1. Howard Wiseman1 and Geoff Pryde1
One-way and reference-frame independent Einstein-Podolsky-Rosen steering
Sabine Wollmann1, Raj Patel1, Michael Hall1, Nathan Walk1,2, Adam Bennet,
Howard Wiseman1 and Geoff Pryde1
1 Centre for Quantum Computation and Communication Technology and Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia
2 Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom

2. Entanglement for Quantum information
Quantum communication
Quantum computation
Entanglement
Entanglement is a key resource for many projects in Quantum Information Science. In our lab, we use it for Quantum communication, Quantum computation, Quantum metrology, or Quantum control.
17/08/16:
-As well as being fundamental interesting, entanglement is a resource for various experiments in Quantum Information Science.
Quantum metrology
Quantum control

3. A bit of good old history
EPR
‘steering’
Alice
Precise formulation very important on this slide; this slide sets the scene for the following distinction of the different classes of nonlocality 17/08/16
- Entanglement is a well-known and unintuitive feature in QM
In this thought experiment here, we have a joint preparation of maximally-entangled state which we share between two observers, Alice and Bob. If Alice makes a measurement on her system, she forces Bob’s into a set of basis states, with the basis depending on her choice of measurement.
This feature was famously discussed by Einstein, Podolsky and Rosen in They described this feature and argued that QM itself must be incomplete.
In the same year, Erwin Schrodinger responded and coined the term ‘steering’
The EPR paper advocated the possibility of local hidden variables.
It was proven by John Bell in 1964 that there exist predictions of quantum mechanics for which no possible local hidden variable model could account
Entanglement and Bell nonlocality have been defined for decades, however it was not until recently in 2007 that the class of nonlocality described by EPR was formalized
17/08/2016
The kind of experiments we talk about have a joint preparation of a bipartite state which is shared between the observers, Alice and Bob
Both make a set of measurements and report their classical outcomes to compare the correlations
EPR considered a thought experiment of this kind and described the feature as spooky actions at a distance
Because they couldn’t describe the correlation in a local realistic framework, it looked like Alice’s measurements caused an instanteanous action at a distance
Erwin Schroedinger responded to their publication and described the effect as ‘steering’
30 years later, Bell mathematically formalized a robust test known as Bell nonlocality
Local hidden variables
Joint preparation
correlations
Bob
A. Einstein et al., Phys. Rev. 47, 777 (1935), E. Schrodinger, Proc.Camb. Philos.Soc. 31, 555 (1935), J. S. Bell, Physics 1, 195 (1964)

4. Different nonlocality classes
Bell
The classes of quantum nonlocality require different level of trust
EPR
The class of nonlocality can be found within Bell nonlocality and Entanglement
In Bell’s nonlocality, the two observers, Alice and Bob, share an entangled state. The referee Charlie, is neither trusting Alice and Bob to make a genuine announcement of their measurements
Another example is entanglement, which requires less correlated states. In entanglement Charlie trusts both parties Alice and Bob.
EPR can be distinguished by its asymmetry in terms of trust. Here, we trust Bob, but we do not trust Alice. As she could cheat, it is necessary to verify that she is able to steer Bob’s state.
To demonstrate steering, we perform a EPR-steering task
Here, you want to emphasize that the key difference between steering and the other two quantum correlations (entanglement and Bell nonlocality) is that steering is defined asymmetrically. In the diagram Bob is trusted and Alice is untrusted, but it could be the other way around. We can consider two separate tasks: Can A steer B? and can B steer A?
Our work is about answering the question: Do there exist quantum states for which can only be steered in one direction, and can we observe? It seems intuitive that such states should exist, and examples do exist if Alice and Bob are restricted to certain kinds of measurements. But we want to answer the question in complete generality.
17/08/2016:
- In Bell nonlocality we don’t assume QM or anything about measurements of the parties or their trustworthiness
In the class of entanglement witness we assume (QM) is trues and try to measure the entanglement witness
This requires trust and is inappropriuate in an adverbial scenario (pop up arrow)
Intermediate to this two classes is EPR
It is a asymmetric scenario where we (Charlie) assume Bob’s apparatus is QM
Entanglement witness
S. J. Jones et al., PRA 76, (2007), H. M. Wiseman et al., PRL 98, (2007), A. Acin et al., PRL 98, (2007), L. Masanes et al., Nat. Com. 2, 238 (2011)

