Jeff Tollaksen, Director, Institute for Quantum Studies (Quantum.Chapman.Edu)

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Jeff Tollaksen, Director, Institute for Quantum Studies (quantum.chapman.edu)

Courtesy Thaller EmQM15 Emergent Quantum Mechanics Vienna, Austria October 24, 2015 Paradigms shield themselves against falsification as part of normal science Thomas Samuel Kuhn (1922 –1996): The paradigm determines how you see the facts! The Structure of Scientific Revolutions Strong vs Weak emergence • Strong emergence: higher levels have causal efficacy over lower levels; whole-part causation • No room at the bottom–the laws of physics @ micro-level completely determine everything (?) • New emergent laws lead to over-determination

Complex Emergent systems laws

Downward causation

fundamental Basic laws particles of physics Methodology to explore Emergent QM: start with more intuitive axioms for QM Quantum Mechanics

Nonlocality Relativity Methodology to explore Emergent QM: start with more intuitive axioms for QM • Those axioms recalls a Woody Allen joke: This guy goes to a psychiatrist and says, “Doc, my brother’s crazy – he thinks he’s a chicken!” The doctor says, “Well, why don’t you turn him in?” The guy says, “I would, but I need the eggs!” • We say, “Quantum theory is crazy – but we need the eggs!” • Not intuitive: “It’s like trying to derive special relativity from the wrong axioms.” – Yakir Aharonov Fast objects contract in the direction of their motion Moving clocks slow down Observers determine the results of measurements Methodology to explore Emergent QM: start with more intuitive axioms for QM • Question: What, indeed, is so “special” about special relativity? • • Answer: The two axioms so nearly contradict each other that only a unique theory reconciles them. Methodology to explore Emergent QM: start with more intuitive axioms for QM Y. Aharonov and (independently) A. Shimony: Quantum mechanics, as well, reconciles two things that nearly contradict each other:

• Can we derive a part of quantum mechanics from these axioms? • Aharonov: Quantum mechanics must include uncertainty. Methodology to explore Emergent QM: start with more intuitive axioms for QM

Quantum Mechanics

Nonlocality Relativity

Why uncertainty? Traditional answer: nature is capricious Methodology to explore Emergent QM: start with more intuitive axioms for QM Nonlocality Relativity ? Quantum Mechanics Can we derive quantum mechanics from these two axioms? 1. What is nonlocality? Nonlocal correlations? Aharonov-Bohm effect? “Modular” dynamical variables? 2. What does “no signaling” mean? What is left of “no signaling” in the limit c . of nonrelativistic quantum mechanics? Methodology to explore Emergent QM: start with more intuitive axioms for QM Nonlocality Relativity ? Quantum Mechanics Can we derive quantum mechanics from these two axioms? 1. What is nonlocality? Nonlocal correlations?

2. What does “no signaling” mean? “No signaling” at any speed!

S. Popescu and D. Rohrlich, Found. Phys. 24 (1994) 379 Monogamy of CHSH correlations

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1J V1J7 > QR RQVJQ ]C:7R1HV8? Time-Symmetric formulation of Quantum Mechanics TSQM • We ask: “Why does God play dice?” • Traditional answer: nature is capricious • Alternative: allows quantum mechanics to independently select both the initial and final states of a single system • TSQM: the state of a system at a given moment is described by two wave-functions, one evolving from the past to the future, and one evolving from the future to the past Time-Symmetric formulation of Quantum Mechanics TSQM • We ask: “Why does God play dice?” • Traditional answer: nature is capricious t 2 Φ • Alternative: allows quantum mechanics to independently select both the initial and final states of a single system t ? • TSQM: the state of a system at a given moment is described by t Ψ two wave-functions, one 1 evolving from the past to the The two-state vector future, and one evolving from the future to the past Φ Ψ Q%JR:`7HQJR1 1QJ7HC:1H:C0_%:J %I

