Controlling quantum state and its coherence Tanumoy Pramanik S - - PowerPoint PPT Presentation

Controlling quantum state and its coherence

Tanumoy Pramanik S N Bose National Centre for Basic Sciences

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770

Motivation

|ψi = X

i

ki |ii

State Superposition coherence

⌦ ⌦

Controlling State Controlling coherence

⌦

?

Outline of talk

Quantum correlations Uncertainty relations Quantum coherence

Steering of quantum state Steering of quantum coherence Example: Pure state and mixed state

Quantum Correlations

Entanglement Steering Bell Nonlocality

Entanglement

A B

ρAB

It is the weakest form of non-local correlation

ρAB 6= X

i

pi ρA

i ⌦ ρB i

Quantum

State tomography

Bell-nonlocal correlation

A B

ρAB

It is the strongest form of non-local correlation

It violates any Bell-CHSH inequality

P(aA, bB) A B a b

Steering

A B

ρAB Steerable, if Alice can control the state of Bob’s system

Quantum

• S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76, 052116 (2007).

Quantum Correlations

Entanglement Steering Bell-nonlocal

i

i

• S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76, 052116 (2007).

Uncertainty relation

Uncertainty relation

Two non-commuting observables can not be measured simultaneously with arbitrary precision.

Coarse grained form : Heisenberg uncertainty relation (HUR) and entropic form of uncertainty relation (EUR). Fine-grained form : Fine-grained uncertainty relation (FUR).

HUR

Here, uncertainty is measured by standard deviation which is coarse grained measure of uncertainty.

∆A ∆B

• |h[A, B]i|

2

∆A = sX

a

a2 pa − X

a

(a pa)2

W . Heisenberg, Z. Phys. 43, 172 (1927); E. H. Kennard, Z. Phys. 44, 326 (1927).

EUR

Here, uncertainty is measured by Shannon entropy.

• H. Maassen, and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).

H (A) + H (B) ≥ log 1 c

H (A) = − X

a

pa log pa and the complementarity, 1/c, is defined as c = max

a,b |ha|bi|2

Fine-grained Uncertainty relation

Here, uncertainty is measured by probability of a particular measurement outcome or a combination of measurement outcomes, i.e., fine-graining of all possible outcomes.

• J. Oppenheim, and S. Wehner, Science 330, 1072 (2010).

FUR

• J. Oppenheim, and S. Wehner, Science 330, 1072 (2010).

ρB b

0 ⌘ | "i 1 ⌘ | #i σz σx σy

P(σi) = 1 3

PSuccess =

3

X

i=1

P(σi) P(bσi = 0) ≤ PSuccess = max

ρB

PSuccess

FUR

ρB

b

0 ⌘ | "i 1 ⌘ | #i σz σx σy

P(σi) = 1 3

ρCertain = 1 2(I + 1 √ 3 (σx + σy + σz))

P(0σx) + P(0σy) + P(0σz) ≤ 3 2 + 3 2 √ 3

Quantum coherence

Quantum coherence

Zero Coherence: All off-diagonal terms are zero.

ρB(i, j|i 6= j) = 0 ρB(i, j|i 6= j) = 0

ρB =        x1 x2 x3 ... xn       

Measures of Quantum coherence

l1-norm:

Cl1(ρB) = X

i,j,i6=j

|ρB(i, j)|

Relative entropy of coherence: CE (ρB) = S(ρD

B) − S(ρB)

ρD

B : diagonal matrix formed with diagonal element

• f .

ρB

• T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).

Measures of Quantum coherence

Skew information: the coherence of the state in the basis of eigenvectors of the observable

• S. Luo, Phys. Rev. Lett. 91, 180403 (2003); Theor. Math. Phys. 143, 681 (2005).

CS

B(ρB) = −1

2 Tr [√ρB, B]2

Quantum coherence

Is it possible to measure quantum coherence with arbitrary precision in all possible mutually non- commuting basis, simultaneously?

Coherence complementarity relations

Coherence complementarity relations

: is calculated by writing the state in basis .

Cl1

x (ρ) + Cl1 y (ρ) + Cl1 z (ρ) ≤

√ 6 CE

x (ρ) + CE y (ρ) + CE z (ρ) ≤ 2.23

σk

ρC

max = 1

2 ✓ I + 1 √ 3 (σx + σy + σz) ◆

CE (l1)

k

(ρ)

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770

Coherence complementarity relations

: is measured in basis .

σk

ρC

max = 1

2 ✓ I + 1 √ 3 (σx + σy + σz) ◆ CS

x (ρ) + CS y (ρ) + CS z (ρ) ≤ 2

CS

k (ρ)

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770

Quantum Steering

Quantum steering

EPR paradox Entanglement is used to put question about incompleteness of quantum physics by Einstein, podolsky and Rosen. Steering Schrodinger re-expressed EPR paradox as the power to control of one system by distantly located system.