5. EPR-steering task
Alice
Joint preparation
correlations
Bob

6. EPR-steering task Step 1 Step 2 Step 3 Step 4
|ΨAB˃
Step 1
|ΨA˃
Step 2
Step 3
Bob measures his state and secretly records his result Bk
Step 4
Specifically, that’s how the EPR task works:
Bob receives his qubit. He may receive an unentangled single qubit, or he may receive one-half of a bipartite entangled state. At this state, he cannot distinguish the former from the latter.
Bob announces to Alice his choice of measurement setting k, corresponding to an observable sigma_k_B drawn from the pre-determined set {sigma_k_B}n that Alice can measure in the appropriate direction. Unlike a Bell measurement, you can use multiple measurements per side.
Bob measures his state and secretly records his result B_k
Alice announces her measurement outcome A_k. The announcement may be a genuine measurement outcome, or a fabricated measurement outcome.
Steps 1-3 are repeated to gather measurement statistics so that Bob may calculate Sn.
Bob calculates steering parameter Sn from measurement statistics:
Sn > Cn
Sn < Cn
𝑆n ≡ 1 𝑛 𝑘=1 𝑛 Ak 𝜎kB ≤𝐶n
They must share entanglement!
can’t be sure!
A. Bennet et al., PRX 2, (2012), D. Saunders et al., Nat. Phys. 6, 845 (2010)

7. Some EPR-steering demonstrations
‘Experimental EPR-steering using Bell-local states’
D.J. Saunders, S.J. Jones, H.M. Wiseman and G.J. Pryde, Nature Physics 6, 845 (2010)
2010
‘Aribtrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole’
A.J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, Phys. Rev. X 2, (2012)
2012
Been shown to be more tolerant to noise and photon loss than Bell test

8. Can steering be one-way?
NO TRUST
Alice
Asymmetric state
Steering demonstrated
TRUST
In EPR steering, we share a symmetric state between the trusted party, Bob, and the distrusted party, Alice.
Can we take the working protocol to a non-working protocol by exchanging the partys? And thus, we can demonstrate that Alice is able to steer Bob’s state but not vice versa?
17/08/16:
Can we make some sort of asymmetric state that in one configuration the steering can be completed but in another not?
So can steering be one-way?
Bob

9. Can steering be one-way?
YES
TRUST
Alice
Asymmetric state
Steering fails
No TRUST
In EPR steering, we share a symmetric state between the trusted party, Bob, and the distrusted party, Alice.
Can we take the working protocol to a non-working protocol by exchanging the partys? And thus, we can demonstrate that Alice is able to steer Bob’s state but not vice versa?
17/08/16:
Can we make some sort of asymmetric state that in one configuration the steering can be completed but in another not?
So can steering be one-way?
Bob

10. Can steering be one-way?
NO TRUST
Bob
Non-working protocol
working protocol
Photon source
TRUST
In EPR steering, we share a symmetric state between the trusted party, Bob, and the distrusted party, Alice.
Can we take the working protocol to a non-working protocol by exchanging the partys? And thus, we can demonstrate that Alice is able to steer Bob’s state but not vice versa for specific states?
Alice

11. Do any states exist which are one-way steerable?
YES! ∞
Alice
Photon source
Entangled modes
The answer to question if such one-way steerable states exist, is YES!
In this example, shown by Roman Schnabel’s group, Alice and Bob share entangled modes where they can perform gaussian measurements on
Bob
V. Handchen et al., Nat. Photon 6, 598 (2012)