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t2 Y. Aharonov, P. G. Bergman and J. L. Lebowitz, Phys. Rev. 134, 1410 (1964) Time-Symmetric formulation of QM (TSQM) To be useful and interesting, any re-formulation of QM should meet several criteria, for example: 1) TSQM is consistent with standard QM

2) TSQM brings out features in QM re emergence that were missed before (e.g. weak values) 3) TSQM stimulated discoveries in other fields re emergence 4) TSQM suggests generalizations re emergence 2 The two-state vector description of a quantum system Measurements performed on a pre- & post-selected system described by the two-state vector: Φ Ψ Strong measurement: The Aharonov-Bergmann-Lebowitz (ABL) formula:

2 PΦ = 1 t2 ΦΨPC= c Φ Prob(C= c ) = 2 ∑ ΦΨPC c t C = ? = i i Ψ

t1 PΨ = 1 Weak measurement: The Aharonov-Albert-Vaidman effect: Phys. Rev. Lett. 60, 1351 (1988). ΦΨC Weak value C ≡ w ΦΨ

Graphic courtesy Vaidman 2.a Weak Measurements - spin

Y Aharonov, D Albert, L Vaidman, Phys. Rev. Lett.60 (1988), 1351 2 Weak values and causality: game of errors

Weak value sum rule 2 Weak values and causality: game of errors

• Probability to obtain weak value as an error of the measuring device is greater than the probability to obtain weak value • Uncertainty (playing of dice) derived from nonlocality in time, causality • Led to new approach to information we call “weak information” 2.a Emergence through hierarchical entanglement • 1J$VJ :J$CVIVJ 11 .:.1$.V`.1V`:`H.1H:CCV0VC5 .V`VI:7GV 1IV1J V`0:C11 .JQ1JRV]VJRVJ C7 V0QC01J$%GR7 VI • 8V8]`Q]V` 1V`%JH 1QJRV]VJRQJ1.QCV

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ΦΨC PΦ = 1 C ≡ t2 w Φ ΦΨ t C = ? (ABAB+) = + Ψ w w w ABAB t1 ( )w ≠ w w PΨ = 1

The weak value of a product of observables is not equal to the product of their weak values. In some sense, new properties emerge with complexity! 2.a Non-locality of EPR from perspective of TSQM 1 ↑x↑ z (↑ ↓ − ↓ ↑ ) AB 2 ABAB σ = 1 x σ z = 1 σ zA = −1 t2 ↑ σ = −1 ↑ x z xB σ = ? σ = ? t zA xB σ σzA xB = −1 1 ( ↑ ↓ − ↓ ↑ ) 2 t1 A B

2 ↑ ↑P ↑ ↓ − ↓ ↑ x z (σ σzA xB =− 1) ( ) Prob(σ σzA xB = − 1) = 2 2 =1 ↑↑PP ↑↓−↓↑+↑↑ ↑↓−↓↑ x z (σ σzA xB = 1) ()x z (σ σzA xB =− 1) () 2.a Elitzur & Vaidman Interaction Free Measurements

• However, the phase difference + D+ C between the left and right path can be altered by the presence of an object in which case detector D+ may be triggered

• If detector D+ is triggered one may then conclude: – The particle was not blocked by the object – The object must have been in the path 2.a Hardy’s Paradox • With 2 MZI’s, there is overlap region D- C- D+ C+ • Each MZI can be said to measure whether or not the other MZI’s particle is in the overlapping path, otherwise nothing would have disturbed the electron, and the electron couldn't have ended in D- • But, if detectors D- and D+ both click then intuition leads to a paradox. e- e+ • If D- clicks, then e+ must have gone through the overlapping arm: PARADOX: they should - + never reach the D e overlapping arm detectors because e+ • Conversely if D+ clicks then e- must and e- overlap and have gone through overlapping arm therefore annhilate D+ e- overlapping arm 2.a Hardy’s Paradox • Suppose we try to measure the position of e- by inserting a detector - - D0 in the overlapping arm of the e MZI. • e- is always in the overlapping arm, - - - however D0 disturbs e & e could end up in the D-detector even if no e+ is present in the overlapping arm • Cannot infer from D- that e+ was in the overlapping arm disturbing e-. The paradox disappears. 2.b Weak measurements and counterfactuals

