State independent Uncertainty Relations From Eigenvalue minimization - - PowerPoint PPT Presentation

State independent Uncertainty Relations From Eigenvalue minimization

• P. Giorda, L. Maccone
• A. Riccardi

QUit - GROUP, UNIVERSITY OF PAVIA

Torino, May 2019

Outline

1

Motivation Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations

2

First Result: Mapping onto eigenvalue problem Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors

3

Wrap up

Outline

1

Motivation Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations

2

First Result: Mapping onto eigenvalue problem Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors

3

Wrap up

Uncertainty Relations

Superposition principle ! Complementarity

I Complementary Properties cannot have joint definite values for a given

state |ψi of the system

I Preparation uncertainty

Effects of Complementarity can be quantified depending on the aim, protocol, functional used e.g

I Variance based uncertainity relations I Entropy based uncertainty relations

In each framework the effect of complementarity is described in a different way

Uncertainty Relations

Superposition principle ! Complementarity

I Complementary Properties cannot have joint definite values for a given

state |ψi of the system

I Preparation uncertainty

Effects of Complementarity can be quantified depending on the aim, protocol, functional used e.g

I Variance based uncertainity relations I Entropy based uncertainty relations

In each framework the effect of complementarity is described in a different way

Uncertainty Relations

Superposition principle ! Complementarity

I Complementary Properties cannot have joint definite values for a given

state |ψi of the system

I Preparation uncertainty

Effects of Complementarity can be quantified depending on the aim, protocol, functional used e.g

I Variance based uncertainity relations I Entropy based uncertainty relations

In each framework the effect of complementarity is described in a different way

Uncertainty Relations

Hystorically URs have been thouroghly studied and different ways to quantify complementarity have been proposed: given two Observables A, B Heisenberg-Robertson ∆2A∆2B

• |hψ|[A,B]|ψi|2 /4

(Heisenberg, Z. Phys. 1927, H. Robertson Phys. Rev 1929)

Bialynicki-Birula (position and momentum), Deutsch (bounded

• perators),

H (A)+H (B)

• c (A,B)

(Białynicki-Birula, Mycielski 1975, Comm. Math. Phys; Deutsch 1983, Phys Rev Lett)

Uncertainty Relations

Hystorically URs have been thouroghly studied and different ways to quantify complementarity have been proposed: given two Observables A, B Heisenberg-Robertson ∆2A∆2B

• |hψ|[A,B]|ψi|2 /4

(Heisenberg, Z. Phys. 1927, H. Robertson Phys. Rev 1929)

Bialynicki-Birula (position and momentum), Deutsch (bounded

• perators),

H (A)+H (B)

• c (A,B)

(Białynicki-Birula, Mycielski 1975, Comm. Math. Phys; Deutsch 1983, Phys Rev Lett)

Uncertainty Relations

Hystorically URs have been thouroghly studied and different ways to quantify complementarity have been proposed: given two Observables A, B Heisenberg-Robertson ∆2A∆2B

• |hψ|[A,B]|ψi|2 /4

(Heisenberg, Z. Phys. 1927, H. Robertson Phys. Rev 1929)

Bialynicki-Birula (position and momentum), Deutsch (bounded

• perators),

H (A)+H (B)

• c (A,B)

(Białynicki-Birula, Mycielski 1975, Comm. Math. Phys; Deutsch 1983, Phys Rev Lett)

Uncertainty Relations

The two approaches have different purposes, applications, possible caveats Variance based URs ! information about SPECTRUM OF A, B ∆2A∆2B

• |hψ|[A,B]|ψi|2 /4

I they can give trivial results I strongly dependent on the “relabeling” of the outcomes I this is also their usefulness in experiments and for theoretical purposes: F entenglement detection F spin squeezing

Entropy based URs ! information about the SHAPE of probability distributions {p(an)},{p(bn)} H (A)+H (B)

• c (A,B)

independent on the “relabeling” of the outcomes

I good candidates for protocols in which the amont of information about

measurments outcomes is THE relevant issue

I fundamental in protocols such as F quantum cryptography

(PJ Coles, M Berta, M Tomamichel, S Wehner - Reviews of Modern Physics, 2017)

