quantum metrology 2

QUANTUM METROLOGY 2.0

Janek Kolodynski ICFO – The Institute of Photonic Sciences, Barcelona, Spain

Centre for Theoretical Physics, Warsaw, December 2016

THEORETICAL QUANTUM METROLOGY 2.0

Janek Kolodynski ICFO – The Institute of Photonic Sciences, Barcelona, Spain

Centre for Theoretical Physics, Warsaw, December 2016

• Andrea Smirne, Janek Kolodynski, Susana F. Huelga, Rafal Demkowicz-Dobrzanski “The ultimate precision limits for noisy frequency estimation” , Phys. Rev. Lett. 116, 120801 (2016)

• Pavel Sekatski, Michalis Skotiniotis, Janek Kolodynski, Wolfgang Duer “Quantum metrology with full and fast quantum control” , arXiv:1603.08944 [quant-ph] (2016)

MY SUBJECTIVE CHRONOLOGY: THERE IS MUCH (MUCH!) MORE...

1) THE CONCEPT OF ENHANCING MEASUREMENT PRECISION BY EMPLOYING QUANTUM PROBES: • Phase difference measurement in interferometry by employing squeezed light:

C. Caves “Quantum-mechanical noise in interferometer”, PRD 23, 1693 (1981).

• Transition frequency in atomic Ramsey spectroscopy by employing spin-squeezed states of atoms: J.J. Bollinger et al “Optimal frequency measurements with maximally correlated states”, PRA 54, R4649 (1996).

• Quantum Information Theory track—but one should not forget the books by Helstrom and Holevo: S. Braunstein and C. Caves “Statistical Distance and the Geometry of Quantum States”, PRL 72, 3439 (1994). V. Giovannetti et al “Quantum Metrology”, PRL 96, 010401 (2006).

2) IMPACT OF NOISE ON QUANTUM METROLOGY PROTOCOLS • Mathematical physics: differential geometry applied to parametrised CPTP maps:

A. Fujiwara and H. Imai “A fibre bundle over manifolds of quantum channels…”, JPhysA 41, 255304 (2008).

• Quantum Information Theory techniques applied to noisy quantum metrology protocols: B. M. Escher et al “General framework for noisy quantum-enhanced metrology”, NatPhys 7, 406 (2011). R. Demkowicz-Dobrzanski et al “The elusive Heisenberg Limit in quantum metrology”, NatCommun 3, 1063 (2012).

3) FIGHTING THE NOISE IN QUANTUM METROLOGY (QM 2.0) • Allowing for positive memory effects in the noise, i.e., the non-Markovian quantum dynamics:

A. W. Chin et al “Quantum Metrology in Non-Markovian Environments”, PRL 109, 233601 (2012).

• Optimising the geometry of noise with respect to the parameter encoding: R. Chaves et al “Noisy Metrology beyond the Standard Quantum Limit”, PRL 111, 120401 (2013).

• Employing error correction and quantum control techniques: E. Kessler et al “Quantum Error Correction for Metrology”, PRL 112, 150802 (2014). W. Duer et al “Improved Quantum Metrology Using Quantum Error Correction”, PRL 112, 080801 (2014).


MY SUBJECTIVE CHRONOLOGY: THERE IS MUCH (MUCH!) MORE...

1) THE CONCEPT OF ENHANCING MEASUREMENT PRECISION BY EMPLOYING QUANTUM PROBES: • Phase difference measurement in interferometry by employing squeezed light:

C. Caves “Quantum-mechanical noise in interferometer”, PRD 23, 1693 (1981).

• Transition frequency in atomic Ramsey spectroscopy by employing spin-squeezed states of atoms: J.J. Bollinger et al “Optimal frequency measurements with maximally correlated states”, PRA 54, R4649 (1996).

• Quantum Information Theory track—but one should not forget the books by Helstrom and Holevo: S. Braunstein and C. Caves “Statistical Distance and the Geometry of Quantum States”, PRL 72, 3439 (1994). V. Giovannetti et al “Quantum Metrology”, PRL 96, 010401 (2006).

