Margaret Reid
Centre for Atom Optics and Ultrafast Spectroscopy Swinburne University of Technology Melbourne, Australia
Thank you! Outline
• Non-locality, reality and quantum mechanics:
Einstein-Podolsky-Rosen (EPR) paradox Schrodinger cat Bell’s theorem: Bell inequalities Entanglement and Steering Experiments
• Macro-scopic reality
EPR and Schrodinger cat Genuine multipartite nonlocality: GHZ states CV EPR Entangled atoms Macroscopic realism: Leggett- Garg inequalities EPR paradox 1935
• Einstein, Podolsky and Rosen argument
• Einstein was unhappy about quantum mechanics
• Believed it was correct but incomplete: Quantum mechanics and reality
1 Ψ = x + x ʹ 2 ( )
• Principle of superposition
• Not one or the other until measured: Dirac
€ • Cannot view things as existing until they are measured?
• But why would this be a problem? Quantum mechanics and reality
1 Ψ = x + x ʹ 2 ( )
• You might argue….
• Fundamental indeterminacy in nature? Heisenberg microscope
€ • Interaction of a microscopic system with any measurement apparatus?
• But this is not a resolution 2 problems put forward Problem 1: Schrodinger’s cat 1935
1 Ψ = dead + alive 2 ( ) • Quantum mechanics predicts macroscopic superpositions
• How does “not one or the other until measured” work for macroscopic€ superpositions? Dead and alive? Diosi/ Penrose theories propose collapse mechanism for massive objects
Diosi Penrose decoherence time for massive object m Problem 2: Einstein-Podolsky-Rosen (EPR) paradox 1935 X Nonlocal measurements B A Entanglement and correlation
Spatial separation
1 Ψ = dead ↓ + alive ↑ 2 ( )
• Entangled superposition state • Alice’s spin€ measurement is correlated with Bob’s cat being dead or alive • EPR assume: (“no action-at-a-distance”) Local realism • So, EPR argue, Bob’s cat was dead or alive (all along) so it seems we need predetermined “hidden variables” to complete QM? The quantum mixture
Bell
• But, …..this case arises all the time • We understand correlation well- caused by past events The quantum mixture
1 Ψ = dead ↓ + alive ↑ 2 ( )
1 ρ = dead ↓ ↓ dead mix 2 ( € + alive ↑ ↑ alive ) • We would say, the cat is in the probabilistic mixture ….dead or alive • Or the€ superposition is equivalent to such a mixture- so realism holds… BUT - for some quantum states, EPR showed differently….. EPR Entangled states Entangled states are non-separable: 2 classic “EPR entanglement” states 1 Ψ = ( ↑↓ − ↓↑ ) 2 Bell state, Bohm’s EPR paradox
δ(xA − xB )δ(pA + pB ) € • Alice can predict both Bob’s x (and p) with no fuzziness - despite uncertainty relation! € • Both conditional variances are zero:
2 2 Δ (xB | xA ) →0 Δ (pB | pA ) →0
€ € EPR paradox: 2 elements of reality
δ(xA − xB )δ(pA + pB )
€ Hidden variables Simple experimental criterion for EPR paradox
EPR criterion Bohm’s qubit version EPR paradox
1 Ψ = ( ↑↓ − ↓↑ ) 2
€ EPR’s hopes of a local hidden variable (LHV) theory Bell’s theorem- no Local Hidden Variable theories consistent with QM
Consider experiment to measure spin correlation: spin ½ system
A B E(θ,φ) = Jθ Jφ
IF we assign local hidden variables to each spin: € CHSH-Bell inequality S = E(θ,φ) − E(θ ʹ, φ) + E(θ,φ ʹ) + E(θ ʹ, φ ʹ) ≤ 2
Quantum Mechanics predicts a violation of Bell’s inequality! 1 Ψ = ( ↑↓ − ↓↑ ) € 2 ⇒ S = 2 2 (Tsirelson) maximum QM value
€ € Experiments confirm Bell’s nonlocality
Clauser, Aspect, Zeilinger
A B E(θ,φ) = Jθ Jφ = cos2(φ −θ) ≡ cos2(b − a)
€ Schrodinger’s cat and macroscopic reality Harmonic Oscillator- coherent states
• Define quadratures- position momentum X =a+ + a P = (a − a+ )/i • Define the coherent (Gaussian) state
