Quantum Measurement Uncertainty Reading Heisenbergs mind or invoking - - PowerPoint PPT Presentation

▶
quantum measurement uncertainty
SMART_READER_LITE
LIVE PREVIEW

Quantum Measurement Uncertainty Reading Heisenbergs mind or invoking - - PowerPoint PPT Presentation

Aug 17, 2023 221 likes •642 views

Quantum Measurement Uncertainty Reading Heisenbergs mind or invoking his spirit? Paul Busch Department of Mathematics Quantum Physics and Logic QPL 2015, Oxford, 13-17 July 2015 Paul Busch (York) Quantum Measurement Uncertainty 1 / 40

slide-1
SLIDE 1

Quantum Measurement Uncertainty

Reading Heisenberg’s mind or invoking his spirit? Paul Busch

Department of Mathematics Quantum Physics and Logic – QPL 2015, Oxford, 13-17 July 2015

Paul Busch (York) Quantum Measurement Uncertainty 1 / 40

slide-2
SLIDE 2

Peter Mittelstaedt 1929-2014 Paul Busch (York) Quantum Measurement Uncertainty 2 / 40

slide-3
SLIDE 3

Outline

1

Introduction: two varieties of quantum uncertainty

2

(Approximate) Joint Measurements

3

Quantifying measurement error and disturbance

4

Uncertainty Relations for Qubits

5

Conclusion

Paul Busch (York) Quantum Measurement Uncertainty 3 / 40

slide-4
SLIDE 4

Introduction: two varieties of quantum uncertainty

Introduction

Paul Busch (York) Quantum Measurement Uncertainty 4 / 40

slide-5
SLIDE 5

Introduction: two varieties of quantum uncertainty

Heisenberg 1927

Essence of the quantum mechanical world view: quantum uncertainty & Heisenberg effect

Paul Busch (York) Quantum Measurement Uncertainty 5 / 40

slide-6
SLIDE 6

Introduction: two varieties of quantum uncertainty

Heisenberg 1927

quantum uncertainty: limitations to what can be known about the physical world Preparation Uncertainty Relation: PUR For any wave function ψ: (Width of Q distribution) · (Width of P distribution) ∼ (Heisenberg just discusses a Gaussian wave packet.) Later generalisation: ∆ρA ∆ρB ≥

1 2

  • [A, B]
  • ρ
  • (Heisenberg didn’t state this...)

Paul Busch (York) Quantum Measurement Uncertainty 6 / 40

slide-7
SLIDE 7

Introduction: two varieties of quantum uncertainty

Heisenberg 1927

Heisenberg effect – reason for quantum uncertainty? any measurement disturbs the object: uncontrollable state change measurements disturb each other: quantum incompatibility Measurement Uncertainty Relation: MUR (Error of Q measurement) · (Error of P) ∼ (Error of Q measurement) · (Disturbance of P) ∼

Paul Busch (York) Quantum Measurement Uncertainty 7 / 40

slide-8
SLIDE 8

Introduction: two varieties of quantum uncertainty

Reading Heisenberg’s thoughts?

Heisenberg allegedly claimed (and proved): ε(A, ρ) ε(B, ρ) ≥

1 2

  • [A, B]
  • ρ
  • ???
  • Paul Busch (York)

Quantum Measurement Uncertainty 8 / 40

slide-9
SLIDE 9

Introduction: two varieties of quantum uncertainty

MUR made precise?

Heisenberg’s thoughts – or Heisenberg’s spirit? ...or: what measurement limitations are there according to quantum mechanics?

combined joint measurement errors for A, B ≥ incompatibility of A, B

  • True of false? Needed:

precise notions of approximate measurement measure of approximation error measure of disturbance

Paul Busch (York) Quantum Measurement Uncertainty 9 / 40

slide-10
SLIDE 10

Introduction: two varieties of quantum uncertainty

Quantum uncertainty challenged

Paul Busch (York) Quantum Measurement Uncertainty 10 / 40

slide-11
SLIDE 11

Introduction: two varieties of quantum uncertainty

Quantum uncertainty challenged

Paul Busch (York) Quantum Measurement Uncertainty 11 / 40

slide-12
SLIDE 12

(Approximate) Joint Measurements

(Approximate) Joint Measurements

Paul Busch (York) Quantum Measurement Uncertainty 12 / 40

slide-13
SLIDE 13

(Approximate) Joint Measurements

Quantum Measurement Statistics – Observables as POVMs

[π] ∼ ρ, [σ] ∼ E = {ωi → Ei} : pσ

π(ωi) = tr[ρEi] = pE ρ (ωi)