Uncertainty Relations

The two approaches have different purposes, applications, possible caveats Variance based URs ! information about SPECTRUM OF A, B ∆2A∆2B

• |hψ|[A,B]|ψi|2 /4

I they can give trivial results I strongly dependent on the “relabeling” of the outcomes I this is also their usefulness in experiments and for theoretical purposes: F entenglement detection F spin squeezing

Entropy based URs ! information about the SHAPE of probability distributions {p(an)},{p(bn)} H (A)+H (B)

• c (A,B)

independent on the “relabeling” of the outcomes

I good candidates for protocols in which the amont of information about

measurments outcomes is THE relevant issue

I fundamental in protocols such as F quantum cryptography

(PJ Coles, M Berta, M Tomamichel, S Wehner - Reviews of Modern Physics, 2017)

Uncertainty Relations

The two approaches have different purposes, applications, possible caveats Variance based URs ! information about SPECTRUM OF A, B ∆2A∆2B

• |hψ|[A,B]|ψi|2 /4

I they can give trivial results I strongly dependent on the “relabeling” of the outcomes I this is also their usefulness in experiments and for theoretical purposes: F entenglement detection F spin squeezing

Entropy based URs ! information about the SHAPE of probability distributions {p(an)},{p(bn)} H (A)+H (B)

• c (A,B)

independent on the “relabeling” of the outcomes

I good candidates for protocols in which the amont of information about

measurments outcomes is THE relevant issue

I fundamental in protocols such as F quantum cryptography

(PJ Coles, M Berta, M Tomamichel, S Wehner - Reviews of Modern Physics, 2017)

Outline

1

Motivation Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations

2

First Result: Mapping onto eigenvalue problem Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors

3

Wrap up

Sum uncertainty relations

• L. Maccone, A. K. Pati, PRL 113 (26), 260401 (2014)

Given N An bounded operators acting on HM VTot (|ψi) =

N

∑

n=1

∆2

|ψiAn lB

the problem of trivial results is removed the goal is now to find a STATE INDEPENDENT LOWER BOUND lB direct numerical minimization HARD for large dimension M

Differnt strategies have been proposed

Analyitc: An generators of Lie algebra (Mde Guise, H.,et al PRA 2018) Elegant Mapping of the problem into a geometrical one: joint numerical range (R. Schwonnek, et al., Phys. Rev. Lett. 119, 170404 (2017);

• K. Szymański and K. Życzkowski. arXiv:1804.06191 (2018).)

Open problems: ARBITRARY NO OF OBSERVABLES, UNBOUNDED OBSERVABLES

Sum uncertainty relations

• L. Maccone, A. K. Pati, PRL 113 (26), 260401 (2014)

Given N An bounded operators acting on HM VTot (|ψi) =

N

∑

n=1

∆2

|ψiAn lB

the problem of trivial results is removed the goal is now to find a STATE INDEPENDENT LOWER BOUND lB direct numerical minimization HARD for large dimension M

Differnt strategies have been proposed

Analyitc: An generators of Lie algebra (Mde Guise, H.,et al PRA 2018) Elegant Mapping of the problem into a geometrical one: joint numerical range (R. Schwonnek, et al., Phys. Rev. Lett. 119, 170404 (2017);

• K. Szymański and K. Życzkowski. arXiv:1804.06191 (2018).)

Open problems: ARBITRARY NO OF OBSERVABLES, UNBOUNDED OBSERVABLES

Outline

1

Motivation Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations

2

First Result: Mapping onto eigenvalue problem Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors

3

Wrap up

Sum uncertainty relations: OUR SOLUTION Basic Mapping :

For each operator An define on HM ⌦HM Hn = A2

n ⌦I+I⌦A2 n

2 An ⌦An. (1) then HTot = ∑n Hn VTot (|ψi) = hψ|hψ|

N

∑

n=1

Hn|ψi|ψi εGS = lB (2) state independent lB from ground state energy of HTot = ∑n Hn

Exact lowerbound

Generators of su(2) VXYZ = ∆2JX +∆2JY +∆2JZ j (3)

Sum uncertainty relations: OUR SOLUTION Basic Mapping :

