CQT PhD Thesis DefenceZaw Lin Htoo20 May 2025
Precession ProtocolsCertifying Non-classicality withOscillations and Rotations
Box with a harmonic oscillator
(period of oscillation known)
Measurement device
(that you trust)
When can we be certain that we have a quantum harmonic oscillator?
What is
quantum about a quantum harmonic oscillator?not
Discrete Energy Levels
Zero-point Motion
Heisenberg Uncertainty
But possible for classical systems to exhibit the same behaviour
Classical Time Evolution
Quantum Time Evolution
By the Ehrenfest theorem,
Time evolution of the quantum harmonic oscillator is as classical as it gets!
Go beyond linear dynamics
Higher-order Dynamics
Hybrid Systems
Or look at correlations
Bell InequalitiesCorrelations in space
Leggett–Garg InequalitiesCorrelations in time
From Tsirelson’s dusty arXivs, we ask the question:
“How often is the coordinate of a harmonic oscillator positive?”
uniformly-precessing system
Assumption: is uniformly precessing with period T
For each round,
1. Prepare the system in some state
2. Wait for a duration (chosen randomly)
3. Measure the coordinate of the system: Keep track of the sign of
After many rounds: sum of average sign of
Comment: No simultaneous or sequential measurements!
where
angular momentum
harmonic oscillator
Classical bound of
For all classical ensembles,
The quantum case
Wigner function takes the place of the joint probability density
Time evolution is precession in phase space
Quantum score can be worked out to be
Then, what’s the difference between the two?
Violating the classical bound
Position and momentum are incompatible observables : is a quasiprobability distribution
Negative values are allowed, if marginals are probabilities
What if we concentrate the negativity into ?
Largest quantum violation for
Quantum harmonic oscillators can beat theclassical bound!
(from normalisation)
Protocol with K times
Straightforward extension: probe the system different times at
For odd, analogous arguments for theclassical trajectories give ; for example, for ,
Similarly, negativities can be placed insome regions of the Wigner function toviolate the classical bound, e.g. for
Is there always a quantum-classical gap?
(lower bound on )
How large can the quantum-classical gap be?
(upper bound on )
Uniform precessions of effective oscillators
Two harmonic oscillators with coupling
• Normal mode coordinates precess uniformly with period
• What would the score tell us?
• Perhaps something about non-classical correlations?
Positive partial transpose test of entanglement
One of the most well-known ways to detect entanglement is the positive partial transpose test
• For a chosen basis, partial transpose is
• If is separable, will still be a valid state, and hence positive semidefinite
is entangled
In continuous variable systems, the action of transposition is the same as “time reversal”
transpose
Detection of non-Gaussian entanglement in harmonic oscillators
Local coordinates
Local phase-space coordinates
Let partial transpose of over be
Collective coordinates
Collective phase-space coordinates
Let partial trace of over be
For the special case , Wigner negativity ⇒ PPT negativity
If the centre of mass acts nonclassically, its constituents must be entangled
if the score of the centre of mass violates the classical bound
Anharmonicity: beyond harmonic systems
Approximately harmonic
Kerr nonlinear system
Simple pendulum
Imperfections or practical considerations means most systems are not perfectly harmonic
Problematic only if the energies are too large
Inspired by this, introduce assumption about energy bounds
How should the energy bounds be restricted?
Consider “bad cases”
“longest time a classical trajectory with energy spends in region ”
“shortest time a classical trajectory with energy E spends in region ”
Preventing these “bad cases” ensures classical bound unaffected
Choose energy bounds so that
Assumptions:1. Dynamics of the system is known to be given by 2. Energy of the system is boundedIf , the system is quantumProtocol:Perform the precession protocol with probing time such that
TL;DR: Precession protocols and harmonic oscillators
The precession protocols with harmonicoscillators involve measuring the score
Classical scores bounded by ;violated by quantum harmonic oscillatorswith Wigner negativity in correct regions
Witnesses entanglement when used oncollective coordinate of two oscillators
Robust under presence of anharmonicitywith assumptions on maximum energy
Precession of angular momentum in real space
• Classical bound determined by precessing dynamics: for any two uniformly-precessing variables,
• In particular, for angular momentum with
Apply the protocol to precessions in real space with
Consider spin particle with outcomes
with respect to spin
1) if
2) Seems to approach
3) Peak at
4) Repeats every points
Uniform precessions of effective oscillators
Two fixed spins and evolving under the Hamiltonian
• Total angular momentum precesses uniformly
• Perform protocol on total angular momentum
• Maximally-violating state is always entangled!