Aharonov, Botero, Popescu, Reznik, JT, “Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values” Physics Letters A, v301, 130 Paradoxical reality implied counter-factually has new experimentally accessible consequences in terms of weak measurements, which allow us to test - to some extent - assertions that have been otherwise regarded as counter-factual

weak value of ‘number’ of particle- pairs is negative Experiments: J.S. Lundeen and A.M. Steinberg Phys Rev Lett 102:020404, (2009) K. Yokota, T. Yamamoto, M. Koashi, N. lmoto New Journal of Physics 11 (2009) 033011 2.a Weak values obey a simple intuitive & self-consistent logic D - C - D + C + • If there is 1 e+ in overlapping arm - BS 2 BS 2 + - - + • & 1 e- in other overlapping arm NO O O + NO

- BS + • How can there be ZERO e+e- BS 1 1 pairs in both overlapping arms? e - e + • The particles are both definitely in D - C - D + C + BS - + the overlapping arms, but they 2 BS 2 NO - O - O + NO + are not there together? BS - BS + • Ans: the weak value of a product 1 1 of observables is not equal to the e - e + - - + + product of their weak values. D C D C - BS 2 BS 2 + NO - O - O + NO +

- + Aharonov, Botero, Popescu, Reznik, Tollaksen, BS 1 BS 1 Physics Letters A 301 (3-4): 130-138, 2001. e - e + 2.a Weak values obey a simple intuitive & self-consistent logic • But we also have the statements: D- C- D+ C+ - - BS BS+ – e must be in the overlapping arm 2 2 + - - + otherwise e couldn't have ended at D- & NO O O+ NO – e+ must be in the non-overlapping arm - since there was no annihilation BS BS+ 1 1 – These & the opposite are confirmed

e- e+ • These 2 statements are at odds w/ the fact D- C- D+ C+ that there is just one electron-positron pair - BS BS+ 2 2

- - + • QM solves the paradox in a remarkable way NO O O+ NO - it tells us that there is minus one electron-

- positron pair in the non-overlapping arms BS+ BS 1 which brings the total down to a single pair! 1

e- e+ 2.a Atom of Emergence/Holism • While there is no particle in either outer path D- C- D+ C+ • Nevertheless, the interaction - + between the 2 paths tells us that BS 2 BS 2 - + there is minus one electron- NO O- O + NO positron pair in the non-

overlapping arms - + BS 1 BS 1

e- e+

No particle No particle here here 2.a Generalization of Atom of Emergence/Holism

t2 - - - Φ t Ψ … t1 + + +

particle 1 particle 2 particle 5 2.a Generalization of Atom of Emergence/Holism

t2 - - - Φ Strongly measure projection t operator for 1 Ψ … t1 + + +

particle 1 particle 2 particle 5 2.a Generalization of Atom of Emergence/Holism

t2 - - - Φ Strongly measure projection t operator for 1 & 2 Ψ … t1 + + +

particle 1 particle 2 particle 5 2.a Generalization of Atom of Emergence/Holism

Weak value is 2N ; we pick up something that we don’t normally think is there; it is a very strong, stable effect t3

t2 - - - Φ t Ψ … t1 + + +

particle 1 particle 2 particle 5 Only when strongly measure projection operator for all 5 2.a Generalization of Atom of Emergence/Holism

t2 - - - Φ t Ψ … t1 + + +

particle 1 particle 2 particle 5 2.b %:J %I.V .1`V: • “Well I've often seen a cat without a grin," thought Alice “but a grin without a cat! It’s the most curious thing I ever saw in all my life”

• As if you were “separating a particle from its properties” • what seemed to be a “whole inseparable system" can in fact be physically separated into distinct parts residing at different locations. This opens a new door on the relationship between wholes and parts • New computational/information resource