For each operator An define on HM ⌦HM Hn = A2

n ⌦I+I⌦A2 n

2 An ⌦An. (1) then HTot = ∑n Hn VTot (|ψi) = hψ|hψ|

N

∑

n=1

Hn|ψi|ψi εGS = lB (2) state independent lB from ground state energy of HTot = ∑n Hn

Exact lowerbound

Generators of su(2) VXYZ = ∆2JX +∆2JY +∆2JZ j (3)

Outline

1

Motivation Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations

2

First Result: Mapping onto eigenvalue problem Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors

3

Wrap up

First Extension of the method for εGS = 0

Ground state

HTot has a unique ground state that can be written in the eigenbasis of any

• f the An {|˜

an,ii|˜ an,ii}M

i=1 as the maximally entangled state

|εgsi = 1 p M ∑

i

|˜ an,ii|˜ an,ii (4)

First exicted state energy bound:

Given ε1 on has VTot (|ψi)

• ε1

✓ 1 1 M ◆ improvement but typically not very tight

First Extension of the method for εGS = 0

Ground state

HTot has a unique ground state that can be written in the eigenbasis of any

• f the An {|˜

an,ii|˜ an,ii}M

i=1 as the maximally entangled state

|εgsi = 1 p M ∑

i

|˜ an,ii|˜ an,ii (4)

First exicted state energy bound:

Given ε1 on has VTot (|ψi)

• ε1

✓ 1 1 M ◆ improvement but typically not very tight

Second Extension of the method for εGS = 0

Fix n

Given An define the operator Aα

n

= An αI then 8α 2 [an,min,an,Max] one has that ∆2Aα

n = ∆2An

Define the operator Hα

n

= (Aα

n )2 ⌦I+I⌦(Aα n )2

2 Aα

n ⌦Aα n

and the global operator where Hα

Tot = ∑m6=n Hm +Hα n

For each n and α:

Hα

Tot has non-zero ground state energy εα gs,n > 0

which is a lower bound for the set of states Sα

n = {|φi 2 HM|hφ |An|φi = α};

Second Extension of the method for εGS = 0

Fix n

Given An define the operator Aα

n

= An αI then 8α 2 [an,min,an,Max] one has that ∆2Aα

n = ∆2An

Define the operator Hα

n

= (Aα

n )2 ⌦I+I⌦(Aα n )2

2 Aα

n ⌦Aα n

and the global operator where Hα

Tot = ∑m6=n Hm +Hα n

For each n and α:

Hα

Tot has non-zero ground state energy εα gs,n > 0

which is a lower bound for the set of states Sα

n = {|φi 2 HM|hφ |An|φi = α};

Extension for εGS = 0

Proposition

for fixed n it holds that 8|φi 2 HM VTot (|φi)

• min

α2[an,min,an,Max]

εα

gs,n

and minα εα

gs,n provides a state independent lower-bound;

the best lower bound 8|φi 2 HM is given by max

n

min

α2[an,min,an,Max]

εα

gs,n

>

Strategy to find the lower bound

For fixed n vary 8α 2 [an,min,an,Max] and find minα2[an,1,an,M] εα

gs,n

Optimize over n

Extension for εGS = 0

Proposition

for fixed n it holds that 8|φi 2 HM VTot (|φi)

• min

α2[an,min,an,Max]

εα

gs,n

and minα εα

gs,n provides a state independent lower-bound;

the best lower bound 8|φi 2 HM is given by max

n

min

α2[an,min,an,Max]

εα

gs,n

>

Strategy to find the lower bound

For fixed n vary 8α 2 [an,min,an,Max] and find minα2[an,1,an,M] εα

gs,n

Optimize over n

Example

Dim HM = 3, N=4 operators

A1 = ✓ 0 1 1 i i ◆ , A2 = ✓ 1 1 ◆ A3 = ✓ 1 1 1 1 1 1 ◆ , A4 = ✓ 1 i i 1 ◆

Outline

1

Motivation Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations

2

First Result: Mapping onto eigenvalue problem Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors

3

Wrap up

Extension for εGS = 0

It would be desirable to find a state |ψsati such that VTot (|ψsati) ' lB

Ground state corresponding to best lower bound

it can be shown that the Schmidt decompostion is |εGSi = ∑

n

λn|λni|λni therefore as tentative |ψsati one can choose best lower bound 8|φi 2 HM is given by |ψsati = |λMaxi λMax = max

n λn

In the most favorable case λMax = O(1) and λMax λn, 8λn 6= λMax

Relative errors

Given εGS best lower bound the procedure allows to find lB with δ = (VTot (|ψsati)εgs)/(2εgs)

Extension for εGS = 0

It would be desirable to find a state |ψsati such that VTot (|ψsati) ' lB

Ground state corresponding to best lower bound

it can be shown that the Schmidt decompostion is |εGSi = ∑

n

λn|λni|λni therefore as tentative |ψsati one can choose best lower bound 8|φi 2 HM is given by |ψsati = |λMaxi λMax = max

n λn

In the most favorable case λMax = O(1) and λMax λn, 8λn 6= λMax

Relative errors

Given εGS best lower bound the procedure allows to find lB with δ = (VTot (|ψsati)εgs)/(2εgs)

Example: Dim HM = 2j +1, N=2 operators

Sum of two spin components: VXZ = ∆2JX +∆2JZ lB (j)

1 Relevant for Planar Spin Squeezing and phase estimation

∆φ V 1

XZ ⇠ j2/3

2 Symmetry of Var(|ψi) can be used and limit the search for

α = hψ|JZ|ψi = 0

3 We determine lB (j) and

|ψsati ⇡ eξ(J2

+J2 )|ji

and find the lower bound with δ ⇡ 3% for all j

Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î Î

j

VXZ

20 40 60 80 100 2 4 6 8 10 12

Example: unbounded operators

Harmonic oscillator sum of number and position Vnx = ∆2n +∆2x 0.412721

We again make use of the symmetry of Var(|ψi) and limit the search for α = hψ|x|ψi = 0 We find that |ψsati ⇡ eξsat

h a2(a†)

2i

|0i is the squeezed vacuum We determine the lower bound with relative error δ ⇡ 0.5%

WRAP UP

SOLUTION FOR THE PROBLEM VTot (|ψi) = ∑N

n=1 ∆2 |ψiAn lB

1 It works for ARBITRARY NUMBER OF OPERATORS 2 CAN GIVE ANALYTICAL RESULTS 3 NUMERICAL RESULTS 1

MINIMIZATION ! obtained with WELL KNOWN AND STABLE NUMERICAL ROUTINES

2

• ne can APPLY DIAGONALIZATION TRICKS (ANALYTICAL E.G.

SIMMETRIES, NUMERICAL E.G. SPARSE MATRICES)

3

QUITE GOOD APPROXIMATIONS

OPEN PROBLEMS

1 OTHER MAPPINGS? 2 DEEPEN STUDY OF UNBOUNDED OPERATORS 3 HOW TO FORSEE IN ADVANCE THE PRECISION OF THE

APPROXIMATION?

1

HOW TO GET BETTER APPROXIMATIONS?

4 APPLY METHOD FOR UPPER BOUNDS 5 APPLICATION TO ENTANGLEMENT DETECTION

Criticality enhanced Estimation

Hλ = H0 +λV , λ 2 R such that at λc QPT, and

• |nλi,E λ

n

QFI(λ) = 4∑n>0 |hnλ |V |0λ i|

2

(E λ

0 E λ n ) 2 = 4gFS

λ

and the scaling gλ ⇠ Lν∆Q+d is dictated by ∆Q = 2∆V 2ζ d with dynamical exponent ζ and the scaling exponent of the correlation length ν, and scaling exponent ∆V of the operator V ∆V is such that ∆Q < 0 i.e., if V is “sufficiently” relevant, gλ scales in a super-extensive way and so does the QFI, thus allowing for an enhancement of the estimation precision. |0λ+δλi ⇡ |0λi+|~ vλi with |~ vλi = ∑n>0