12. Homodyne detection of Gaussian states
Successful Gaussian one-way steering with two-mode squeezed states
But:
Gaussian measurements are insufficient to capture the full nonlocality of Gaussian states
Explicit examples of one-way steerable Gaussian states are two-way steerable for appropriate measurements
S. Wollmann et al., Phys. Rev. Lett. 116,  (2016)
Their main result shows, the successful demonstration of one-way steerability of Gaussian states
By alternating the vaccum in Bob’s state, they could control the steering regimes
Blue in is Bob, red line is Alice
HOWEVER:
Gaussian measurements are insufficient to capture the full nonlocality of Gaussian states
And for this particular demonstration, it is possible to find explicit examples of Gaussian states which are one-way steerable, but two-way steerable when certain well-chosen non-Gaussuan measurements
As the presence of one-way steerability under a restricted class of measurements does not imply one-way steerability of the state, we ask, do states exist which are one-way steerable for arbitrary measurements?
Do states exist which are one-way steerable for arbitrary measurements?
V. Handchen et al., Nat. Photon 6, 598 (2012)

13. Do any genuinely one-way steerable states exist? YES!
Theoretical proof for infinite-setting POVMs
And the answer is YES
Nicolas Brunner’s group successfully proofed the existence genuine one-way steerable states for infinite-setting positive-operator-valued-measures, to which I will refer as arbitrary measurements
They chose an exotic family of states to demonstrate the effect over an extremely small parameter range, which is unsuitable for experimental observation
Independently to that, David Evans and Howard Wiseman showed one-way steerability for arbitrary projective measurements for more practical, singlet states with symmetric noise
One-way steerable state for projective measurements
J. Bowles et al., Rev. Lett. 112, (2014), P. Skrzypczyk et al., PRL 112, (2014), R. F. Werner, PRA 40, 4277 (1989).

14. What is a genuine one-way steerable state?
symmetric or asymmetric state
Using the theorem of Quintino et al. to extend to arbitrary measurements ρ
𝜌 𝐴𝐵 = 1−𝑝 3 𝜌 𝑊 + 𝑝 𝐼 𝐴 2 ⊗ 𝑣 𝐵 ⟨𝑣 | 𝐵
Werner state for steering scenario
𝜌 𝑊 =𝜇 | 𝜓 𝑠 ⟩⟨ 𝜓 𝑠 |+ 1−𝜇 4 𝐼 4
one-way steerable for arbitrary measurements if
with μ = [0,1]
In our work we ask if we can extend to a state which is steerable in one direction but cannot be steered in the other direction even for the case of arbitrary measurements and infinite settings.
We consider to observers, Alice and Bob, performing local measurements on their shared state rho
The classical strings j and k, label and record the measurements Ma and Mb they choose to perform
As shared state, we consider a Werner state, which is a singlet state parametrised by the Werner parameter mu. If mu is 1, we have a singlet state. Lower values of mu add symmetric noise to the state
By using the theorem of Quintino, which led to adding assymetric loss into Bob’s arm, here represented by the probability p of a vacuum state, we were able to extend this state to arbitrary measurements, if we can fulfil this condition for loss p
𝑝> 2𝜇+1 3
S. Wollmann et al., Phys. Rev. Lett. 116,  (2016)

15. Experimental generation of steerable state
cw pumped Sagnac Ring interferometer with λp = 410 nm Fidelity of F=99.6% and Tangle of T=98.6% with ideal singlet state
‘We did this test starting with a singlet state encoded in polarization
This state is generated in a continuous-wave Sagnac ring interferometer
17/08/2016
Highlight:
Two photon experiment
Polarization encoded
T. Kim et al., PRA 73, (2006), A. Fedrizzi et al., Opt. Exp. 15, (2007)