Experiment: Denkmayr, Geppert, Sponar, Lemmel, Matzkin, JT, Hasegawa, Nature Comm (2014) Theory: JT 2001 PhD thesis; Aharonov & Rohrlich 2005; Aharonov, Y., Popescu, S., Rohrlich, D., & Skrzypczyk, P. (2013). Quantum cheshire cats. New Journal of Physics, 15(11), 113015. 2.b %:J %IH.V 1`V H: TV6]V`1IVJ :C0V`1`1H: 1QJ

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Mirror Beam Splitter Selective Potential Barrier

2 2 1+σ cos α sin α V()() x=z V Θ x − L 2 0

σ x = +1

x= − L x = 0 x= L x The electron starts with momentum p, such that: Its state is given by Upon hitting the beam splitter:

Now there are two cases: 1. The electron has spin-z up and therefore feels the potential 2. The electron has spin-z down and therefore does not feel the potential Y. Aharonov, E. Cohen, S. Popescu, arXiv:1510.03087 2.b : CV %``VJ Q`]1J

Mirror Beam Splitter Selective Potential Barrier

2 2 1+σ cos α sin α V()() x=z V Θ x − L 2 0

σ x = +1

x= − L x = 0 x= L x 1. The electron has spin-z up and therefore feels the potential After n period times

Hence if the electron would move to the right side.

2. Electron has spin-z down and therefore does not feel the potential After n period times the electron would stay on the left (the amplitudes did not sum up coherently). Y. Aharonov, E. Cohen, S. Popescu, arXiv:1510.03087 2.b : CV %``VJ Q`]1J

Mirror Beam Splitter Selective Potential Barrier

2 2 1+σ cos α sin α V()() x=z V Θ x − L 2 0

σ x = +1

x= − L x = 0 x= L x Therefore, if after n period times we find the electron on the left, we immediately conclude that it has .

But how can its spin change if it was all the time at the left side? The electron’s spin left its mass behind and traveled to the right side!

Y. Aharonov, E. Cohen, S. Popescu, arXiv:1510.03087 2.b : CV %``VJ Q`]1J

Mirror Beam Splitter Selective Potential Barrier

2 2 1+σ cos α sin α V()() x=z V Θ x − L 2 0

σ x = +1

x= − L x = 0 x= L x The weak values tell the full story: The spin along the z-direction was up at all times

The spin along the x-direction moved from left to right, as indicated also by its time derivative Local current of massless spin can account for seemingly nonlocal effects! Y. Aharonov, E. Cohen, S. Popescu, arXiv:1510.03087 2.b IV`$VJ J :J$CVIVJ When the selective potential barrier has spin, some more surprises are expected!

Y. Aharonov, E. Cohen, S. Popescu, arXiv:1510.03087 2.b IV`$VJ J :J$CVIVJ

A product state If the spins are aligned, the Otherwise, the particle right amplitude is coherently stays on the left built Becomes Maximally entangled! Here the particle and barrier become entangled by virtue of a massless current of entangled spins! Time-Symmetric formulation of QM (TSQM) To be useful and interesting, any re-formulation of QM should meet several criteria, for example: 1) TSQM is consistent with standard QM

•Novel proof that pre-and-post-selected QM is contextual •Weak value signature that can be tested experimentally •Conjectured that anomalous weak values constitute proofs of the incompatibility of quantum theory with non- contextual ontological models

•JT, Journal of Physics A, 40 (2007) 9033-9066) 3.b Weak Values and Contextuality

M Waegell, JT "Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values," arXiv: 1505.00098 3.d Quantum pigeonhole principle & nature of quantum correlations

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%:J %I]1$VQJ HQI]CVIVJ :`7 Q 3.d Quantum pigeonhole principle & nature of quantum correlations

Aharonov, Colombo, Popescu, Sabadini, JT; arXiv1407.3194 3.d Quantum pigeonhole principle & nature of quantum correlations