hnλ |V |0λ i

(E λ

0 E λ n )|nλi and with

|ˆ vλi = |~ vλi/|~ v|. If one now consideres the orthonormal basis B0,v =

• |0λi±|ˆ

vλi

• /

p 2 S{|αni}Ld

n=2 one has that

4gFS

λ

= h ∂ 2

δλCohB0,v

⇣ |0λi ⌘i

δλ=0

Single Qubit

Measure δλ = λtrue λest ⌧ 1 Measurement basis Bˆ

b defined by ˆ

b = {sinθ cosφ,sinθ sinφ,cosθ} a generic Bloch vector The Fisher information is F(Bˆ

b,|ψδλihψδλ|) ⇡ 4γ2 sin2 φ γ3

16cos2 φ sinφ cotθ

• δλ

Pure states

FB: |ψλ+dλi = |0i+|vidλ +|widλ 2 +O(dλ 3) with |0i ⌘ |ψλi, |vi ⌘ d

dλ |ψλ+dλi

• dλ=0 and |wi ⌘

⇣

d2 dλ 2 |ψλ+dλi

⌘

dλ=0

. one has Lλ = |0ihv?|+|v?ih0| with |v?i = |vih0|vi|0i suh that the QFI QFI = 4hv?|v?i = 4

• hv|vi|hv|0i|2

. Now the SLD eigenbasis is |±i =

1 p 2(|0i± 1 hv?|v?i1/2 |v?i) such that

the estimation happens in subspace H2 = span

• |0i,|v?i

.

• ne has that

pλ+dλ

±

= 1

2(1±2hv?|v?i1/2dλ ±2Re hw|v?i hv?|v?i1/2 dλ 2 +O(dλ 3)) such

that ⇣ pλ+dλ

±

⌘

dλ=0 = pλ ± = 1/2,

⇣ ∂dλpλ+dλ

±

⌘

dλ=0 = ±hv?|v?i1/2,

⇣ ∂ 2pλ+

±

(5)

QFI: single qubit vs (dynamical change o f) correlations

Theorem

N even ρ0 = ∑n pn|nihn| full rank hn|G|mi 2 R 8n,m slide.

• 1. HN = H2~

⌦HN/2;

• 2. Lλ = σy ˜

⌦ON/2: ⇢α±,k = ±αk 2 R |α±,ki = |±i˜ ⌦|ki, k = 1,..N/2

• 3. CohBλ

α local maximum :

h ∂δλCohBλ

α

i

δλ=0 = 0

• 4. QFI =

h ∂ 2

λ CohBλ

α

i

δλ=0 = FI2 +

⇣ ∂ 2

δλM λ+δλ Lλ

⌘

δλ=0

⇢FI2 {|±i} on ρλ

2 = TrN/2

⇥ ρλ⇤ M λ

L0 = M λ L0

• σy ⌦ON/2
• Classical MI σy ˜

⌦ON/2

Corollary

Lower Bound on QFI: ρλ

2 = TrN/2

⇥ ρλ⇤ FI2  QFI

• ρλ

2

•  QFI

QFI: single qubit vs (dynamical change o f) correlations

Theorem

N even ρ0 = ∑n pn|nihn| full rank hn|G|mi 2 R 8n,m slide.

• 1. HN = H2~

⌦HN/2;

• 2. Lλ = σy ˜

⌦ON/2: ⇢α±,k = ±αk 2 R |α±,ki = |±i˜ ⌦|ki, k = 1,..N/2

• 3. CohBλ

α local maximum :

h ∂δλCohBλ

α

i

δλ=0 = 0

• 4. QFI =

h ∂ 2

λ CohBλ

α

i

δλ=0 = FI2 +

⇣ ∂ 2

δλM λ+δλ Lλ

⌘

δλ=0

⇢FI2 {|±i} on ρλ

2 = TrN/2

⇥ ρλ⇤ M λ

L0 = M λ L0

• σy ⌦ON/2
• Classical MI σy ˜

⌦ON/2

Corollary

Lower Bound on QFI: ρλ

2 = TrN/2

⇥ ρλ⇤ FI2  QFI

• ρλ

2

•  QFI

For Further Reading I

• A. Author.

Handbook of Everything. Some Press, 1990.

• S. Someone.

On this and that. Journal on This and That. 2(1):50–100, 2000.