Download Quantum theoretical generalization of the root-mean

Hong Kong Workshop on Quantum Information and Foundations
Hong Kong, 4–6 May 2016
Quantum theoretical generalization of the
root-mean-square error for measurements
MASANAO OZAWA
Nagoya University
Supported by JSPS KAKENHI.
1
Why “State-Dependent Formulation” to understand the uncertainty principle?
• Conventional Understanding of Heisenberg’s Uncertainty Principle:
SIMULTANEOUS MEASURABILITY ⇔ COMMUTATIVITY
• The State-Dependent Formulation challenges this.
• Einstein challenged Heisenberg’s uncertainty principle and FOUND a
violation.
• Schrödinger’s Modification of Einstein’s Challenge to Heisenberg:
Canonically conjugate observables Q ⊗ I and P ⊗ I are simultaneously measurable in the EPR state (Schrödinger 1935).
• Prior information circumvents the uncertainty principle (Hall 2004).
2
Achievements in the State-Dependent Formulation
• First Universal Joint Measurement Uncertainty Relation (Ozawa 2003):
ε(A)ε(B) + σ(A)ε(B) + ε(A)σ(B) ≥
1
2
| h[A, B]i |.
• In the EPR case, we have
ε(Q ⊗ I)ε(P ⊗ I) + σ(Q ⊗ I)ε(P ⊗ I) + σ(P ⊗ I)ε(Q ⊗ I) ≥
h̄
2
• In the EPR state, we have σ(Q ⊗ I) → ∞ and σ(P ⊗ I) → ∞, so
that it is allowed to have ε(Q ⊗ I) → 0 and ε(P ⊗ I) → 0.
• The Universal Joint Measurement Uncertainty Relation is consistent
with Schrödinger’s Modification of Einstein’s Challenge to Heisenberg.
3
.
Heisenberg’s Error-Disturbance Relation is StateDependent!
• Common misconception: The resolution power is determined by the
wave length independent of the measured electron. Thus, the error of
position measurement using a microscope is state-independent.
• The resolution power is defined only in the state where the electron is
suitably localized inside of the scope.
• If the electron’s wave function spreads outside of the scope, the γ ray
microscope cannot be considered as a measuring apparatus with its resolution power.
• Thus, the resolution power is a state-dependent notion.
4
Classical Root-Mean-Square Error
• Suppose a measurement of a classical observable A is replaced by a direct measurement of another classical observable M .
• Then, the error N is defined by N = M − A. How should we define
the mean error?
• Gauss’s mean error (Gauss 1873): hN 2 i
1/2
= h(M − A)2 i
1/2
[C. F. Gauss, Theoria Combinationis Observationum Erroribus Miinimis Obnoxiae,
Pars Prior, Pars Posterior, Supplementum (1873)]
• Obviously, hN 2 i
1/2
= 0 implies A = M with probability 1.
5
Measuring Processes
• A measuring process M(x) = (K, σ, U, M ) determines:
Time evolution: A(0) = A ⊗ I 7→ A(∆t) = U † (A ⊗ I)U ,
M (0) = I ⊗ M 7→ M (∆t) = U † (I ⊗ M )U
Quantum instrument:
POVM:
I(x)ρ = TrK [E M (∆t) (x)(ρ ⊗ σ)]
Π(x) = I(x)∗ I = TrK [E M (∆t) (x)(I ⊗ σ)]
Pr{x = xkρ} = Tr[I(x)ρ]
I(x)ρ
Quantum state reduction: ρ 7→ ρ{x=x} =
Tr[I(x)ρ]
[Ozawa, JMP 25, 79 (1984)]
Output distribution:
• By the measuring process M(x) = (K, σ, U, M ) a measurement of an
observable A(0) is replaced by a direct measurement of M (∆t).
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Root-Mean-Square of Noise Operator
• Definition. For any measuring process M(x) = (K, σ, U, M ) the noise
operator N (A) for measuring A is defined by
N (A) = M (∆t) − A(0).
• The rms of noise operator for measuring A in ρ is defined by
ε(A, ρ) = hN (A)2 i
1/2
= Tr[(M (∆t) − A(0))2 (ρ ⊗ σ)]1/2 .
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Quantum Root Mean Square Errors
• Requirements for quantum generalization of root-mean-square error.
(i) Device-independent definability: The error measure should be definable
by the POVM of the measuring process, the observable to be measured,
and the state of the object.
(ii) Correspondence principle: The error measure should be identical with
the classical rms error if M (∆t) and A(0) commute.
(iii) Soundness: The error measure should take the value zero for any precise
measurements.
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Device-Independent Definability
• RMS of noise operator, ε(A, ρ), satisfies the device-independent definability.
• Moment operator of POVM Π: m(n) (Π) =
∑
x∈R
xn Π(x)
• ε(A, ρ) satisfies:
ε(A, ρ)2 = Tr[(m(2) (Π) − Am(Π) − m(Π)A + A2 )ρ].
2
ε(A, ρ) =
∑
(m − a)2 <Tr[E A (a)Π(m)ρ]
a,m
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Correspondence Principle
• RMS of noise operator, ε(A, ρ), satisfies the correspondence principle.
• If M (∆t) and A(0) commute, there exists the joint probability distribution
Pr{M (∆t) = m, A(0) = a} = Tr[E M (∆t) (m)E A(0) (a)(ρ ⊗ σ)],
and we have
2
ε(A, ρ) =
∑
(m − a)2 Pr{M (∆t) = m, A(0) = a}.
a,m
• Note that any error notions having been proposed based on the distance
between the probability distributions do not satisfy (ii).