2) IMPACT OF NOISE ON QUANTUM METROLOGY PROTOCOLS • Mathematical physics: differential geometry applied to parametrised CPTP maps:

A. Fujiwara and H. Imai “A fibre bundle over manifolds of quantum channels…”, JPhysA 41, 255304 (2008).

• Quantum Information Theory techniques applied to noisy quantum metrology protocols: B. M. Escher et al “General framework for noisy quantum-enhanced metrology”, NatPhys 7, 406 (2011). R. Demkowicz-Dobrzanski et al “The elusive Heisenberg Limit in quantum metrology”, NatCommun 3, 1063 (2012).

3) FIGHTING THE NOISE IN QUANTUM METROLOGY (QM 2.0) • Allowing for positive memory effects in the noise, i.e., the non-Markovian quantum dynamics:

A. W. Chin et al “Quantum Metrology in Non-Markovian Environments”, PRL 109, 233601 (2012).

• Optimising the geometry of noise with respect to the parameter encoding: R. Chaves et al “Noisy Metrology beyond the Standard Quantum Limit”, PRL 111, 120401 (2013).

• Employing error correction and quantum control techniques: E. Kessler et al “Quantum Error Correction for Metrology”, PRL 112, 150802 (2014). W. Duer et al “Improved Quantum Metrology Using Quantum Error Correction”, PRL 112, 080801 (2014).


FUTURE? MY BET... CHANGING THE RULES OF THE GAME ADJUSTING FOR THE ACTUAL EXPERIMENTS:

Atomic clocks (Allan variance), Gravitational-wave detection (waveform estimation, measurement back-action), Continuous measurements with feedback (filtering, e.g., Kalman or Wiener filters)

QUANTUM METROLOGY EXPERIMENTS OPTICAL (MACH-ZEHNDER) INTERFEROMETRY

ATOMIC (RAMSEY) SPECTROSCOPY

ESTIMATION STAGE (Data Inference)

Techniques of Statistics

CLASSICAL VS QUANTUM METROLOGY CLASSICAL SETTING (uncorrelated particles)

Central Limit Theorem

Standard Quantum Limit

(SQL)

QUANTUM SETTING (entangled particles)

Heisenberg Limit (HL)

A.S. – GHZ state:

O.I. – N00N state:

SENSING

CLASSICAL VS QUANTUM METROLOGY CLASSICAL SETTING (uncorrelated particles)

Central Limit Theorem

Standard Quantum Limit

(SQL)

QUANTUM SETTING (entangled particles)

Heisenberg Limit (HL)

A.S. – GHZ state:

O.I. – N00N state:

OK, BUT: 1) In the classical case, maybe one could do collective measurements on all particles...

2) In the quantum case, the probability distribution (and actually the quantum state itself!) is periodic. I cannot distinguish between φ and φ+2π/N !!! Hence, I need to know the phase with precision at the Heisenberg Limit already in advance… ?

SENSING

The outcomes of an experiment repeated ν times are described by ν random variables, each of which is distributed with a PDF that depends on the parameter to be estimated.

Frequentist approach (SENSING):

• The estimated parameter φ is a deterministic variable which value I may assume to be known.

• Moreover, one is interested in the limit of infinitely large statistics of ν ∞.

• One seeks the optimal strategy allowing to sense with highest resolution the fluctuations of a parameter around a given value.

estimator (e.g. mean)

estimator, in principle, depends on the true value…

FREQUENTIST VS BAYESIAN PARADIGMS

Minimise the Mean Squared Error for a given true parameter value:

In the asymptotic ν limit, the ultimate resolution of any unbiased estimator is dictated by the Cramer-Rao Bound:

Fisher Information clearly additive…


Bayesian approach (ESTIMATION):

• The estimated parameter φ is also a random variable distributed according to a prior distribution.

• Moreover, one interested in the single-shot scenario of ν =1.

• One seeks the optimal strategy allowing to estimate on average the parameter in a single repetition, where averaging is performed over the prior representing the knowledge about the parameter.



Minimise the Average Mean Squared Error for a given prior distribution and a single shot (ν =1):

optimal estimator depends on the choice of prior

The ultimate precision of estimation is always lower-bounded by the Bayesian Cramer-Rao Bound:

if prior does not provide any

information about φ

SENSING is more optimistic than

ESTIMATION if the FI is parameter-value

independent


Bayesian approach (ESTIMATION):

• The estimated parameter φ is also a random variable distributed according to a prior distribution.

• Moreover, one interested in the single-shot scenario of ν =1.

