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Realization of the ContextualityNonlocality Tradeoff with a QutritQubit Photon Pair
Peng Xue
the Department of Physics, Southeast University
[email protected]
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Outlines
Brief review on nonlocality and
contextuality
Idea of monogamy relation on
contextuality vs nonlocality
Realization of contextuality-nonlocality
tradeoff with a qubit-qutrit photon pair
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Einstein-Poldosky-Rosen
Elements of Reality
...
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Einstein-Poldosky-Rosen
Elements of Reality
Einstein, Podolsky and Rosen’s question
•Is quantum mechanics complete? Or is there some
hidden variable theory behind it?
Bell’s answer
•Both a local realistic picture and quantum mechanics
can explain the perfect correlations observed.
•If a hidden variable model is local, it can be ruled out
experimentally.
•Bell inequality states that certain statistical
correlations predicted by quantum mechanics for
measurements on two-particle ensembles cannot be
understood within a realistic picture based on local
properties of each individual particle.
J. S. Bell, Physics 1, 195 (1964).
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CHSH inequality and violation of
local realism
Generalized Bell inequality — CHSH inequality (easier for
experimental test),
S  E  a, b   E  a, b '  E  a ', b   E  a ', b ' 
If E=1, perfect correlation.
If E=-1, perfect anticorrelation.
Local Reality prediction:
S MAX  2
Quantum Mechanics prediction:
S MAX  2 2
Corresponding experimental tests support the
necessity of quantum mechanics by violating CHSH
inequality.
J. Clauser et al., Phys. Rev. Lett. 23, 880 (1969).
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Contextuality
Q: What if one has only one quantum system?
A: The Kochen-Specker theorem states that
noncontextual theories are incompatible with
quantum mechanics. Noncontextuality means
that the value for an observable predicted by
such a theory does not depend on the
experimental context which other comeasurable （compatible） observables are
measured simultaneously.
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Contextuality （KCBS inequality）
Definition of compatibility: two measurements A
and B are called compatible if they can be
measured simultaneously or in any order without
disturbance.
KCBS inequality: Alice randomly chooses two
compatible measurements from five
measurements {Ai} (i=1,…,5) and performs them on
her system. Each two of Ai and A(i+1)mod5 are
compatible.
Non-contextual hidden variable model:
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KCBS inequality
Quantum mechanical prediction
KCBS inequality is violated based on
QM prediction!
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Monogamy relation
• Two spatial separated systems:
Quantum theory
local reality
• One quantum system:
Quantum theory
noncontextual reality
• Q：Are two realities independent？
• A：There is a monogamy relation on contextuality
versus nonlocality, i.e., in the same quantum system,
two inequalities can not be violated at the same time.
Ref: P. Kurzyński et al., PRL 106 180402 (2014)
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No-disturbance (ND) principle
• The ND principle is a generalization of the no-signaling
principle that refers to compatible observables instead of
spacelike separated observables
• Based on the ND principle, the probabilities of outcomes of
the measurement Ai do not depend on if Ai is measured
with A(i+1)mod5.
• The ND principle imposes a nontrivial tradeoff between the
violation of CHSH and KCBS inequalities.
• Quantum theory is one kind of the ND principle.
Ref: P. Kurzyński et al., PRL 106 180402 (2014)
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Stronger monogamy relation imposed
by quantum theory
• The monogamy relation holds in any theory satisfying the
ND principle such as quantum theory.
• Quantum theory imposes a more stringent monogamy
relation between quantum contextual and nonlocal
correlations.
• Consider Alice and Bob share a qutrit-qubit system.
Alice’s measurements are
, with
• In particular, the state is assumed to be
• Bob’s observables are chosen to be two Pauli operators
B1=Z and B2=X.
Ref: P. Kurzyński et al., PRL 106 180402 (2014)
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Stronger monogamy relation imposed
by quantum theory
• The more stringent
monogamy relation
makes the quantum
region smaller than that
imposed by the ND
principle.
• Arbitrary quantum state
corresponds a point
inside the quantum
region.
• The boundary of the
quantum region can be
produced by the states
taking the form.
<CHSH>=-2.08,
<KCBS>=-2.92
Ref: P. Kurzyński et al., PRL 106 180402 (2014)
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Realization of the Contextuality-Nonlocality
Tradeoff with a Qutrit-Qubit Photon Pair
X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B. C. Sanders, and PX, Phys. Rev. Lett. 116, 090401
(2016)
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Measurements
• Alice randomly chooses two compatible
measurements from five measurements {Ai}
(i=1,..,5) and performs on her system. Each
two of five measurements are compatible.
• Bob chooses one of two incompatible
measurements, B1 and B2 and performs on
his system.
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Experimental demonstration
A qutrit-qubit state preparation: entangle photons are
generated via type-I spontaneous parameteric downconversion (SPDC) and the initial state is prepared in
this form
Entangled
photon source
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Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016)
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Experimental demonstration
Bob’s measurements: setting Hb=0, he performs
projective measurement along the z axis; setting
Hb=22.5, he performs projective measurement
along the x axis. Dh clicks, the result of the
measurement is +1; Dv clicks, the result of the
measurement is -1.
Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016)
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Experimental demonstration
• Alice’s measurements via three steps
• Step1: performs Ai via BD1-2 and HWP2-6
• Step2: recreates the eigenstate of Ai via BD3 and BD6,
HWP7-10, HWP15-17
• Step3: performs Ai+1 via BD4-5, BD7-8, HWP11-14,
HWP18-21.