Aharonov, Colombo, Popescu, Sabadini, JT; arXiv1407.3194 3.b Kochen-Specker contextuality can be localized and observed through weak measurements •using pre- and post-selected states (PPS) along with many existing proofs of the Kochen-Specker (KS) theorem, it is possible to localize the violation of noncontextuality to specific observables where it can be probed using weak measurements. •Several important examples are discussed in detail, and a framework for a more general set of experimental tests based on known proofs of the KS theorem is given. •The underlying ontological models that are used in these arguments are explored detail, and connections are made to PPS paradoxes such as the 3-box paradox, the quantum Cheshire Cat, and the quantum pigeonhole principle, as well as to the Mean King’s problem. "Kochen-Specker contextuality can be localized and observed through weak measurements," Mordecai Waegell , JT, arXiv:1505.00098 3.c V1 QJ; C:1 7CQH:CV_%: 1QJ Q`IQ 1QJ

JHC:1H:C].71H5 .V`Q`HV.: Q:H 1J .V:IV]C:HV1.V`V .V]:` 1HCV1 3.c QJCQH:C1 77 @1JVI: 1H 08 R7J:I1H Bell-inequality violations follow from the Hilbert-space structure of quantum mechanics; they are purely kinematic Aharonov-Bohm effect demonstrates dynamical non-locality , i.e. in the quantum equations of motion 3.c Quantum interference, modular variables & weak measurements

D 2D

JT et al,New Jrnl Physics 12 (2010) 013023; see experiment Spence, Parks arXiv:1010.3289, Foundations of Physics, 2011 Quantum interference, modular variables & weak measurements

JT, Aharonov, Casher, Kaufherr, Nussino,New Jrnl Physics 12 (2010) 013023 3.c Having your cake & eating it-measuring nonlocal interactions w/o violating causality Post-select an t4 eigenstate of parity

t3 Open or close slit

ΦΨC Weakly measure t C ≡ modular variable exp(ipD//h) 2 ? w ΦΨ

t Ψ Pre-select all particles 1 at right slit

JT et al,New Jrnl Physics 12 (2010) 013023; see experiment Spence, Parks arXiv:1010.3289, Foundations of Physics, 2011 3.c Schrödinger vs Heisenberg Conclusions • if your only tool is a hammer, then you tend to treat everything as if it were a nail • To grasp the world more fully by grasping it gently

See Quantum.chapman.edu

Aharonov Members Include: Englert Davies www.birkhauser.ch/QSMF

Gross Leggett Susskind Berry 3.c C:1H:C C1I1 Q` _%:J %I Q] 1H7JQ 1.: 1 VVI : `1` 1$.

Y Aharonov, A Botero, S Nussinov, S Popescu, JT, L Vaidman, New Journal of Physics 15 093006, 3 Sept 2013 3.c C:1H:C C1I1 Q` _%:J %I Q] 1H7JQ 1.: 1 VVI : `1` 1$.

Y Aharonov, A Botero, S Nussinov, S Popescu, JT, L Vaidman, New Journal of Physics 15 093006, 3 Sept 2013 4.a QM Generalization: Each moment of time a new universe

• New ability to obtain a post-selected state of one particle that is completely correlated to the pre-selected state of a second particle:

• stack N particles on top of another along the time axis: 4.a QM Generalization: Each moment of time a new universe

• “Collapse” does not necessarily imply arrow of time at microscopic level

Aharonov, JT, 2010, Visions of Discovery; Aharonov, Popescu, JT, arXiv:1305.1615 Aharonov, Popescu, JT, Vaidman, Phys Rev A 79, 052110 (May 1, 2009) 4.a Each moment of time is a new universe