16. Experimental generation of steerable state
cw pumped Sagnac Ring interferometer with λp = 410 nm Fidelity of F=99.6% and Tangle of T=98.6% with ideal singlet state
For EPR steering, we use a Sagnac Ring interferometer as source for generating entangled photons.
source of entanglement
T. Kim et al., PRA 73, (2006), A. Fedrizzi et al., Opt. Exp. 15, (2007)

17. Experimental generation of steerable state
cw pumped Sagnac Ring interferometer with λp = 410 nm Fidelity of F=99.6% and Tangle of T=98.6% with ideal singlet state
Alice
For EPR steering, we use a Sagnac Ring interferometer as source for generating entangled photons.
source of entanglement
T. Kim et al., PRA 73, (2006), A. Fedrizzi et al., Opt. Exp. 15, (2007)

18. Experimental generation of steerable state
loss
cw pumped Sagnac Ring interferometer with λp = 410 nm Fidelity of F=99.6% and Tangle of T=98.6% with ideal singlet state
Bob
This setup allowed us to generate states with high fidelity and efficient transmission before loss that no bandpass filter is required
source of entanglement
T. Kim et al., PRA 73, (2006), A. Fedrizzi et al., Opt. Exp. 15, (2007)

19. Experimental generation of steerable state
loss
cw pumped Sagnac Ring interferometer with λp = 410 nm Fidelity of F=99.6% and Tangle of T=98.6% with ideal singlet state
Alice
Bob
This setup allowed us to generate states with high fidelity and efficient transmission before loss that no bandpass filter is required
source of entanglement
T. Kim et al., PRA 73, (2006), A. Fedrizzi et al., Opt. Exp. 15, (2007)

20. Steering regimes A B A B A B A B A B A B one-way steerable for
Two-way steerable
selected measurements
arbitrary measurements A B A B A B A B A B A B
S. Wollmann, N.Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, Phys. Rev. Lett. 116,  (2016)

21. One-way steerable regime for arbitrary measurements
Two-way steering
𝑆16 ≡ 𝑘=1 𝑛 Ak 𝜎kB ≤𝐶16
Werner parameter
Two-way steerable regime
n=∞
n=16
One-way steerable regime for projective measurements
Steering parameter S
One-way steerable regime for arbitrary measurements
For Alice S16 = ± at ηA = (16.98 ± 0.02)%
For Bob S16 = ± at ηB = (16.94 ± 0.02)%
for n measurement directions on Bloch sphere, here: n=16

22. One-way steering for selected measurements
𝑆16 ≡ 𝑘=1 𝑛 Ak 𝜎kB ≤𝐶16
Two-way steerable regime
n=∞
n=16
One-way steerable regime for projective measurements
One-way steerable regime for projective measurements
Steering parameter S
One-way steerable regime for arbitrary measurements
For Alice S16 = ± at ηA = (17.11 ± 0.07)%
For Bob S16 = ± no violation
for n measurement directions on Bloch sphere, here: n=16

23. One-way steering for arbitrary measurements
𝑆16 ≡ 𝑘=1 𝑛 Ak 𝜎kB ≤𝐶16
Two-way steerable regime
n=∞
n=16
One-way steerable regime for projective measurements
Steering parameter S
One-way steerable regime for arbitrary measurements
One-way steerable regime for POVM
For Alice S16 = ± at ηA = (17.17 ± 0.04)%
For Bob S16 = ± no violation
for n measurement directions on Bloch sphere, here: n=16

24. Conclusion
Construction of a readily-accessible class of states to observe truly asymmetric nonlocality
Successful demonstration of genuine one-way steering for arbitrary measurements
In conclusion:
We constructed an experimentally accessible class of states to observe asymmetric nonlocality
And successfully demonstrated genuine one-way steering for arbitrary measurements
We investigated the reference frame dependence of EPR steering and found the most optimal inequality for two measurements per side
And we showed that this inequality allows to demonstrate steering for even maximal misaligned measurement directions

25. Reference frame independent EPR-steering
And now we are going to another steering experiment

26. Reference frames in a typical EPR-steering protocolx
NO TRUST
Alice y
demonstrate steering
Joint preparation x
TRUST
In a typical EPR-steering we assume that the reference frames for Alice and Bob’s measurements are aligned with each other
Bob y

27. Is quantum steering possible without a reference frame?x
NO TRUST y
Alice
demonstrate steering?
Joint preparation x
TRUST
So we asked ourselves what happens to a working EPR-steering protocol if this not the case. Do we still have a working protocol?
Bob y
Can nonlocality be demonstrated without establishing a common reference frame?