POVM : E = {E1, E2, · · · , En}, 0 ≤ O ≤ Ei ≤ I ,

  • Ei = I

state changes: instrument ωi, ρ → Ii(ρ) measurement processes: measurement scheme M = Ha, φ, U, Za

Paul Busch (York) Quantum Measurement Uncertainty 13 / 40

slide-14
SLIDE 14

(Approximate) Joint Measurements

Signature of an observable: its statistics

pC

ρ

= pA

ρ

for all ρ ⇐ ⇒ C = A Minimal indicator for a measurement of C to be a good approximate measurement of A: pC

ρ

≃ pA

ρ

for all ρ Unbiased approximation – absence of systematic error: C[1] =

  • j

cjCj = A[1] =

  • i

aiAi = A ... often taken as sole criterion for a good measurement

Paul Busch (York) Quantum Measurement Uncertainty 14 / 40

slide-15
SLIDE 15

(Approximate) Joint Measurements

Joint Measurability/Compatibility

Definition: joint measurability (compatibility) Observables C = {C+, C−}, D = {D+, D−} are jointly measurable if they are margins of an observable G = {G++, G+−, G−+, G−−}: Ck = Gk+ + Gk−, Dℓ = G+ℓ + G−ℓ

Joint measurability in general Pairs of unsharp observables may be jointly measurable – even when they do not commute!

Paul Busch (York) Quantum Measurement Uncertainty 15 / 40

slide-16
SLIDE 16

(Approximate) Joint Measurements

Approximate joint measurement: concept

G

  • C
  • D
  • A

B joint observable approximator observables (compatible) target observable Task: find suitable measures of approximation errors Measure of disturbance: instance of joint measurement approximation error

Paul Busch (York) Quantum Measurement Uncertainty 16 / 40

slide-17
SLIDE 17

Quantifying measurement error and disturbance

Quantifying Measurement Error

Paul Busch (York) Quantum Measurement Uncertainty 17 / 40

slide-18
SLIDE 18

Quantifying measurement error and disturbance

Approximation error

(vc) value comparison (e.g. rms) deviation of outcomes of a joint measurement: accurate reference measurement together with measurement to be calibrated, on same system (dc) distribution comparison (e.g. rms) deviation between distributions of separate measurements: accurate reference measurement and measurement to be calibrated, applied to separate but identically prepared ensembles alternative measures of deviation: error bar width; relative entropy; etc. ... Crucial: Value comparison is of limited applicability in quantum mechanics!

Paul Busch (York) Quantum Measurement Uncertainty 18 / 40

slide-19
SLIDE 19

Quantifying measurement error and disturbance

Approximation error – Take 1: value comparison Measurements/observables to be compared: A = {A1, A2, . . . , Am}, C = {C1, C2, . . . , Cn} where A is a sharp (target) observable and C an (approximator) observable representing an approximate measurement of A Protocol: measure both A and C jointly on each system of an ensemble of identically prepared systems Proviso: This requires A and C to be compatible, hence commuting. δvc(C, A; ρ)2 =

  • i

(ai − cj)2 tr[ρAiCj] (Ozawa 1991)

Paul Busch (York) Quantum Measurement Uncertainty 19 / 40

slide-20
SLIDE 20

Quantifying measurement error and disturbance

Issue: δvc is of limited use! Attempted generalisation: measurement noise (Ozawa 2003) δvc(C, A; ρ)2 =

C[2] − C[1]2

ρ +

(C[1] − A)2

ρ = εmn(C, A; ρ)2

where C[k] =

j ck j Cj, A = A[1] are the kth moment operators...

...then give up assumption of commutativity of A, C Critique (BLW 2013, 2014) If A, C do not commute, then: δvc(C, A; ρ) loses its meaning as rms value deviation and becomes unreliable as error indicator – e.g., it is possible to have εmn(C, A; ρ) = 0 where A, C may not even have the same value sets.

Paul Busch (York) Quantum Measurement Uncertainty 20 / 40

slide-21
SLIDE 21

Quantifying measurement error and disturbance

Measurement noise as approximation error?

ε(C, A; ρ) =

(Zτ − A)21/2

ρ⊗σ

=

C[2] − C[1]2

ρ +

(C[1] − A)2

ρ

1/2

adopted from noise concept of quantum optical theory of linear amplifiers first term describes intrinsic noise of POVM C, that is, its deviation from being sharp, projection valued second term intended to capture deviation between target observable A and approximator observable C State dependence – a virtue? Then incoherent to offer three-state method. C[1], A ma not commute: C[1] − A incompatible with C[1], A.