• A. Einstein, D. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
• E. Schrodinger, Proc. Cambridge Philos. Soc. 31, 553 (1935); 32, 446 (1936).

EPR-Steering : Physical interpretation

A B

Steerability : Alice’s control on the state of Bob’s system, i.e., Bob can know his system with higher precision than allowed by uncertainty principle. ρAB R = σz S = σx

EPR-Steering : Physical interpretation

A B

R = σz S = σx

|ΨiAB = | " #iAB | # "iAB 2

Alice’ s measurement outcome State of Bob’ s system

| "iz (x) | #iz (x) | #iz (x) | "iz (x)

Bob’ s Uncertainty of system B is zero when Alice communicates her results

EPR-Steering : Mathematical interpretation

Steerability : Absence of local hidden state (LHS) model for Bob’s system.

P(aA, bB) 6= X

λ

P(λ) P(aA|λ) PQ(bB|λ) ρAB 6= X

λ

P(λ) ρA

λ ⌦ ρB λ, Q

• S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76, 052116 (2007).

Local Hidden State model

LHS model Local : Alice prepares system B in a state quantumly uncorrelated with other systems possessed by Alice. Hidden : Bob has no information about the state of B.

Result : Once Alice sends the system B to Bob, Alice does not have any control on the state of system B.

ρAB = X

λ

P(λ) ρA

λ ⊗ ρB λ, Q

Steering criterion : Intuition

When Alice and Bob share steerable state, Alice can reduce Bob’s uncertainty about his system by controlling its state.

It should violate some local uncertainty relation satisfied by

P(bB|aA) P(bB)

Steerability of quantum state

Fine-grained steering criteria

P(aA, bB) = X

λ

P(λ)P(aA|λ) PQ(bB|λ)

+

qmin X

i

pi ≤ X

i

pi qi ≤ qmax X

i

pi

• T. Pramanik, M. Kaplan, and A. S Majumdar, Phys. Rev. A 90, 050305(R) (2014).

P(bσx) + P(bσy) + P(bσz) ≤ 3 2 + 3 2 √ 3

P(bσx|aA1) + P(bσy|aA2) + P(bσz|aA3) ≤ 3 2 + 3 2 √ 3

α

Steerability of pure entailed state

• T. Pramanik, M. Kaplan, and A. S Majumdar, Phys. Rev. A 90, 050305(R) (2014).

ρP = pα |00i + p 1 α |11i

All pure entangled states are maximally steerable

P(bσx|aA1) + P(bσy|aA2) + P(bσz|aA3) = 3

Steerability of werner state

ρW = p ρS + (1 − p) I 4

ρS = |01i |10i p 2

Steerability of werner state

ρW = p ρS + (1 − p) I 4

p > 1 3 p > 1 2 p > 1 √ 3 p > 1 √ 2

: The state is entangled. : The state is steerable with infinite measurement settings. : The state is steerable with three measurement settings. : The state is Bell nonlocal and steerable with two settings.

Steerability of quantum coherence

Steerability of quantum coherence

A B

ηAB Πa

σi

ηB|Πa

σi

Steerable, if it violates coherence complementarity relation

Steerability of quantum coherence

CE

x (ηB|Πa

y (z)) + CE

y (ηB|Πa

z (x)) + CE

z (ηB|Πa

x (y)) > 2.23

Cl1

x (ηB|Πa

y (z)) + Cl1

y (ηB|Πa

z (x)) + Cl1

z (ηB|Πa

x (y)) >

√ 6 CS

x (ηB|Πa

y (z)) + CS

y (ηB|Πa

z (x)) + CS

z (ηB|Πa

x (y)) > 2

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

Steerability of pure entailed state

ρP = pα |00i + p 1 α |11i

All pure entangled states are maximally steerable

Cx(ηB|Πa

y (z)) + Cy(ηB|Πa z (x)) + Cz(ηB|Πa x (y)) = 3

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

Steerability of werner state

ρW = p ρS + (1 − p) I 4

: Steerable under l1-norm. : Steerable under relative entropy coherence : Steerable under skew information

p > 0.82 p > 0.91 p > 0.94

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

State Steerability : p > 0.58

Steerability of werner state

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

Summary

Steering is a kind of non-local correlation where

• ne of the systems is not trusted as quantum system.

Fine-grained steering criterion overcomes the limitations of coarse grained form of steering criteria. Coherence complementary relation : No single quantum state is fully coherence under all non- commuting basis. With three measurement settings, Werner state is steerable (state property) for p > 0.58. With three measurement settings, Werner state is steerable (coherence property) for p > 0.82.

• D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

Thank You