28. What to do if there is no reference frame?
1. Establish a common reference frame, e.g. compensation of polarisation of photons in optical fibre
2. Share states which do not require a common reference frame
How do we usually handle these situatons?
Well, we can either try to establish a common reference frame which can be a nontrivial issue in experimental situations
We can share states encoded in specific degrees of freedom or large, complicated entangled states which make a reference frame unnecessary
Both options are not very feasible so asked ‘Can nonlocality be demonstrated without establishining a common reference frame’??
Can nonlocality be demonstrated without establishing a common reference frame?

29. And this question was answered…
… for Bell nonlocality
Theoretically:
Experimentally:
A. Laing et al., PRA 82, (2010)
Y.-C. Liang et al., PRL 104, (2010)
J. J. Wallman et al., PRA 83, (2011)
M. S. Palsson et al., PRA 86, (2012)
-And the answer is yes
- It was the first time answered for Bell nonlocality which can be demonstrated with high probability for a large class of entangled states, when the parties have one or no shared reference direction
However the degree of observed nonlocality can is measurement-orientation dependent and can be arbitrarily small
So we asked if this work can be extended to EPR-steering
P. Shadbolt et al., Sci. Rep. 2, 470 (2012)
Observed nonlocality is measurement-orientation dependent and can be arbitrarily small
So what about EPR-steering?

30. Frame-independent EPR-steering
Necessary-and-sufficient EPR-steering inequality
And Eric Cavalcanti showed that it is possible for two orthogonal measurement settings per side
But we were asking if there is a steering inequality which is rotation-invariant?
For orthogonal measurements in m=2 directions for Alice and n=2 directions for Bob
But is there an inequality which is rotation-invariant?

31. Rotation invariant steering inequality
with
𝑚× 𝑛
correlation matrix
𝑀 𝑗𝑘 ≔〈 𝐴 𝑗 𝐵 𝑘 〉
Best possible rotationally invariant steering inequality for m=n=2 and for m=n=3
And we theoretically prove the existence of such
Which is the trace norm of the correlation matrix M created by Alice and Bob’s measurement results in m and n directions
We found that this inequality is the most optimal inequality when Alice and Bob have two and three measurement settings
So we asked us how do we do compared to Cavalcanti’s necessary-and-sufficint inequality
How do we do compared with the necessary-and-sufficient condition?
S. Wollmann, M.J.W. Hall, R.B. Patel, H.M. Wiseman & G.J. Pryde, manuscript in preparation

32. In our experiment…

33. Why m=n=2 ? Two settings on each side Locally orthogonal
Line of sight:
Alice’s and Bob’s measurement directions are in same plane
Relative orientation within this plane given by 𝛼
We first considered the case where Alice and Bob share a single reference direction and where they use m =n = 2 measurement settings—the minimal set size. The measurement directions lie in a plane orthogonal (on the Bloch sphere) to the shared direction, and the two settings on each side are locally orthogonal. This is a natural physical situation because the shared reference direction may be determined reliably by line of sight between the parties or by the propagation axis of light as it emerges from an optical fibre, for example. Furthermore, it is natural to assume that Alice and Bob can reliably set local measurement directions. Although Alice and Bob’s measurement directions will lie in the same plane, their relative orientation within this plane may be unknown. α