Paul Busch (York) Quantum Measurement Uncertainty 21 / 40

slide-22
SLIDE 22

Quantifying measurement error and disturbance

Ozawa and Branciard inequalities

ε(A, ρ) ε(B, ρ) + ε(A, ρ)∆ρB , + ∆ρAε(B, ρ) ≥

1 2

  • [A, B]
  • ρ ,

ε(A)2(∆ρB)2 + ε(B)2(∆ρA)2 + 2

  • (∆ρA)2(∆ρB)2 − 1

4|[A, B]ρ|2 ε(A)ε(B) ≥ 1 4|[A, B]ρ|2

Does allow for ε(A; ρ) ε(B; ρ) <

1 2|[A, B]ρ|.

Branciard’s inequality is known to be tight for pure states. Not purely error tradeoff relations! (BLW 2014)

Paul Busch (York) Quantum Measurement Uncertainty 22 / 40

slide-23
SLIDE 23

Quantifying measurement error and disturbance

Measurement Noise – some oddities

Take two identical systems, probe in state σ, measurement coupling U = SWAP, pointer Z = A. Then C = A and D = Bσ I. η(D, B; ρ)2 = (∆ρB)2 + (∆σB)2 +

Bρ − Bσ 2

η(D, B; ρ)2 contains a contribution from preparation uncertainty – not solely a measure of disturbance. For ρ = σ: η(D, B; σ) = √ 2∆(Bσ); i.e., distorted observable D is statistically independent of B. Note η(D, B; σ) = 0, despite the fact that the state has not changed (no disturbance).

Paul Busch (York) Quantum Measurement Uncertainty 23 / 40

slide-24
SLIDE 24

Quantifying measurement error and disturbance

Approximation error – Take 2: distribution comparison Protocol: compare distributions of A and C as they are obtained in separate runs of measurements on two ensembles of systems in state ρ δγ(pC

ρ , pA ρ )α = ij(ai − cj)αγ(i, j)

(1 ≤ α < ∞) where γ is any joint distribution of the values of A and C with marginal distributions pA

ρ , pC ρ

∆α(pC

ρ , pA ρ ) = inf γ δγ(pC ρ , pA ρ )

Wasserstein-α distance – scales with distances between points. ∆α(C, A) = sup

ρ ∆α(pC ρ , pA ρ )

quantum rms error: α = 2

Paul Busch (York) Quantum Measurement Uncertainty 24 / 40

slide-25
SLIDE 25

Uncertainty Relations for Qubits

Qubit Uncertainty

Paul Busch (York) Quantum Measurement Uncertainty 25 / 40

slide-26
SLIDE 26

Uncertainty Relations for Qubits

Qubits

σ = (σ1, σ2, σ3) (Pauli matrices acting on C2) States: ρ = 1

2

I + r · σ ,

|r| ≤ 1 Effects: A = 1

2(a0I + a · σ) ∈ [O, I],

0 ≤ 1

2

a0 ± |a| ≤ 1

  • bservables: (Ω = {+1, −1})

A : ±1 → A± = 1

2(I ± a · σ)

|a| = 1 B : ±1 → B± = 1

2(I ± b · σ)

|b| = 1 C : ±1 → C± = 1

2(1 ± γ) I ± 1 2c · σ

|γ| + |c| ≤ 1 D : ±1 → D± = 1

2(1 ± δ) I ± 1 2d · σ

|δ| + |d| ≤ 1 symmetric: γ = 0 sharp: γ = 0, |c| = 1; → unsharpness: U(C)2 = 1 − |c|2

Paul Busch (York) Quantum Measurement Uncertainty 26 / 40

slide-27
SLIDE 27

Uncertainty Relations for Qubits

Joint measurability of C, D

Symmetric case (sufficient for optimal compatible approximations): Proposition C = {C± = 1