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Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016)
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Results
Direct experimental evidence of a tradeoff between locally
contextual correlations and spatially separated correlations
In the same quantum system, two inequalities can not be
violated at the same time.
Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016)
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Meaning
• The contextuality-nonlocality monogamy suggests the
existence of a quantum resource of which
entanglement is just a particular form.
• That is, to violate the locality inequality costs
entanglement as a resource, while to violate the
noncontextuality inequality costs contextuality as a
resource. In a quantum system, only one of the two
inequalities can be violated because nothing is left to
violate the other one.
• The resource required to violate the noncontextuality
inequality and that required to violate the locality
inequality are fungible through entanglement.
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Meaning
• Nonlocality and contextuality are both just different
manifestations of a more fundamental concept, the
assumption of realism.
• The reason for the nonlocality-contextuality tradeoff
arises from the fact that both properties have the
same root: the assumption of realism, which is the
assumption that the physical world exists
independent of our observations, and that the act of
observation does not change it.
• Since nonlocality and contextuality can be thought of
as two different manifestations of the basic
assumption of realism, then one of them can be
transformed into the other, but both cannot exist at
the same time because they are essentially the same
thing.
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Violation of a generalized non-contextuality
inequality with single-photon qubit
• A qutrit and five projectors are required for a proof of
KS-contextuality (in a state-dependent manner)
• A qutrit and thirteen projectors are required for a proof of
KS-contextuality (in a state-independent manner)
• A qubit (smallest quantum system) and three unsharp
binary qubit measurements are enough to violate a
generalized non-contexutality---Liang, Spekkens,
Wiseman (LSW) inequality (state-dependent)
Ref: Y. C. Liang, R. W. Spekkens, and H. M. Wiseman, Physics Reports 506 1-39 (2011);
Ref: R. Kunjwal and S. Ghosh, Phys. Rev. A 89, 042118 (2014)
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Violation of a generalized non-contextuality
inequality with single-photon qubit
• A violation of LSW inequality is interesting because (1)
more stringent than that set by the KCBS (or KS) noncontextuality---larger upper bound (1-η/3 v.s. 2/3 with
0<η<1) to rule out the non-contextual models; (2) less
requirements---a smallest quantum system (a qubit) and
three unsharp measurements.
• The key point: how to realize unsharp measurements
Ref: Y. C. Liang, R. W. Spekkens, and H. M. Wiseman, Physics Reports 506 1-39 (2011);
Ref: R. Kunjwal and S. Ghosh, Phys. Rev. A 89, 042118 (2014)
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Vertices---measurement, edges---jointly measurable contextuality.
KCBS contextuality scenario: A
qutrit and five projectors;
Upper bound 2/3 of the
probability of anticorrelations
LSW contextuality scenario:
A qubit and three unsharp
measurements;
Upper bound 1-η/3 (0<η<1) of the
probability of anticorrelations;
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G+-
generalized noncontextuality
inequality
Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted
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Realization of unsharp measurements
• State being measured
• LSW inequality concerns average
probability of anticorrelations (η is
sharpness )
• 3 unsharp measurements
can be constructed and
realized by the joint
POVMs, each of which
has four elements
• Joint POVM via a 5-step
quantum walk with sitedependent coin rotations
Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted
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Realization of joint POVM
• The key point is to construct and realize the joint
POVMs each of which has four elements
• Joint POVM via a 5-step QW with site-dependent coin
rotations
Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted
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Joint POVMs
Single photona qubit
G- -
G+G-+
G++
sandwich-type QWP-HWP-QWP sets
HWP
• The probabilities of the clicks on the detectors D4, D3, D2, D1
correspond to those of the joint POVMs elements
on the polarization state of single photons
Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted
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Violation of generalized noncontextual
inequality with a smallest quantum system
The measured average probability of anticorrelations
violates the boundary
(
) and
is in a good agreement with quantum prediction 0.8075. Support the
necessity of quantum machines. Quantum machines is proven to be
complete even with a smallest quantum system---a single qubit.
Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted
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Conclusion
Realization of contextualitynonlocality tradeoff with a qubitqutrit photon pair
Violation of a generalized
noncontextuality inequality
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Collaborators:
Xiang Zhan, Zhihao
Bian, Kunkun Wang,
Xin Zhang, Jian Li @
Southeast Univ.;
Barry C. Sanders @
USTC & Univ. of
Calgary
Yongsheng Zhang @
USTC
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Thank you for your attention…
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Implementation of POVM
{E1,…,En}, where E i =λi |ψi>< ψi|
1 initialize the walker at x=0 with coin state corresponding to the
qubit state to be measured |φo>
2 set i:=1
3 while i<n (n is the number of the elements of the POVM) do
(a)For each odd step, apply coin operation
at position x=0 and identity elsewhere and then apply position
shift operation
(b)For each even step, apply coin operation
at x =1, NOT gate at x=-1 and identity elsewhere and then
position shift operation
(c) i:=i+1, next round
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Implementation of POVM {E1,…,En}, where
E i =λi |ψi>< ψi|
coin operation C(1)i is chosen to guarantee that after the step 3(a)
the unitary operation at position x=1 on the ‘initial’ state is
proportional to |ψi><ψi| (one of the elements of POVM E i=λi
|ψi><ψi| ), i.e. mapping the state of the horizontal photons to E i
|φo>
coin operation C(2)i is chosen to guarantee after the step 3(b) the
probability of click at x=2 is the probability of the element of
POVM E i applied on the system of interest λiTr(|ψi><ψi|φo><φo|) ,
where |φo> is the state to be measured.
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