• One seeks the optimal strategy allowing to estimate on average the parameter in a single repetition, where averaging is performed over the prior representing the knowledge about the parameter.



Minimise the Average Mean Squared Error for a given prior distribution and a single shot (ν =1):

optimal estimator depends on the choice of prior

The ultimate precision of estimation is always lower-bounded by the Bayesian Cramer-Rao Bound:

if prior is flat

SENSING is more optimistic than

ESTIMATION

CONCLUSION:

In both paradigms the Fisher Information may be used to bound the attainable precision.

Such conclusion also holds in case of quantum protocols.

if the FI is parameter-value

independent

QUANTUM ATOMIC SPECTROSCOPY with uncorrelated atoms and no noise

the true Standard Quantum Limit

without cheating

N two-level atoms (qubits) in a separable state

unitary rotation

output state (separable)

uncorrelated measurement – POVM:

independent processes:

Cramer-Rao Bound

Quantum Fisher Information maximised over all (also collective!) measurements—POVMs:

Classical Fisher Information

Quantum Cramer-Rao Bound

• QFI of a pure state :

• QFI of a mixed state :

◦ Evaluation requires eigen-decomposition of the density matrix, which size grows exponentially, e.g. d=2N for N qubits. ◦ Projective measurement onto the SLD eigenbasis serves as the measurement saturating the Quantum CRB.

• Geometric interpretation – QFI is a local quantity

The necessity of the asymptotic limit of repetitions ν → ∞ is a consequence of locality.

• Purification-based definition of the QFI

• In case of output of N parallel channels acting on any state of N particles, i.e., for with being a valid Kraus representation channel we have

QUANTUM FISHER INFORMATION

– Symmetric Logarithmic Derivative

Two PDFs :

Two q. states :

[Fujiwara & Imai, J. Phys. A 41(25), 255304 (2008)] –

Due to locality of the QFI need to consider only purifications that differ in:

Last inequality, if applicable, imposes SQL-scaling of precision. Both inequalities can be evaluated

with help of SDP.

distorted unitary rotation

constant factor improvement over SQL (shot noise)

optimal pure state

mixed output state

In practice, need to optimize for particular model and N

with dephasing noise added:

[achievable with spin-squeezed states]

complexity of computation grows exponentially with N

QUANTUM ATOMIC SPECTROSCOPY with dephasing noise

phase-covariant type of noise

Phase as parameter:

Frequency as parameter and :

Andrea Smirne, Janek Kolodynski, Susana F. Huelga, Rafal Demkowicz-Dobrzanski “The ultimate precision limits for noisy frequency estimation” ,

Phys. Rev. Lett. 116, 120801 (2016)

Pavel Sekatski, Michalis Skotiniotis, Janek Kolodynski, Wolfgang Duer “Quantum metrology with full and fast quantum control” ,

arXiv:1603.08944 [quant-ph] (2016)

SUMMARY OF RECENT RESULTS

• Results generalized to schemes involving entanglement with extra ancillary particles:

• Most general form of qubit phase-covariant noise considered (note arbitrary t-dependence)

• The so-called Zeno regime manifested by: yields

• Note that in such a regime the corresponding CPTP map describing the evolution is divisible, so that no non-Markovianity (memory) is manifested at the level of the reduced dynamics.

ANCILLA-ASSISTED QUBIT ATOMIC SPECTROSCOPY with correlated atoms and phase-covariant noise

Super-classical scaling achievable with GHZ states

• Results generalized to schemes involving also interrupting global control operations:

• Most general form of qubit semigroup noise considered (note t-independence)

• RESULTS:

o FFQC allows to restore Heisenberg Scaling, 1/N2 , for any rank-one Pauli noise that is not parallel to the frequency-encoding Hamiltonian (along z in the above parametrisation).

o A simple two (probe+ancilla) qubit protocol with fast error correction achieves the goal.

o All other noise-types unavoidably yield the SQL-like scaling, const/N, of precision.

o All the above statements hold both in the frequentist and Bayesian paradigms.

QUBIT ATOMIC SPECTROSCOPY with any semigroup noise and full and fast quantum control

with arbitrary

CONCLUSIONS

Thank You very much for Your attention

and

Let me wish You

Merry Christmas and Happy New Year

quantum metrology 2

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