34. Comparing both inequalities…
Let’s consider we share a Werner state between Alice and Bob
necessary-and-sufficient inequality
rotationally invariant inequality
In our experiment we first considered the case of two measurement directions on each side and a shared Werner state
Because of this the necessary-and-sufficient inequality reduces to this (point) and the rotationally invariant inequality reduce to this (point) form
Both depend on the angle phi between and Alice and Bob’s measurement planes
Φ : angle between Alice and Bob’s measurement planes
α : angle between measurement planes and blue measurement direction

35. Two’s company, three’s a crowd…
Rotation in the plane for m=n=2
1 shared direction
Steering parameter α
maximally entangled state
Werner state
Violation of necessary-and-sufficient and rotationally invariant inequality
First we investigated the case where the two orthogonal measurement directions lie in the same plane orthogonal to the shared direction
Although Alice and Bob’s measurement directions lie in the same (sigma_x and sigma_y) plane, but their relative orientation within this plane may be unknown, that’s why consider a rotation by the angle alpha
This situation allows a direct comparison between the two inequalities
And we see that both inequalities violate the inequality and are close to the theoreticalpredicted value of 2.80 which is close to the maximum of 2\sqrt(2) α

36. Two’s company, three’s a crowd… Tilting out of plane
maximally entangled state
Werner state
1 shared direction
Steering parameter
Tilting off the shared direction α
Next we allowed an offset in the shared reference direction
We tilted Alice’s measurement plane by 64 degree, shifting to a regime where the rotationally invariant inequality is not violating the bound of 2 with a value of 1.98
And the necessary-and-sufficient inequality showed an oscillatory behaviour and violation for angles below 20 degree and above 70 degree
Rotationally invariant inequality not violated with 1.98 ± 0.01
Necessary-and-sufficient inequality is oscillating

37. Two’s company, three’s a crowd… Tilting out of plane
maximally entangled state
Werner state
1 shared direction
Tilting off the shared direction
The final case we investigated is for extreme misalignment in the shared reference direction
For this Alice measured in the sigma_z-sigma_y plane
We can see that neither inequality was violated for any rotation angle
While our rotation-invariant inequality stayed insensitive to rotations of Alice’s measurement directions, the necessary-and-sufficient inequality showed an oscillatory behaviour
Finally we asked what are the measurement outcomes for the rotationally invariant inequality if we extend the number of measurement settings for Alice and Bob to 3
no shared direction
No inequality was violated

38. Two’s company, three’s a crowd… Tilting out of plane
maximally entangled state
Werner state
1 shared direction
Tilting off the shared direction
The final case we investigated is for extreme misalignment in the shared reference direction
For this Alice measured in the sigma_z-sigma_y plane
We can see that neither inequality was violated for any rotation angle
While our rotation-invariant inequality stayed insensitive to rotations of Alice’s measurement directions, the necessary-and-sufficient inequality showed an oscillatory behaviour
Finally we asked what are the measurement outcomes for the rotationally invariant inequality if we extend the number of measurement settings for Alice and Bob to 3
no shared direction
What happens for more # of measurement settings?

39. Rotation-invariant steering inequality for m=n = 3
2.93 ± 0.01 > 3 ≈1.73
perfect alignment
2.21 ± 0.01 > 3 ≈1.73
miscalibration
For this scenario we investigated three different cases:
First we investigated the case where Alice and Bob’s directions are perfectly aligned
With a steering parameter of 2.93, we successfully violate the bound of sqrt(3)
Then we studied the case where Alice and Bob’s measurement settings are strongly misaligned.
Even in this case the steering value of 2.21 is well above the bound of sqrt(3)
Finally we observed if steering is possible if Alice is measuring in nonorthogonal directions
We measured a steering value of 2.74 successfully violating the bound
nonorthogonal directions
2.74 ± 0.01 > 3 ≈1.73