Violation with total angular momentum seems related to some type of entanglement
Detection of genuine multipartite entanglement in spins
Can be made more general and quantitative for an ensemble of spins
not GMEnot GMEGME
• A state that is not a mixture of biseparable states is genuinely multipartite entangled (GME)
• Possible to show that if the classical bound is violated by the total angular momentum beyond some threshold value, the spins must be GME
spins are GME
• Notably, in the paradigm of detecting GME with only collective measurements, this solved the open question of detecting GHZ states for an odd number > 3 of qubits
A black-box approach to the precession protocol
Study Tsirelson’s precession protocol in a theory-independent (“black-box”) way
is uniformly precessing at if
some (weaker) notion of precession
With this, we can perform Tsirelson’s precession protocol on
some notion ofstate preparation
some notion of measurement settings
some notion ofmeasurement outcomes(discrete, real numbers)
some notion of the passage of time
represents the observable that describes measuring at time , whereas
is the expectation value of given state
What type of objects are and ?
How to calculate given and ?
Depends on the theory!
Observables in general theories
Observables depend on the theory, but it is reasonable to expect them to obey one property:
Linearity of the expectation value
• We believe that observables in any theory should have these properties
• In particular, all general probabilistic theories (GPTs) satisfy linearity
• But less restrictive than GPT axioms (e.g. probabilistic theory on a torus)
Classical clock precesses uniformly and has discrete measurement spectrum
Same geometric argumentgives
Classical example of uniform precession with discrete variables
Calculating the general theory-independent bound
By linearity,
The score is , so the general bound is
s.t.
Hence, theory-independent bound is a linear programme—can be solved numerically!
An example with the spin-3/2 spectrum
Consider solving the linear programme for the spin-3/2 spectrum
using LinearAlgebra, JuMP, COSMOa⃗ = [-3/2,-1/2,1/2,3/2]model = Model(COSMO.Optimizer)set_attributes(model, "eps_abs" => 1e-9, "eps_rel" => 1e-9)@variable(model, 0 ≤ p⃗0[1:length(a⃗)] ≤ 1)@variable(model, 0 ≤ p⃗1[1:length(a⃗)] ≤ 1)@variable(model, 0 ≤ p⃗2[1:length(a⃗)] ≤ 1)@variable(model, y0)@constraint(model, p⃗1⋅a⃗ == cos(2π/3)*p⃗0⋅a⃗ + sin(2π/3)*y0)@constraint(model, p⃗2⋅a⃗ == cos(4π/3)*p⃗0⋅a⃗ + sin(4π/3)*y0)@constraint(model, sum(p⃗0) == 1)@constraint(model, sum(p⃗1) == 1)@constraint(model, sum(p⃗2) == 1)@objective(model, Max, (aₙ.≥0)⋅(p⃗0 + p⃗1 + p⃗2)/3)optimize!(model)objective_value(model)
Theory-independent bound
Quantum spin-3/2 particle saturates the general bound for the spectrum
Spin Angular Momentum
Analytical solution of the general bound
The linear programme can be solved analytically for a closed-form expression
where
For a given spectrum, it is always possible to construct a uniformly-precessingquantum observable such that
e.g. quantum clock achieves
Quantum mechanics saturates the general bound for discrete variable systems
For discrete variable systems,
What is the maximum score achievable in quantum theory,and can any alternate theories do better?
more manageable
An open question
TL;DR: Precession protocols and angular momenta
Precession protocols can be applied toangular momenta of rotating systems
Classical scores bounded by ; violatedby quantum systems for almost all values of spin
Genuine multipartite entanglementwitnessed on total angular momentum
The theory-independent bound for equally-spacedthree-angle protocol saturated by quantum theory
Things not covered due to time constraints
Protocol with arbitrary times
Conditional probability spaces [Phys. Rev. A 109, 062216 (2024)]
Contributions to this thesis and other relevant works
PRA 106032222(2022)190201(2025)PRL 134 2411.03132(2024)arXivChapters1, 2, 3, 4, 7Chapters3, 4, 5Chapters8, 9
Chapter 6Chapter 5Chapter 8042402(2024)PRA 109050201(2024)PRL 133PRA 108022211(2023)
062408(2024)PRA 110160201(2023)PRL 130Newton 1100017(2025)
TL;DR TL;DR: Certifying non-classicality with the precession protocols
The system is uniformly precessingAssumption
Is it really quantum rather than classical?Problem
Just look at the signs of the coordinates at different times;violation of the classical bound certifies non-classicalityPrecessionprotocols
Violation related to Wigner negativity, and non-Gaussian or genuine multipartite entanglement•
Robust even if precession not perfectly uniformunder assumptions about energy bounds•
•Equally-spaced three-angle precession protocolmaximally violated by quantum theory
Acknowledgements
Thank you!Any questions?At a quarantine centre at Trento, Italy, after catchingcovid at a quantum info workshop, wbere we had firstread Tsirelson’s preprint to pass the time
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