Asymptotic theory of quantum channel estimation
Sisi Zhou with Liang Jiang
arXiv: 2003.10559
YaleUniversity
2021/02/03
QIP 2021
Outline
โข Introduction to quantum metrology
โข Asymptotic quantum channel estimation
โข Examples
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Classical estimation theory
Estimation precision:
ฮ๐ = ๐ผ %๐ ๐ฅ โ๐ !"!
For unbiased estimators ๐ผ %๐ ๐ฅ = ๐, we have the Cramรฉr-Rao bound:
ฮ๐ โฅ1
๐#$%& โ ๐น(๐(๐ฅ;๐))
๐#$%&: number of experiments
๐น(๐(๐ฅ;๐)): Fisher information
The bound is saturable asymptotically using the maximum likelihood estimator.
๐
Probing
๐(๐ฅ; ๐)
Estimator: %๐ ๐ฅ
๐(๐)๐(๐ + ๐๐)
๐ฅ
๐(๐ฅ; ๐)
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Quantum estimation theory
The quantum Cramรฉr-Rao bound:
ฮ๐ โฅ1
๐#$%& โ ๐น(๐')
๐#$%&: number of experiments
๐น(๐'): quantum Fisher information
๐น ๐' = max{)!}
๐น(๐(๐ฅ:๐))
The quantum Cramรฉr-Rao bound is also saturable asymptotically.
๐ ๐!
Probing
๐(๐ฅ; ๐)
POVM {๐)}
Estimator: %๐ ๐ฅ
Helstrom 1976, Holevo 1982Braunstein & Caves 1994
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Quantum Fisher information
Quantum Fisher information (QFI): ๐น ๐' = max{)!}
๐น(๐(๐ฅ:๐))
๐น ๐' = Tr ๐'๐ฟ'! , +","-,"+"! = ๐'๐'
โข Faithfulness: ๐น ๐' โฅ 0 and ๐น ๐' = 0 if and only if ๐'๐' = 0.
โข Monotonicity: ๐น ๐ฉ(๐') โค ๐น ๐' , ๐ฉ is an arbitrary CPTP map.
โข Additivity: ๐น ๐'โ๐' = ๐น ๐' +๐น(๐'). โข Convexity: ๐น ๐"๐' +๐!๐' โค ๐"๐น ๐' +๐!๐น ๐' .
โข Connection to Fidelity: ".๐น ๐' ๐๐! = ๐/0! ๐', ๐'-2' = 2โ 2๐น๐๐(๐', ๐'-2').
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Quantum Fisher information
Example 1 (pure state):
๐' = |๐'โฉโจ๐'|, ๐' = ๐34'56 ๐7 ,
๐น ๐' = 4๐ก! ฮ!๐ป = ฮ ๐ก! ,
where ฮ!๐ป = ๐7 ๐ป! ๐7 โ ๐7 ๐ป ๐7 ! .
โข For a single qubit state, when ๐ป = ๐/2, the optimal initial state is
๐7 =0 + 1
2.
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Quantum Fisher information
Example 2 (single-qubit dephasing):๐๐๐๐ก = โ๐
๐๐2 , ๐ +
๐พ2 ๐๐๐ โ ๐
Input state: ๐7 = 0 + 1 / 2.
Noiseless case (๐พ = 0): ๐น ๐' ๐ก = ๐ก! = ฮ(๐ก!).
Noisy case (๐พ > 0): ๐น ๐' ๐ก = ๐ก!๐3!86.
If we measure and renew the qubit every constant time, we get only QFI= ฮ(๐ก).
Noiseless๐! ๐ ๐ก |๐!โฉ
Noisy๐! ๐ ๐ก |๐!โฉ
Quantum Fisher information
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Example 3 (๐-qubit dephasing):
Input state (GHZ state):
๐7 =0 โ: + 1 โ:
2Noiseless case (ฮณ = 0): ๐น ๐' ๐ก = ๐!๐ก! = ฮ(๐!).
Noisy case (ฮณ > 0): ๐น ๐' ๐ก = ๐!๐ก!๐3!:86.
Average QFI over time: max6;7
๐น ๐' ๐ก /๐ก = ๐/2๐๐พ = ฮ(๐).
Noiseless๐! ๐ ๐ก |๐!โฉ
Noisy๐! ๐ ๐ก |๐!โฉ
Quantum Fisher information
In quantum metrology, the resource we care about is the number of channels used๐ or the probing time ๐ก. There are two types of estimation precision limits:
โข The Heisenberg limit (HL): QFI = ฮ(๐!) or ฮ(๐ก!)
โ the ultimate estimation precision limit allowed by quantum mechanics.
โข The standard quantum limit (SQL): QFI = ฮ(๐) or ฮ(๐ก)
โ achievable using โclassicalโ strategies (no need to maintain the coherence in & between probes for a long time).
