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 X | ψ i = k i | i i i 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 ρ AB Quantum State tomography X p i ρ A i ⌦ ρ B ρ AB 6 = i i It is the weakest form of non-local correlation

Bell-nonlocal correlation A B A B ρ AB P ( a A , b B ) b a It violates any Bell-CHSH inequality It is the strongest form of non-local correlation

Steering A B ρ AB Quantum Steerable, if Alice can control the state of Bob’s system S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76 , 052116 (2007).

Quantum Correlations i i Entanglement Steering Bell-nonlocal 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 | h [ A , B ] i | ∆ A ∆ B � 2 Here, uncertainty is measured by standard deviation which is coarse grained measure of uncertainty. sX X ∆ A = a 2 p a − ( a p a ) 2 a a W . Heisenberg, Z. Phys. 43, 172 (1927); E. H. Kennard, Z. Phys. 44, 326 (1927).

EUR H ( A ) + H ( B ) ≥ log 1 c Here, uncertainty is measured by Shannon entropy. X H ( A ) = − p a log p a a and the complementarity, 1/c, is defined as a,b | h a | b i | 2 c = max H. Maassen, and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).

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 σ x 0 ⌘ | "i ρ B b σ y P ( σ i ) = 1 1 ⌘ | #i 3 σ z 3 X P S uccess = P ( σ i ) P ( b σ i = 0) ≤ P S uccess = max P S uccess ρ B i =1 J. Oppenheim, and S. Wehner, Science 330, 1072 (2010).

FUR ρ C ertain = 1 1 2( I + 3 ( σ x + σ y + σ z )) √ σ x 0 ⌘ | "i ρ B b σ y P ( σ i ) = 1 1 ⌘ | #i 3 σ z P (0 σ x ) + P (0 σ y ) + P (0 σ z ) ≤ 3 3 2 + √ 2 3

Quantum coherence

Quantum coherence Zero Coherence: All off-diagonal terms are zero. ρ B ( i, j | i 6 = j ) = 0   x 1 x 2     x 3 ρ B ( i, j | i 6 = j ) = 0   ρ B =   ...     x n

Measures of Quantum coherence l 1 -norm: X C l 1 ( ρ B ) = | ρ B ( i, j ) | i,j,i 6 = j Relative entropy of coherence: C E ( ρ B ) = S ( ρ D B ) − S ( ρ B ) B : diagonal matrix formed with diagonal element ρ D of . ρ B 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 B ( ρ B ) = − 1 2 Tr [ √ ρ B , B ] 2 C S S. Luo, Phys. Rev. Lett. 91, 180403 (2003); Theor. Math. Phys. 143, 681 (2005).

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 √ C l 1 x ( ρ ) + C l 1 y ( ρ ) + C l 1 z ( ρ ) ≤ 6 C E x ( ρ ) + C E y ( ρ ) + C E z ( ρ ) ≤ 2 . 23 C E ( l 1 ) : is calculated by writing the state in basis . ( ρ ) σ k k ✓ ◆ max = 1 1 ρ C I + 3 ( σ x + σ y + σ z ) √ 2 D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770

Coherence complementarity relations C S x ( ρ ) + C S y ( ρ ) + C S z ( ρ ) ≤ 2 C S : is measured in basis . k ( ρ ) σ k ✓ ◆ max = 1 1 ρ C I + 3 ( σ x + σ y + σ z ) √ 2 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 R = σ z A B ρ AB S = σ x 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.

EPR-Steering : Physical interpretation R = σ z | Ψ i AB = | " #i AB � | # "i AB B A 2 S = σ x Alice’ s measurement outcome State of Bob’ s system | #i z ( x ) | "i z ( x ) | #i z ( x ) | "i z ( 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. X P ( λ ) ρ A λ ⌦ ρ B ρ AB 6 = λ , Q λ X P ( a A , b B ) 6 = P ( λ ) P ( a A | λ ) P Q ( b B | λ ) λ 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. X P ( λ ) ρ A λ ⊗ ρ B ρ AB = λ , Q λ Result : Once Alice sends the system B to Bob, Alice does not have any control on the state of system B.

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 P ( b B | a A ) by P ( b B )

Steerability of quantum state

Fine-grained steering criteria X P ( a A , b B ) = P ( λ ) P ( a A | λ ) P Q ( b B | λ ) λ + X X X p i ≤ p i q i ≤ q max q min p i i i i P ( b σ x ) + P ( b σ y ) + P ( b σ z ) ≤ 3 3 2 + √ α 2 3 P ( b σ x | a A 1 ) + P ( b σ y | a A 2 ) + P ( b σ z | a A 3 ) ≤ 3 3 2 + √ 2 3 T. Pramanik, M. Kaplan, and A. S Majumdar, Phys. Rev. A 90, 050305(R) (2014).

Steerability of pure entailed state p ρ P = p α | 00 i + 1 � α | 11 i All pure entangled states are maximally steerable P ( b σ x | a A 1 ) + P ( b σ y | a A 2 ) + P ( b σ z | a A 3 ) = 3 T. Pramanik, M. Kaplan, and A. S Majumdar, Phys. Rev. A 90, 050305(R) (2014).

Steerability of werner state ρ W = p ρ S + (1 − p ) I 4 ρ S = | 01 i � | 10 i p 2

Steerability of werner state ρ W = p ρ S + (1 − p ) I 4 p > 1 : The state is entangled . 3 p > 1 : The state is steerable with infinite measurement settings . 2 1 p > : The state is steerable with three measurement settings . √ 3 1 : The state is Bell nonlocal and steerable with two settings . p > √ 2

Steerability of quantum coherence

Steerability of quantum coherence A B η AB Π a η B | Π a σ i σ i Steerable , if it violates coherence complementarity relation

Steerability of quantum coherence √ C l 1 y ( z ) ) + C l 1 z ( x ) ) + C l 1 x ( η B | Π a y ( η B | Π a z ( η B | Π a x ( y ) ) > 6 C E y ( z ) ) + C E z ( x ) ) + C E x ( η B | Π a y ( η B | Π a z ( η B | Π a x ( y ) ) > 2 . 23 C S y ( z ) ) + C S z ( x ) ) + C S x ( η B | Π a y ( η B | Π a z ( η B | Π a x ( y ) ) > 2 D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

Steerability of pure entailed state p ρ P = p α | 00 i + 1 � α | 11 i All pure entangled states are maximally steerable C x ( η B | Π a y ( z ) ) + C y ( η B | Π a z ( x ) ) + C z ( η B | Π a x ( y ) ) = 3 D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

Steerability of werner state ρ W = p ρ S + (1 − p ) I 4 : Steerable under l 1 - norm . p > 0 . 82 : Steerable under relative entropy coherence p > 0 . 91 : Steerable under skew information p > 0 . 94 State Steerability : p > 0.58 D. Mondal, T. Pramanik, and A. K. Pati, arXiv: 1508.03770.

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 one 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.

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