2(I ± c · σ)}, D = {D± = 1 2(I ± d · σ)} are compatible if and

  • nly if

|c + d| + |c − d| ≤ 2. Interpretation: unsharpness U(C)2 = 1 − |c|2; |c × d| = 2

  • [C+, D+]
  • |c + d| + |c − d| ≤ 2 ⇔

1 − |c|21 − |d|2 ≥ |c × d|2

C, D compatible ⇔ U(C)2 × U(D)2 ≥ 4

  • [C+, D+]
  • 2

Unsharpness Relation

Paul Busch (York) Quantum Measurement Uncertainty 27 / 40

slide-28
SLIDE 28

Uncertainty Relations for Qubits

Approximation error

Recall: Observable C is a good approximation to A if pC

ρ ≃ pA ρ

Take here: probabilistic distance dp(C, A) = sup

ρ sup X

  • tr[ρC(X)] − tr[ρA(X)]
  • = sup

ρ

  • pC

ρ − pA ρ

  • 1 = sup

X

  • C(X) − A(X)
  • Qubit case: C+ = 1

2

c0I + c · σ , A+ = 1

2

a0I + a · σ

  • dp(C, A) =
  • C+ − A+
  • = 1

2|c0 − a0| + 1 2|c − a| ≡ da ∈ [0, 1].

Paul Busch (York) Quantum Measurement Uncertainty 28 / 40

slide-29
SLIDE 29

Uncertainty Relations for Qubits

Comparison 1: Wasserstein 2-distance (quantum rms error)

∆2

  • pC

ρ , pA ρ

2 = inf

γ

  • ij

(ai − cj)2γ(i, j) where γ runs through all joint distributions with margins pC

ρ , pA ρ .

∆2(C, A)2 = sup

ρ d2

  • pC

ρ , pA ρ

2 ≡ ∆2

a

Qubit case: ∆2

a = ∆2(C, A)2 = 2|c0 − a0| + 2|c − a|

= 4dp(C, A) = 4da.

Paul Busch (York) Quantum Measurement Uncertainty 29 / 40

slide-30
SLIDE 30

Uncertainty Relations for Qubits

Comparison 2: Measurement noise (Ozawa et al)

ε(C, A; ϕ)2 =

ϕ ⊗ φ

  • (Zτ − A)2ϕ ⊗ φ
  • =

C[2] − C[1]2

ρ +

(C[1] − A)2

ρ ≡ ε2 a

Qubit observables, symmetric case: ε2

a = 1 − |c|2 + |a − c|2 = U(C)2 + 4d2 a

ε(A; ρ) double counts contribution from unsharpness. Virtue of state-dependence all but gone ... for more general approximators C, εa may be zero although C is quite different from A Branciard notices this and considers it an artefact of the definition of εa – you might rather consider it a fatal flaw if the aim is to identify optimal compatible approximations of incompatible observables...

Paul Busch (York) Quantum Measurement Uncertainty 30 / 40

slide-31
SLIDE 31

Uncertainty Relations for Qubits

Optimising approximate joint measurements

Gkℓ

  • ℓ
  • k
  • Ck

dp(C,A)

  • Dℓ

dp(D,B)

  • Ak

Bℓ Goal To make errors dA = dp(C, A), dB = dp(D, B) simultaneously as small as possible, subject to the constraint that C, D are compatible.

Paul Busch (York) Quantum Measurement Uncertainty 31 / 40

slide-32
SLIDE 32

Uncertainty Relations for Qubits

Admissible error region

sin θ = |a × b| (dA, dB) =

dp(C, A), dp(D, B) ∈ [0, 1

2] × [0, 1 2] with C, D compatible

Paul Busch (York) Quantum Measurement Uncertainty 32 / 40

slide-33
SLIDE 33

Uncertainty Relations for Qubits

Qubit Measurement Uncertainty Relation

sin θ = |a × b|

PB, T Heinosaari (2008), arXiv:0706.1415

|c + d| + |c − d| ≤ 2 U(C)2 × U(D)2 ≥ 4[C+, D+]2 dp(C, A) + dp(D, B) ≥

1 2 √ 2 [ |a + b| + |a − b| − 2 ]

|a + b| + |a − b| = 2

  • 1 + |a × b| = 2
  • 1 + 2
  • [A+, B+]
  • Paul Busch (York)

Quantum Measurement Uncertainty 33 / 40

slide-34
SLIDE 34

Uncertainty Relations for Qubits

Qubit Measurement Uncertainty

PB & T Heinosaari (2008), S Yu and CH Oh (2014) Optimiser, case a ⊥ b:

c = |c|a, d = |d|b, 2da = |a − c| = 1 − |c|, 2db = |b − d| = 1 − |d|, Compatibility constraint: |c|2 + |d|2 = 1, i.e., U(C)2 + U(D)2 = 1 (1 − 2da)2 + (1 − 2db)2 = |c|2 + |d|2 = 1