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Quantum channel estimation
โฐ* ๐ = โ+,-. ๐พ*,+๐๐พ*,+0
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Sequential strategy
โ
10
Parallel strategy
Quantum channel estimation
The channel QFI:
๐น โฐ' = max,๐น((โฐ'โ๐) ๐ )
The asymptotic channel QFI:
๐น โฐ'โ: โ ๐!: the Heisenberg limit (HL)
๐น โฐ'โ: โ ๐: the standard quantum limit (SQL)
The asymptotic channel QFI either follows the HL or the SQL.
Examples: unitary channels, depolarizing noise.
The same is true for ๐น(1#=) โฐ', ๐ ----the QFI optimized over all sequential strategies.
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Fujiwara & Imai 2008, Demkowicz-Dobrzanski et al. 2012Demkowicz-Dobrzanski & Maccone 2014, SZ & Jiang 2020
Necessary and sufficient condition
Theorem 1: The HL is achievable using parallel strategies (or sequential strategies) if and only if the Hamiltonian-not-in-Kraus-Span (HNKS) condition is satisfied:
๐ป(โฐ') โ ๐ฎ(โฐ'),
where
Hamiltonian: ๐ป โฐ' โ ๐โ4๐พ',4@ ๐'๐พ',4 = ๐๐@๏ฟฝฬ๏ฟฝ ,
Kraus span: ๐ฎ(โฐ') = span ๐พ',4@ ๐พ',A, โ๐, ๐ .
Remark: For unitary channel ๐' = ๐345', or โฐ' =๐ฉ โ๐ฐ', ๐ป โฐ' = ๐ป.
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Fujiwara & Imai 2008, Demkowicz-Dobrzanski et al. 2012Demkowicz-Dobrzanski & Maccone 2014, SZ & Jiang 2020
๐ =๐พ',"โฎ
๐พ',B
The asymptotic QFI
๐นCD โฐ' = lim:โF
๐น โฐ'โ:
๐! , ๐นGHD โฐ' = lim:โF
๐น โฐ'โ:
๐Theorem 2: When ๐ป โ ๐ฎ,
๐นCD โฐ' = 4minI
๐ฝ ! , ๐ฝ = ๐ป โ๐!โ๐.
Theorem 3: When ๐ป โ ๐ฎ,
๐นGHD โฐ' = 4 minI:KL7
๐ผ , ๐ผ = ๏ฟฝฬ๏ฟฝ โ ๐โ๐ 0(๏ฟฝฬ๏ฟฝ โ ๐โ๐).
In particular, ๐นGHD โฐ' = ๐นGHD1#= โฐ' .
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๐ =๐พ',"โฎ
๐พ',B, Hermitian โ โ โBรB
Fujiwara & Imai 2008, Demkowicz-Dobrzanski et al. 2012Demkowicz-Dobrzanski & Maccone 2014, SZ & Jiang 2020
min!
๐ฝ is a distance between ๐ป and ๐ฎ
Proof of Theorem 2 and 3
๐นCD โฐ' = 4minI
๐ฝ ! , ๐นGHD โฐ' = 4 minI:KL7
๐ผ .
๐ฝ = ๐ป โ๐!โ๐, ๐ผ = ๏ฟฝฬ๏ฟฝ โ ๐โ๐ 0(๏ฟฝฬ๏ฟฝ โ ๐โ๐).
โข Upper bounds:
๐น โฐ'โ: โค 4(๐ ๐ผ +๐(๐ โ 1)โ๐ฝโ!)
๐น(1#=) โฐ', ๐ โค 4(๐ ๐ผ +๐(๐ โ 1) ๐ฝ ( ๐ฝ + ๐ผ + 1))
โข Attainability:
Quantum error correction (QEC) protocols
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Fujiwara & Imai 2008, Demkowicz-Dobrzanski et al. 2012 Demkowicz-Dobrzanski & Maccone 2014
SZ & Jiang 2020
Hamiltonian estimation in master equationsDemkowicz-Dobrzanski et al. 2017, SZ et al. 2018, SZ & Jiang 2019
SZ, QIP 2018
Attainability: the QEC protocol
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The Heisenberg limit is recovered.
The optimal SQL coefficient ๐น"#$ โฐ%is achievable.