40. Conclusion Reference frame independent steering:
2 measurement-settings per side:
our rotationally invariant inequality is the most optimal inequality
3 measurement-settings per side:
the rotationally-invariant inequality is violated for maximal misalignment of the reference frames
In conclusion:
We constructed an experimentally accessible class of states to observe asymmetric nonlocality
And successfully demonstrated genuine one-way steering for arbitrary measurements
We investigated the reference frame dependence of EPR steering and found the most optimal inequality for two measurements per side
And we showed that this inequality allows to demonstrate steering for even maximal misaligned measurement directions

42. 80 years ago…

43. EPR-paradox Showed incompability between concepts of local causality,
Information cannot travel faster than light
Future events have causes in the past
and completeness,
Every element of reality has a counterpart in the theory (Heisenberg’s uncertainty principle)
EPR-steering is a phenomena observable within quantum theory. It is proven by a set of quantum correlations that violate assumptions as realism, completeness and local causality, and as a result, have troubled famous physicists since the conception of quantum mechanics 100 years ago. EPR-steering follows out of the EPR paradox which got formulated by Einstein, Podolsky and Rosen in 1935.
Probably delete the reference to Heisenberg’s UP. Perhaps add in the EPR definition that an “element of reality” can be predicted with 100% certainty.

44. EPR Gedankenexperiment
“Amazing knowledge”
Schrödinger, Proc. Camb. Phil. Soc. 31, 446 (1935)
“key characteristic of quantum mechanics”
introduced the term “entanglement”
Two particles W1 and W2
Interact for a time t and form Wab
Move apart for time T, that local causality is imposed
After T>t, the particles are measured in different labs by Alice and Bob. They are allowed to repeat their measurements on a large ensemble of identically prepared systems
One party (Alice) can “steer or pilot” another’s state (say Bob) purely by their choice of measurement

45. EPR Gedankenexperiment
|ΨAB˃ = 1/√2 (|0A1B˃ - |1A0B˃)
Alice’s measurement
Bob’s measurement
Depending on Alice’s outcome
in |0˃ and |1˃ basis:
50% |0˃ and 50% |1˃
50% |0˃ and 50% |1˃
Depending on Alice’s outcome
in |+˃ and |-˃ basis:
Two particles W1 and W2
Interact for a time t and form Wab
Move apart for time T, that local causality is imposed
After T>t, the particles are measured in different labs by Alice and Bob. They are allowed to repeat their measurements on a large ensemble of identically prepared systems
%% Nathan’s comments: The contradiction you want to highlight is the following:
considering Bob’s measurements in isolation Heisenberg’s uncertainty principle says that we cannot perfectly predict the outcome of Bob’s measurements in the 0/1 and +/- basis.
2) But, if Alice measures in the 0/1 (or +/-) basis then that information lets us predict the outcome of Bob’s 0/1 (or +/-) measurement with 100% certainty.
3) However if local causality is true (and EPR were sure that it was) then what Alice does shouldn’t matter. This means Bob must actually be able to somehow predict his measurement perfectly. This means his measurement result is an “element of reality” which doesn’t exist in the quantum predictions in 1). Therefore it is an element of reality without a counterpart in the quantum theory and thus quantum mechanics is incomplete.
Einstein believed that there were “hidden variables” that would complete QM and resolve the paradox, but today we know that this isn’t true and you can’t explain all quantum correlations with hidden variable models. Hence, we call QM as non-local.
So maybe what you want to write on the slide is something to express that Bob’s measurement out comes occur in a 50/50 split if Alice *doesn’t* measure but if we know Alice’s out come then it will be one out come or the other with 100% certainty (points 1) and 2) above). Then make the point in 3), that if we accept local causality then we violate completeness.
50% |+˃ and 50% |-˃
50% |+˃ and 50% |-˃
Violation of completeness

46. Generation of steerable state
cw pumped Sagnac Ring interferometer with λp = 410 nm Type-II SPDC process Generation of polarization entangled photons Fidelity of F=99.6% and Tangle of T=98.6% with ideal singlet state
For EPR steering, we use a Sagnac Ring interferometer as source for generating entangled photons.