“block universe on steroids” David Albert

Anthony’s URB v ER

Aharonov, JT, 2010, Visions of Discovery; Aharonov, Popescu, JT, arXiv:1305.1615 Aharonov, Popescu, JT, Vaidman, Phys Rev A 79, 052110 (May 1, 2009) Clock Time, t Time, Clock Becoming Time, ˊ A moment of awareness corresponding to a collapse and a wavepacket of ‘becoming time’ in the universe 2 Measurement of arbitrary observable C & Hint = g() t PMD C Q= g() t C Ψ ()Q t2 Qfin = c2 MD ψ f = c2 t

t1 Qin = 0

0 c c c3 Q Collapse!1 2

α∑α ic i Q=0 → ∑ i c i Q = c i →c2 Q = c 2 i i 2 Weak Measurements of arbitrary observable C t2 t Hint = g() t PMD C Ψ ()Q MD PPMD=0, ∆ MD small Qin = 0 t1 ⇒ Hint is small

0 c1 c2 c3 Q 2 Weak Measurements of arbitrary observable C t2 t H= g() t P C Ψ ()Q int MD Q = 0 MD t 1 PPMD=0, ∆ MD small ⇒ Hint is small

0 c1 c2 c3 Q 2 Weak Measurements of arbitrary observable C t2 QCfin = t Hint = g() t PMD C Ψ ()Q MD PP=0, ∆ small Q = 0 MD MD t 1 ⇒ Hint is small

0 c1 C c2 c3 Q 2 Weak Measurements of arbitrary observable C

The outcomes of weak measurements are weak values

Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector Φ Ψ

PΦ = 1 t2 ΦΨC Φ Cw ≡ t C = ? ΦΨ

t1 PΨ = 1 2 Weak Measurements of C with post-selection t2 t Hint = g() t PMD C

ΨMD ()Q Q = 0 PPMD=0, ∆ MD small t1 ⇒ Hint is small

0 c1 c2 c3 Q 2 Weak Measurements of C with post-selection

QCfin= w t2 PΦ = 1

Hint = g() t PMD C t ΨMD ()Q Q = 0 PPMD=0, ∆ MD small ⇒ t Hint is small 1 PΨ = 1

c 0 Cw c1 c2 3 Q The outcomes of weak measurements are weak values

Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector Φ Ψ

t2 σ = 1 y ΦΨC ↑ y Cw ≡ t σξ = ? ΦΨ

↑ x σ σx+ y t1 σ x = 1 σξ ≡ 2

σ σx+ y ↑σ ↑ ↑y ↑ x σ =yξ x =2 = 2 ()ξ w ↑y ↑ x ↑ y ↑ x Weak measurements performed on a pre- and post-selected ensemble

σ σx+ y Pointer probability distribution Weak Measurement of σ = ξ 2 strong Q 2 − MD 2 ∆ 2 Hint = g() t PMDσξ Ψin ()Q = e

The particle pre-selected σ x = 1

The particle post-selected σ y = 1

weak t 2 σ y = 1 ↑ y

t σξ = ?

↑ x t 1 σ x = 1 σ ; 1.4 ( ξ )w ! Robust weak measurement on a pre- and post-selected single system

The system of 20 particles 20 Pointer probability distribution 1 strong Weak Measurement of ∑σ iξ 20 i=1 20 particles pre-selected σ x = 1 20 particles post-selected σ y = 1 σ i y = 1 t2 20 ↑ ∏ y i i=1 weak 1 20 t ∑σ iξ 20 i=1

20 ↑ ∏ x i t i=1 1 σ i x = 1 1 20  ∑σ iξ  ; 1.4 20 i=1 w ! See also “Robust weak measurement “ JT JOP, (2007) Weak quantum measurement of C t2 Hint = g() t PMD C t PPMD=0, ∆ MD small Ψ ()Q ⇒ MD Hint is small Qin = 0 t1

0 c1 c2 c3 Q Weak quantum measurement of C t2 Hint = g() t PMD C t PPMD=0, ∆ MD small

Ψ MD ()Q ⇒ H is small Q = 0 int t1

0 c1 c2 c3 Q Weak quantum measurement of C t 2 H= g() t P C QCfin = int MD t PPMD=0, ∆ MD small

Ψ MD ()Q ⇒ H is small Q = 0 int t1

0 c1 C c2 c3 Q Weak measurement of C with post-selection t2 Hint = g() t PMD C t PPMD=0, ∆ MD small