Attainability: the QEC protocol
โข Step 1: Asymptotic QFIs for dephasing channels
โข Step 2: Reduction to dephasing channels
โข Step 3: Code optimization
Dephasing channels๐':
๐77 ๐7"๐"7 ๐""
๐" ๐77 ๐๐7"๐โ๐"7 ๐""
When ๐ = 1, ๐นCD ๐' = ๏ฟฝฬ๏ฟฝ !,
When ๐ < 1, ๐นGHD ๐' = Qฬ(
"3 Q (,
9:56 AM16SZ & Jiang 2020
Attainability: the QEC protocol
โข Step 1: Asymptotic QFIs for dephasing channels
โข Step 2: Reduction to dephasing channels
โข Step 3: Code optimization
Two-dimensional code:
Encoding:
0+ = โ4A ๐ถ7 4A ๐ R ๐ S) 0 S(, 1+ = โ4A ๐ถ" 4A ๐ R ๐ S) 1 S(
Decoding:
โ โ = โT๐ T โ ๐ T! , ๐ T = ๐ 7 *๐ T๐ 7 +(
+๐ " *๐ T๐ " +(
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๐7,*
time order
โฐ*
๐8
โ โฐ9:;
SZ & Jiang 2020
Attainability: the QEC protocol
โข Step 1: Asymptotic QFIs for dephasing channels
โข Step 2: Reduction to dephasing channels
โข Step 3: Code optimization
Two-dimensional code:
Encoding:
0+ = โ4A ๐ถ7 4A ๐ R ๐ S) 0 S(, 1+ = โ4A ๐ถ" 4A ๐ R ๐ S) 1 S(
Decoding:
โ โ = โT๐ T โ ๐ T! , ๐ T = ๐ 7 *๐ T๐ 7 +(
+๐ " *๐ T๐ " +(
The optimization of ๐นCD,GHD ๐+,' over ๐ถ7, ๐ถ" and โ gives Theorem 2 and 3.
9:56 AM18SZ & Jiang 2020
๐7,*
time order
โฐ*
๐8
โ โฐ9:;
Example 1: Pauli ๐ signal with bit-flip noise
โฐ' ๐ = ๐ฉ โ๐ฐ' ๐ ,
๐ฉ ๐ = 1โ ๐ ๐ + ๐๐๐๐, ๐' = ๐3,"-( ,
๐ป = ๐ โ ๐ฎ = span{๐ผ, ๐}
Optimal code: 0+ = 0 Rโ 0 S , 1+ = 1 Rโ 1 S.
After error:
(๐โ ๐ผ) 0+ = 1 Rโ 0 S , (๐โ ๐ผ) 1+ = 0 Rโ 1 S.
Recovery:
Measure ๐ = ๐โ๐. If ๐ = โ1, flip the probe with ๐; otherwise do nothing.
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Kessler et al. 2014, Arrad et al. 2014, Dur et al. 2014, Unden et al. 2016
Example 1: Pauli ๐ signal with bit-flip noise
โฐ' ๐ = ๐ฉ โ๐ฐ' ๐ ,
๐ฉ ๐ = 1โ ๐ ๐ + ๐๐๐๐, ๐' = ๐3,"-( ,
๐ป = ๐ โ ๐ฎ = span{๐ผ, ๐}
The asymptotic QFI is: ๐นCD โฐ' = 1.
(equal to the ๐นCD ๐ฐ' )
The single-channel QFI is: ๐น โฐ' = 1.
(equal to the ๐น ๐ฐ' )
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Kessler et al. 2014, Arrad et al. 2014, Dur et al. 2014, Unden et al. 2016
Example 2: Pauli ๐ signal with dephasing noise
๐' ๐ = ๐ฉ โ๐ฐ' ๐ ,
๐ฉ ๐ = 1โ ๐ ๐ + ๐๐๐๐, ๐' = ๐3,"-( ,
๐ป = ๐ โ ๐ฎ = span{๐ผ, ๐}
The asymptotic QFI is:
๐นGHD ๐' =1โ 2๐ !
4๐ 1 โ ๐ = ๐ 1/๐ .
The single-channel QFI is: ๐น ๐' = 1โ 2๐ ! = ๐(1).
QEC is not necessary in this case.
21Ulam-Orgikh & Kitagawa 2001, Demkowicz-Dobrzanski et al. 20129:56 AM
Example 3: Pauli ๐ signal with depolarizing noise
โฐ' ๐ = ๐ฉ โ๐ฐ' ๐ ,
๐ฉ ๐ = 1โ U.๐ ๐ + V
.(๐๐๐ + ๐๐๐ + ๐๐๐), ๐' = ๐3
,"-( ,
๐ป = ๐ โ ๐ฎ = span{๐ผ, ๐, ๐, ๐}
The asymptotic QFI is:
๐นGHD โฐ' =2 1โ ๐ !
๐(3 โ 2๐) = ๐ 1/๐ .
The single-channel QFI is:
๐น โฐ' =2 1โ ๐ !
2 โ ๐ = ๐(1).
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SZ & Jiang 2020
Conclusions and outlook
โข The structure of the noise and the Hamiltonian determines the estimation precision limit in quantum metrology.
โข Quantum error correction is powerful โ recovering the HL, achieving the optimal SQL.
โข Future directions: โข parallel strategies vs sequential strategies when the HL is achievable
โข multi-parameter estimation for general quantum channels
โข fault-tolerant QEC for quantum metrology
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Thank you! Liang Jiang