2 € ⎡ α ⎤ ∞ α n α = exp⎢ − ⎥ ∑ n ⎣ 2 ⎦ n =0 n! • Measure quadrature X position - P(x) € The “cat” is a superposition of 2 coherent states
1 Ψ = −α ↓ + α ↑ 2 ( )
€
Δ inf x =1
€ Distinguishing Schrodinger’s cat from any quantum mixture
P(p)
€
Δ inf p <1
€ EPR paradox with a S cat EPR paradox with a S cat Decoherence- interaction with environment
α
η € out 1 ρmix = ( −α ↓ ↓ −α + α ↑ ↑ α ) € Yurke, Stoler,PRL 2 • The S cat decoheres to a quantum mixture € € Δp =1 • Interference originates from off-diagonal terms in
density matrix 1 ρ = −α ↓ ↓ −α + α ↑ ↑ α + −α ↓ ↑ α + α ↑ ↓ −α €sup 2 ( )
• Greater α€ implies greater sensitivity to decoherence Photon Cats
Haroche Grangier experiments Measuring cat decoherence 3 famous types of entanglement
• Not all entanglement is the same
• Classification of entanglement
Entanglement ⇒ failure of quantum separability ρ = P ρ R ρ R ∑R R A B
A B A B P(xθ ,xφ ) = ∫ ρ(λ) PQ (x θ,λ)PQ (x φ,λ) dλ ⇒ R Bell’s nonlocality: failure of local hidden variables (LHV) € € ie hidden variable separability
A B A B P(xθ ,xφ ) = ∫ ρ(λ) P(x θ,λ)P(x φ,λ) dλ € R
€ Where does EPR paradox fit in?
A B A B P(xθ ,xφ ) = ∫ ρ(λ) P (x θ,λ)PQ (x φ,λ) dλ R EPR steering iff this model fails
€ Wiseman, Jones, Doherty, PRL 2007;
Steering EPR argument Concept introduced in Schrodinger’s Alice can infer Bob’s outcomes: x and p famous reply to EPR paradox, 1935 Local realism implies “elements of reality” for Generalised EPR paradox for different Bob measurements If these “elements of reality” inconsistent with a Alice appears to “steer” Bob’s state quantum state then from distant site Quantum Mechanics is incomplete
Cavalcanti Jones Wiseman and R,PRA 2009; R et al, RMP, 2009 Hierarchy of “quantum nonlocality”
Corresponds to a failure of different separability LHS models:
Entanglement nonlocality: Failure of Local Quantum State (LQS) model
Alice Bob
Werner PRA; Wiseman, Jones, Doherty, PRL 2007; distinct classes of nonlocality Bell’s nonlocality: Failure of Local EPR steering nonlocality: Failure Hidden Variable (LHV) State model of Hybrid LHV-LQS model
Cavalcanti, Jones, Wiseman, R, PRA 2009 Qubit spin nonlocality inequalities Experimental loophole-free demonstration of EPR paradox steering nonlocality
Zeilinger experiment Wittman et al, 2012 Bigger systems predicted to show Bell nonlocality
Higher dimension “qudits”: d outcomes 1 d −1 Ψ = ∑ jj d j =0
Multi-site qubits genuine nonlocality Chen et al, PRA, 2006 €
1 N N Ψ = ↑ ⊗ − ↓ ⊗ GHZ 2 ( ) Mermin, PRL; HDR, PRA 2011
€ Genuine multipartite entanglement
A B 1 N N Ψ = ↑ ⊗ − ↓ ⊗ GHZ ( ) C 2
Verifying genuine tripartite entanglement: need to exclude€ all 2-body entanglement
Leads to criteria:
The Greenberger-Horne-Zeilinger cat state is N-partite entangled GHZ cat states using photons
Pairs of polarization-entangled photons (one photon H polarized and the other V) are generated by a short pulse of light. Observation of the GHZ correlations requires two pairs. The photon registered at T is always H and its partner in b is V. The photon reflected at the polarizing beam-splitter (PBS) in arm a is always V, being turned into equal superposition of V and H by the /2 plate, and its partner in arm b must be H. If all four detectors register at the same time, the two
photons in D1 and D2 must either both have been VV and reflected by the last PBS or HH and transmitted. The photon at D3 was therefore H or Zeilinger experiments V, respectively. N=4 (now ~ 6- 8) GHZ cat states using ion traps (N=14)
Wineland, Blatt experiments BUT how many qubits share a Bell nonlocality?