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State-Dependent Commutativity
• Definition: A and B commute in ρ (A ↔ρ B) iff [f (A), g(B)]ρ = 0
for all polynomials f (A), g(B).
• Definition: A joint probability distribution (JPD) of observables A, B in
state ρ is a probability distribution P A,B,ρ (x, y) on R2 satisfying
Tr[f (A, B)ρ] =
∑
x,y
f (x, y) P A,B,ρ (x, y)
for any polynomial f (A, B).
• Theorem (Gudder 1968): A ↔ρ B ⇔ there exists a JPD of A, B in ρ.
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Weak Joint Probability
• Definition: The weak joint distribution (WJD) PwA,B,ρ (x, y) of observables A, B in state ρ is defined by
PwA,B,ρ (x, y) = Tr[E A (x)E B (y)ρ].
• Remark: The WJD can be measured by weak measurement and postselection, i.e.,
PwA,B,ρ (x, y)
D
B
E
= E (y)
w,A=x,ρ
P A,ρ (x).
• Theorem:
2
ε(A, ρ) =
∑
(m − a)2 <PwM (∆t),A(0),ρ⊗σ (m, a).
a,m
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State-Dependent Identity
• Definition: A and B are perfectly correlated in ρ (A =ρ B)
⇔ A ↔ρ B and the JPD P A,B,ρ satisfies
∑
P A,B,ρ (x, x) = 1.
x
• Theorem (Ozawa 2005):
A =ρ B ⇔ PwA,B,ρ (x, y) = 0
⇔
D
E
E B (y)
w,A=x,ρ
if x 6= y
= δx,y .
• Remark: Perfectly correlation is experimentally accessible.
• Theorem (Ozawa 2005): The relation =ρ is an equivalence relation
among observables. In particular, if A =ρ B and B =ρ C, then
A =ρ C.
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State-Dependent Precise Measurements
• Definition (Ozawa 2005): A measuring apparatus M(x) = (K, σ, U, M )
precisely measures A in ρ ⇔
A(0) =ρ⊗σ M (∆t).
• Common misconception: A measurement of an observable A in a state
ρ is precise if and only if the meter M reproduces the probability distribution of A.
• Counter example: If A(0) and M (∆t) are commuting, identically
distributed, and independent, then reading of M (∆t) gives no information about the measurement of A(0). In this case, we have
√
ε(A, ρ) = 2σ(A) by the definition of classical root-mean-square error.
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Soundness
• The RMS of NO satisfies the soundness condition.
• If A(0) =ρ⊗σ M (∆t) then
PwM (∆t),A(0),ρ⊗σ (m, a) = 0 for x 6= y,
and hence
2
ε(A, ρ) =
∑
(m − a)2 <PwM (∆t),A(0),ρ (m, a) = 0.
a,m
15
Complete Quantum Root Mean Square Errors (1)
• Definition: Quantum rms error is called complete if it satisfies:
(iv) Completeness: The error measure should take the value zero if and only
if the measurement is precise.
• For any t ∈ R, define
εt (A, ρ) = ε(A, e−itA ρeitA ).
We call {εt (A, ρ)} the quantum root mean square error profile.
• If A(0) and M (∆t) commute, then εt (A, ρ) = ε(A, ρ), since
ε(A, e
−itA
itA 2
ρe
)
=
D
itA(0)
e
(M (∆t) − A(0)) e
= h(M (∆t) − A(0))2 i
= ε(A, ρ)2 .
16
2 −itA(0)
E
Complete Quantum Root Mean Square Errors (2)
• The locally uniform rms error is defined by
ε(A) = ε(A, ρ) = supt∈R εt (A, ρ).
• The projective rms error for an invariant mean ηt∈R is defined by
ε̂(A)2 = ε̂(A, ρ)2 = ηt∈R εt (A, ρ)2 .
• Theorem: If A is discrete,
(
ε̂(A, ρ)2 = ε A,
=
∑
∑
)2
E A (a)ρE A (a)
a
(a − m)2 Tr[E A (a)Π(m)E A (a)ρ].
a,m
17
Complete Quantum Root Mean Square Errors (3)
• Definition: For any invertible probability distribution function f , we define f -distributed rms error by
∫
εf (A, ρ)2 =
R
εt (A, ρ)2 f (t)dt.
• Theorem:
(1) ε, ε̂, and εf are complete quantum rms errors.
(2) ε(A) ≤ ε(A),
ε̂(A) ≤ ε(A),
εf (A) ≤ ε(A).
(3) If A2 = I and m(2) (Π) = I, then ε(A) = ε(A) = ε̂(A) = εf (A).
• Theorem: For any quantum rms errors ε0 , the relation
h̄
ε0 (Q)ε0 (P ) ≥
2
is violated.
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Joint Measurement Uncertainty Relations
√
√
• Let CAB = 12 |Tr([A, B]ρ)|, DAB = 12 Tr(| ρ[A, B] ρ|).
• The following relations hold for ε0 (A) = ε0 (A, ρ, M(f (x))) and
ε0 (B) = ε0 (B, ρ, M(g(x))), where ε0 = ε or ε0 = ε.
• 0 (A)0 (B) + σ(B)0 (A) + σ(A)0 (B) ≥ CAB .
2
2
• σ(B)2 0 (A)2 + σ(A)
(B)
0
√
2
2
≥ CAB
.
+20 (A)0 (B) σ(A)2 σ(B)2 − CAB
2
2
• σ(B)2 0 (A)2 + σ(A)
(B)
0
√
2
2
+20 (A)0 (B) σ(A)2 σ(B)2 − DAB
≥ DAB
.
19
(Ozawa 2003)
(Branciard 2013)
(Ozawa 2014)