Ψ MD ()Q ⇒ H is small Q = 0 int t1

0 c1 c2 c3 Q Weak measurement of C with post-selection QC= fin w Hint = g() t PMD C t2 PΦ = 1 PPMD=0, ∆ MD small t Ψ ()Q ⇒ MD Hint is small Q = 0 t 1 PΨ = 1

c 0 Cw c1 c2 3 Q 4.a QM Generalization: Each moment of time a new universe

Aharonov, JT, 2010, Visions of Discovery; Aharonov, Popescu, JT, arXiv:1305.1615 Aharonov, Popescu, JT, Vaidman, Phys Rev A 79, 052110 (May 1, 2009) 4.a QM Generalization: Each moment of time a new universe

Aharonov, JT, 2010, Visions of Discovery; Aharonov, Popescu, JT, arXiv:1305.1615 Aharonov, Popescu, JT, Vaidman, Phys Rev A 79, 052110 (May 1, 2009) 4.d V1`Q`I Q`.QC1 ILVIV`$VJHV

Q]RRQ1J:]]`Q:H.- 4.a QM Generalization: Each moment of time a new universe

• New ability to obtain a post-selected state of one particle that is completely correlated to the pre-selected state of a second particle:

• stack N particles on top of another along the time axis: Ascertaining the results of products of the 9 observables (Tollaksen, Jrnl of Phys A, 40 (2007) 9033)

• If we measure the operators corresponding to the first 2 observables of row 3 given the PPS

• then the measurements interfere with each other Diagonal PPS – generic feature Ascertaining the results of products of the 9 observables (Tollaksen, Jrnl of Phys A, 40 (2007) 9033)

• Given one PPS, the subset of observables circled (and the products of those circled observables) can be assigned eigenvalues in a way that satisfies the product rule

• But, the product of the other observables can only be ascertained (given this particular PPS) using information from both the pre- and post-selected vector in a diagonal sense, and will thus violate the product rule. Physical reason for restrictions on these assignments (Tollaksen, Jrnl of Phys A, 40 (2007) 9033)

• All sets of boundary conditions are needed • However, when the first observable is ascertained, then it will depend on both the pre- and the post-selection measurement (i.e. it will be diagonal-PPS) and will collapse the entire configuration onto a subset of the PPSs, thereby disturbing the terms of the multiple-time state.

• Mermin's statement: ``Alice's other two `results' have nothing to do with any properties of the particle or the results of any measurement actually performed." C= c If C = c i w/ prob 1 then w i This theorem circumvent the need to consider measurements that are temporal successors in the PPS paradox as counterfactual alternatives in the proof of contextuality Theorem: Logical-PPS-paradoxes imply ``quantum contextuality" through weak values • Quantum Contextuality: For any initial quantum state which exhibits a breakdown of non-contextuality in the associated HVT for a certain set of operators (i.e. for which ABL assigns definite values of 0/1), one can find at least one post-selected state which will show how the function composition rule (i.e. sum and product rules) is violated. • Applied this analysis to Mermin, EPR, GHZ: in each case, eccentric WVs outside EV spectrum • Post-selections suggests a physical picture for why the assignments cannot be made. In addition, the existence of strange WVs demonstrate a new way that QM copes with contextuality. Summary

• Used PPS to probe contextuality-can be tested experimentally. As Mermin states, this is not “theorizing about `hidden variables'. It is a rock solid quantum mechanical effort to answer a perfectly legitimate quantum mechanical question." • weak values go outside the spectrum with contextuality – Mermin contextuality: even though separately and – these 3 outcomes can be measured weakly without contradiction because the product of WVs is not equal to the WV of the product • With the assumption that a WM does not considerably modify the hidden variable, then this strengthens a hidden variables proof of contextuality Weak Values and ContextualityM Waegell, JT "Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values,"arXiv: 1505.00098 Weak Values and ContextualityM Waegell, JT "Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values,"arXiv: 1505.00098 Weak Values and ContextualityM Waegell, JT "Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values,"arXiv: 1505.00098