47. EPR-steering Werner State 𝑊μ= 𝜇 𝛹− Ψ− + 1−𝜇 𝐼 4 |ΨAB˃
For n=6 measurements
S6= ± at ε = 20.99% A B
Werner State 𝑊μ= 𝜇 𝛹− Ψ− + 1−𝜇 𝐼 4
𝑆n ≡ 1 𝑛 𝑘=1 𝑛 Ak 𝜎kB ≤𝐶n (𝜀)
Saunders et al., Nat. Phys. 6, (2010)
Bennet et al., PRX 2, (2012)

48. One-way EPR-steering Werner State 𝑊p= 1−𝑝 𝑊μ+𝑝 0 0 𝑥 𝐼 2 |ΨAB˃
For n=6 measurements
Two-way steerable A B
One way steerable for projective measurements
Werner State 𝑊p= 1−𝑝 𝑊μ+𝑝 𝑥 𝐼 2
The history goes something like this.
One way steering exists and has been observed in continuous variable systems, if we restrict Alice and Bob to particular projective measurements.
First Brunner [2], and then David Evans and Howard [3] then proved that there were certain qubit states that were one-way steerable allowing arbitrary projective measurements. However Howard’s state was much more practical, being a combination of loss and Werner noise given by the blue line in the graph. It is also more intuitive how adding extra loss allows Alice to cheat (although you may not have any time to explain why this is so).
In this paper, we combine [3] with another Brunner result [4] (Basically [4] gives a recipe for taking a state for which a result holds for projective measurements, in our case the result of [3], and tells you how to construct a new state where the result holds for POVM’s.) to identify a new region of loss and Werner mixing that is one way steerable even for the most general measurements of all, POVM’s.
[1] Händchen, V., Eberle, T., Steinlechner, S., Samblowski, A., Franz, T., Werner, R. F., & Schnabel, R. (2012). Observation of one-way Einstein–Podolsky–Rosen steering. Nature Photonics, 6(9), 598–601.
[2] Bowles, J., Vertesi, T., Quintino, M. T., & Brunner, N. (2014). One-way Einstein-Podolsky-Rosen Steering. Physical Review Letters, 112(20),
[3] Evans, D. A., & Wiseman, H. M. (2014). Optimal measurements for tests of Einstein-Podolsky-Rosen steering with no detection loophole using two-qubit Werner states. Physical Review A, 90(1),
[4] Marco Túlio Quintino, Tamas Vertesi, Daniel Cavalcanti, Remigiusz Augusiak, Maciej Demianowicz, Antonio Acín, and Nicolas Brunner, (2015). Entanglement, steering, and Bell nonlocality are inequivalent for general measurements. arXiv:
One way steerable for POVM’s

49. EPR-steering protocol
|ΨAB˃
Step 1
This is probably a good spot to explicitly talk about cheating?
Step 2
Bob measures his state and secretly records his result Bk
Step 3
Step 4
Bob receives his qubit. He may receive an unentangled single qubit, or he may receive one-half of a bipartite entangled state. At this state, he cannot distinguish the former from the latter.
Bob announces to Alice his choice of measurement setting k, corresponding to an observable sigma_k_B drawn from the pre-determined set {sigma_k_B}n that Alice can measure in the appropriate direction.
Bob measures his state and secretly records his result B_k
Alice announces her measurement outcome A_k. The announcement may be a genuine measurement outcome, or a fabricated measurement outcome.
Steps 1-3 are repeated to gather measurement statistics so that Bob may calculate Sn.
Sn has to be violated to demonstrate successful EPR-steering with no detection loophole
Bob calculates steering parameter Sn from measurement statistics
𝑆n ≡ 1 𝑛 𝑘=1 𝑛 Ak 𝜎kB ≤𝐶n (𝜀)
Bennet et al., PRX 2, (2012)