1 N N Ψ = ↑ ⊗ − ↓ ⊗ GHZ 2 ( )
Verifying genuine tripartite Bell nonlocality € need to exclude all 2-body Bell nonlocality
Leads to criteria: Svetlichny’s Bell inequality The Greenberger-Horne-Zeilinger state is N-partite Bell nonlocal BUT not yet shown for N>3? Continuous Variable (CV) Nonlocality
δ(xA − xB )δ(pA + pB )
• Two coupled harmonic oscillators (fields a and b) • Define X and P for each ΔX ΔP ≥1 • Squeezed quadratures€ when ΔXθ <1 • EPR entanglement€ when 2 2 D = Δ(X A − X B ) + Δ(PA + PB ) < 4
• EPR steering paradox when
€ ε = Δ(X B | X A )Δ(PB | PA ) <1 €
€ How is CV EPR entanglement generated?
Two-mode squeezed state H = κE(a+b+ + ab)
Gross et al, Nature, 2010
Optical parametric down conversion (OPA) € 2 2 −κ 't Δ(X A − X B ) = Δ(PA + PB ) = e 2 2 κ 't Δ(X A + X B ) = Δ(PA − PB ) = e SQUEEZING!
€ EPR entanglement using squeezing
2 optical Parametric amplifiers (oscillators)
EPR fields
Kimble, Bachor, Lam, Leuchs experiments Entanglement shows as noise reduction
Optical Parametric Oscillator (OPO or OPA)
Vacuum noise level (coherent state)
Squeezed noise level
2 2 D = Δ(X A − X B ) + Δ(PA + PB ) < 4
ε = Δ(X B | X A )Δ(PB | PA ) <1
€ € CV EPR steering paradox – how much spooky action at a distance?
EPR criterion
Modified PREMISE: Assume Alice’s measurement can affect Bob’s state, but only up to δ, no more
measure
Premise violated when CV EPR steering nonlocality experiments
Nonlocal shift δ is normalised to vacuum level (graduation assumes Gaussian statistics) EPR spooky action-at-a-distance made larger using spin measurements
11 JX ~ N ~ 10 photons Different sort of homodyne measurement- Uses polariser beam splitters- amplification occurs before choice of spin angle
€ Bowen et al, PRL
a+a
~ N + € b b B + + JZ = (a a − b b)/2 € € € CV Bell nonlocality- Falsifying Local Hidden Variable theories for CV measurements
Superposition of correlated coherent states X B alive 1 Ψ = ( −α A −α B + α A α B ) 2 €
€ dead
X A
Gilchrist, Deuar, R,PRL and PRA
Quadrature outcomes XA and XB are correlated Binned as +1 or -1 (alive/ dead) € Reveal violation of CHSH Bell inequality when |α|~1 CV Bell nonlocality
Superposition of correlated coherent states X B
1 Ψ = ( −α A −α B + α A α B ) 2 €
€
X A
Gilchrist, Deuar, R,PRL Violations reduce as α increases € What about macroscopic reality? Leggett Garg inequalities with BEC ground state
1 Ψ = dead + alive 2 ( )
€
1 Ψ = N 0 + 0 N 2 ( ) Leggett Garg premises NOON states 1. Macroscopic realism: System is in one state or the other €(cat is dead OR alive)
2. Macroscopic noninvasive measurability Possible at least in principle to determine which of the states cat is in, with an arbitrarily small influence on subsequent dynamics Macroscopic reality: Leggett Garg
dead alive
Measure at successive times ti
I(ti ) = +1 or −1 Kij = I(ti )I(t j )
Leggett Garg premises
Leggett Garg inequality- three€ successive times € LG = K12 + K23 − K13 ≤1
Leggett Garg, PRL
€ BEC NOON states- we solve
g H = κ(a+b + b+a) + a+a+aa + b+b+bb 2 [ ] Ng/κ >>1
€
€
Leggett Garg inequality violated- LG=1.5 Two-state tunneling regime Long tunnelling times Fragile to decoherence LG = K12 + K23 − K13 ≤1
€ Generalised LG inequalities
g H = κ(a+b + b+a) + a+a+aa + b+b+bb 2 [ ]
€
Generalised Leggett Garg inequality violated: LG=1.42 Violation for more realistic parameters T~0.3 s l l u LG = K 12 + K 23 − K 13 ≤1
€ SUMMARY
. Predictions of the “unreality” of Quantum Mechanics extremely well verified . But these are still very tiny quantum systems- Penrose Diosi times . EPR steering nonlocality for S cats Failure of mesoscopic quantum reality . Testing Mesoscopic Local Reality directly for S cats remains a challenge