Optimal Estimation of Single Qubit Quantum Evolution Parameters

Transcription

1 Optimal Estimation of Single Qubit Quantum Evolution Parameters David Collins Department of Physical and Environmental Sciences, Mesa State College, Grand Junction, CO Michael Frey Department of Mathematics, Bucknell University, Lewisburg, PA

2 Parameter-dependent Unitary Evolution. Quantum system evolution is described via unitary operations. Ψ i (λ) Ψ f (λ) = (λ) Ψ i Task: Knowing nature of evolution, estimate parameters as accurately as possible by subjecting quantum system to evolution. Mixed states:. ˆρ f (λ) = (λ) ˆρ i (λ) 1

3 Statistical Nature of Measurement Outcomes and Estimation Quantum Measurements Estimation via repeated measurement. Measurements described in terms of POVMs. Possible measurement outcomes: {m 1,m 2,m 3,...} Associated POVM elements: {ˆΠ 1, ˆΠ 2, ˆΠ 3,...} ˆρ i ˆρ i ˆρ f (λ) ˆρ f (λ) Meas. m i1 Meas. m i2 Probabillities: ) Pr(m j ) = Tr(ˆΠjˆρ f (λ) ˆρ i ˆρ f (λ) Estimator for λ: Meas. m i n λ est = λ est (m i1,m i2,...,m i n). 2

4 Classical Parameter Estimation Estimator is probabilistic - distribution determined by measurement outcome distribution. Accuracy assessed via mean square error m.s.e(λ est ) = (λest λ ) 2. Unbiased estimator: λ est = λ gives classical Cramér-Rao bound: m.s.e(λ est ) = var(λ est ) 1 F(λ) where Fisher information is independent of estimator and determined from probability distribution for measurement outcomes by F(λ) = ( ) 2 lnp(mi1,...,m i n λ) λ 3

5 Quantum Parameter Estimation Fisher information depends on system state, ˆρ f (λ), and choice of measurement type via ) ) P(m i1,...,m i n λ) = Tr(ˆΠi1ˆρ f...tr (ˆΠi nˆρ f. Quantum Fisher information gives bound for any conceivable measurement F(λ) H(λ) where quantum Fisher information depends on system state via H(λ) = Tr (ˆρ ) fˆl2 with score operator ˆL satisfying ˆρ f λ = 1 2 (ˆρ fˆl + ˆLˆρf ) Optimal estimation: choose input state so that ˆρ f maximizes quantum Fisher information. 4

6 General Unitary Parameter Estimation Single parameter unitary Unitary on single system = e iĝλ/2 with Ĝ Hermitian. Convexity implies pure input state optimal Fujiwara, et.al. PRA (2001). Pure output state ˆρ f = ψ f (λ) ψ f (λ) Fisher Information For pure output state, Fisher information [ ( ψ ψ H(λ) = 4 λ λ + ψ ψ ) 2 ] λ Fisher information via generator: H(λ) = [ ( ) 2 ] ψ i Ĝ 2 ψ i ψ i Ĝ ψ i. with ψ f (λ) = (λ) ψ i. Fisher information via extreme generator eigenvalues (Giovanetti, et.al. PRL (2006)): H(λ) = g max g min 5

7 Single Qubit Unitary Parameter Estimation Unentangled Input Identical state for each qubit: Entangled Input Entangled state, no ancilla: ψ i ψ f ψ i ψ f ψ i ψ f Optimal input state gives Optimal input state gives H(λ) = n var(λ est ) 1 n H(λ) = n 2 var(λ est ) 1 n 2 6

8 Non-Unitary Parameter Estimation Evolution of quantum system in presence of other external quantum systems described in terms of quantum operations. Quantum Channel and Operation Density operator evolution Example: Bit-Flip Channel Single qubit evolution: ˆρ f (λ) = j Ê j (λ)ˆρ i Ê j (λ) Do nothing with probability λ, flip qubit with probability 1 λ. where Kraus operators satisfy j Ê jêj Î. Estimate λ with minimum number of channel uses. Kraus Operators: Ê 0 = λ Î Ê 1 = 1 λ ˆσ x 7

9 Bit-Flip Parameter Estimation λ λ 1 Trace 0 ˆV Trace ˆρ i ˆσ x ˆρ f ˆρ i ˆσ x ˆρ f Quantum Circuit Optimal Parameter Estimation Parameter via: λ ˆV(λ) = 1 λ 1 λ λ Any measurements (both qubits) gives maximum Fisher information which only depends on ˆV. Optimal channel Fisher information satsifies: Ancilla qubit constrained to input of 0. H channel (λ) H both (λ) = 1 λ(1 λ) 8

10 Optimal Bit-Flip Parameter Estimation Multiple Channel Uses Ancilla bits are constrained to 0. Ancilla bits are unentangled. Channel bits are possibly entangled. Any measurement on all qubits channel qubits irrelevant and optimal Fisher information is H both (λ) = n λ(1 λ). Channel Fisher Information For single channel use with channel pure input state ˆρ i = 1 2 Fisher information is H channel (λ) = For n unentangled uses ] [Î + ˆσy, 1 λ(1 λ) Optimal channel Fisher information bounds: H channel (λ) = n λ(1 λ) H channel (λ) H both (λ) = n λ(1 λ) Entangled channel states give no advantage. 9

11 Conclusions Optimal Fisher information for n uses of single qubit unitary with unentangled inputs: H(λ) = n Optimal Fisher information for n uses of single qubit unitary with entangled inputs: H(λ) = n 2 Optimal Fisher information for n uses of single qubit bit flip with unentangled inputs: H(λ) = n λ(1 λ) Entangled input states offer quadratic advantage for unitary estimation. Entangled channel states give no advantage for bit-flip estimation. 10