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

a⋅b = 0 da d b (d - 1) + (d - 1) = 1

2 2 a b

2 2 + (2 (2

Paul Busch (York) Quantum Measurement Uncertainty 34 / 40

slide-35
SLIDE 35

Uncertainty Relations for Qubits

Ozawa–Branciard (C Branciard 2013, M Ringbauer et al 2014)

a ⊥ b, symmetric approximators C, D: ε2

a

  • 1 − ε2

a

4

  • + ε2

b

  • 1 − ε2

b

4

  • ≥ 1
  • 1 − ε2

a

2

2

+

  • 1 − ε2

b

2

2

≤ 1 ε2

a ≡ 4d′ a,

ε2

b ≡ 4d′ b

(2d′

a − 1)2 + (2d′ b − 1)2 ≤ 1

Optimiser: c = |c|a, d = |d|b, Compatibility constraint: |c|2 + |d|2 = 1, i.e., U(C)2 + U(D)2 = 1 4d′

a = ε2 a = 1 − |c|2 + |a − c|2 = 2|a − c| = 4da,

4d′

b = ε2 b = 4db

(2da − 1)2 + (2db − 1)2 = |c|2 + |d|2 = 1 Experimentally confirmed!

Paul Busch (York) Quantum Measurement Uncertainty 35 / 40

slide-36
SLIDE 36

Uncertainty Relations for Qubits

A twist: Ozawa’s error

Branciard’s inequality has additional optimisers: ε2

a

  • 1 − ε2

a

4

  • + ε2

b

  • 1 − ε2

b

4

  • = 1−|c|2+1−|d|2+|a×c|2+|b×d|2| ≥ 1

M = {M+, M−} = C′ = D′, M± = 1

2(I ± m · σ),

|m| = 1 : Then: 1 − |m|2 + 1 − |m|2 + |a × m|2 + |b × m|2 = 1 m “between” a, b ε(M, A) = ε(M, B) = ε(A, C) = ε(B, D) but 2dp(C, A) = 2dp(D, B) = |a − c| < |a − m| = 2dp(M, A) = 2dp(M, B) In fact, any unit vector m will do!

Paul Busch (York) Quantum Measurement Uncertainty 36 / 40

slide-37
SLIDE 37

Uncertainty Relations for Qubits

Ozawa’s error

Moreover, c = −|c|a, d = −|d|b with |c|2 + |d|2 = 1 is another

  • ptimiser!

Things get worse when a ⊥ b (T Bullock, PB 2015) ⇒ ε(C, A) is unreliable as a guide in finding optimal joint approximations. But still . . . a lucky coincidence that the optimisers “overlap” enough so that the experiments also confirm MUR for probabilistic errors.

Paul Busch (York) Quantum Measurement Uncertainty 37 / 40

slide-38
SLIDE 38

Conclusion

Conclusion

Paul Busch (York) Quantum Measurement Uncertainty 38 / 40

slide-39
SLIDE 39

Conclusion

Conclusion

(1) Heisenberg’s spirit materialised

  • joint measurement errors for A, B
  • ≥
  • incompatibility of A, B
  • unsharpness of compatible C, D
  • ≥
  • noncommutativity of C, D
  • Shown for qubits; also for position and momentum (BLW 2013):

∆2(C, Q) ∆2(D, P) ≥ 2 Generic results: finite dimensional Hilbert spaces, arbitrary discrete, finite-outcome observables (Miyadera 2011) (2) Importance of judicious choice of error measure valid MURs obtained for Wasserstein-2 distance, error bar widths, . . . measurement noise / value comparison – not suited for universal MURs

Paul Busch (York) Quantum Measurement Uncertainty 39 / 40

slide-40
SLIDE 40

Conclusion

References/Acknowledgements

PB (1986): Phys. Rev. D 33, 2253 PB, T. Heinosaari (2008): Quantum Inf. & Comput. 8, 797, arXiv:0706.1415 PB, P. Lahti, R. Werner (2014): Phys. Rev. A 89, 012129, arXiv:1311.0837;

  • Rev. Mod. Phys. 86, 1261, arXiv:1312.4393
  • C. Branciard, PNAS 110 (2013) 6742, arXiv:1304.2071
  • S. Yu, C.H. Oh (2014): arXiv:1402.3785
  • T. Bullock, PB (2015): in preparation

http://demonstrations.wolfram.com/HeisenbergTypeUncertaintyRelationForQubits/

Paul Busch (York) Quantum Measurement Uncertainty 40 / 40