2 2√2 − this setting precise measurements do not violate LG in-
cos(π/8)]. By tracing out the system and q1A, one can equalities by definition. The addition of noise, however,
compute E(q2Aq1B) given above. Therefore BS = √12(1+ distinguishesmodelsincorporatingLGassumptionsfrom
y)2, where B denotes the sequential setting. quantumtheory,sincenon-demolitionmeasurementsim-
S
By the central limit theorem the size of the ensemble ply that the noise and the measured observable are in-
which is requiredto manifest violation of Bell inequality dependent [14]. This is violated in quantum theory even
with z standard deviations is determined by thoughtheeffectofweakmeasurementsonthesystemis
small. An immediate consequence of Eq. (4) is that LG
inequalitiesaremaximallyviolatedbyquantumtheoryin
V(B)
z <B 2, (14) the limit of weak measurements. Note that the realism
s N3 −
assumption is crucial since one can write a joint proba-
bilitydistributiontothe pointersofweakmeasurements,
where V(B) is the variance of B. In the regular non-
which do not correspond to 1 values.
sequentialsetting B is the sum of four independent vari- ±
ables V(B) V(qAqB) = 4(1 + σ2)2 2. In
≡ i,j i j −
case of sequential measurementsthe experiments are de-
pendent, thus V(PB ) CV(qAqB,qAqB), where
S ≡ i,j,k,l i j k l
CV(x,y) = E(xy) E(x)E(y) is the mutual covari-
− P VI. CONCLUSIONS
ance. By a straightforward computation it can be
shown that for i = k and j = l, CV(qAaB,qAaB) =
6 6 i j i l
CV(qiAqjB,qkAqlB)=0, and V(qiAqjB)=(1+σ2)2. There- We propose a new approach to test quantum nonlo-
fore V(B ) = 4(1+σ2)2 1(1+y)4. These results are cality, in which all observables are measured on each
S 2
−
illustrated in fig. 2. system without the need to select randomly a single
Now assume that the two parties perform n CHSH observable. By enlarging the sequence of measurements
sequences before starting the currentCHSH experiment, it is also possible to manifest a simultaneous violation
wherethe ithobservableofeachpartyismaximallynon- of more than one Bell inequality. Our proposal can
commuting with the i 1 one for any i. Then it can be be demonstrated in all systems in which both Bell
−
shown that state preparation and sequential weak measurements
are realizable, for example atomic physics devices [17],
y2n
B (n)= (1+y)2 (15) photons [18, 19], and solid-state qubits [20].
S
√2
and V B (n) =4(1+σ2)2 y2n(1+y)4. These results
S − 2
are illustrated in fig. 3. It can be numerically shown
(cid:0) (cid:1)
that Eq. (15) yields a typical value for randomly chosen
n=2m measurements.
Acknowledgments
V. MAXIMAL VIOLATION OF
LEGGETT-GARG INEQUALITIES WethankP.Skrzypczyk,N.Klinghoffer,J.Kupferman
andG.Ben-Porathforhelpfuldiscussions. Thisworkhas
Our suggestion to manifest violation of Bell inequali- been supported by the Israel Science Foundation grant
ties is related to the time inequalities first suggested by numbers 784/06,920/09.
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