Weak Measurement and Quantum Correlation

Arun Kumar Pati

Quantum Information and Computation Group Harish-Chandra Research Institute Allahabad 211 019, India

Arun Kumar Pati (HRI) 1 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Outlines

Quantum World Weak Measurements Quantum correlation Super Quantum Discord Quantum-Classical Boundary Conclusions

Arun Kumar Pati (HRI) 2 / 24 Quantum World

Even after more than 100 years, quantum theory still continues to surprise us.... Linear superposition: A quantum system can remain simultaneously in all possible allowed states. Entanglement: Two quantum systems can be in a strongly correlated state even if they are far apart (Einstein, Schrodinger).¨ Non-locality: Correlations in entangled state cannot be explained by local-realistic theory (Bell). Quantum Measurement : State vector collapses in a probabilistic way to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24 Quantum World

Even after more than 100 years, quantum theory still continues to surprise us.... Linear superposition: A quantum system can remain simultaneously in all possible allowed states. Entanglement: Two quantum systems can be in a strongly correlated state even if they are far apart (Einstein, Schrodinger).¨ Non-locality: Correlations in entangled state cannot be explained by local-realistic theory (Bell). Quantum Measurement : State vector collapses in a probabilistic way to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24 Quantum World

Even after more than 100 years, quantum theory still continues to surprise us.... Linear superposition: A quantum system can remain simultaneously in all possible allowed states. Entanglement: Two quantum systems can be in a strongly correlated state even if they are far apart (Einstein, Schrodinger).¨ Non-locality: Correlations in entangled state cannot be explained by local-realistic theory (Bell). Quantum Measurement : State vector collapses in a probabilistic way to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24 Quantum World

Even after more than 100 years, quantum theory still continues to surprise us.... Linear superposition: A quantum system can remain simultaneously in all possible allowed states. Entanglement: Two quantum systems can be in a strongly correlated state even if they are far apart (Einstein, Schrodinger).¨ Non-locality: Correlations in entangled state cannot be explained by local-realistic theory (Bell). Quantum Measurement : State vector collapses in a probabilistic way to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24 Quantum World

Even after more than 100 years, quantum theory still continues to surprise us.... Linear superposition: A quantum system can remain simultaneously in all possible allowed states. Entanglement: Two quantum systems can be in a strongly correlated state even if they are far apart (Einstein, Schrodinger).¨ Non-locality: Correlations in entangled state cannot be explained by local-realistic theory (Bell). Quantum Measurement : State vector collapses in a probabilistic way to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24 Quantum World

Even after more than 100 years, quantum theory still continues to surprise us.... Linear superposition: A quantum system can remain simultaneously in all possible allowed states. Entanglement: Two quantum systems can be in a strongly correlated state even if they are far apart (Einstein, Schrodinger).¨ Non-locality: Correlations in entangled state cannot be explained by local-realistic theory (Bell). Quantum Measurement : State vector collapses in a probabilistic way to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24 Quantum Entanglement

Schrodinger¨ (1935): “When two systems ... enter into temporary physical interaction ... and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (the quantum states) have become entangled.

Arun Kumar Pati (HRI) 4 / 24 Quantum Entanglement

Schrodinger¨ (1935): “When two systems ... enter into temporary physical interaction ... and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (the quantum states) have become entangled.

Arun Kumar Pati (HRI) 4 / 24 Quantum Entanglement

Schrodinger¨ (1935): “When two systems ... enter into temporary physical interaction ... and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (the quantum states) have become entangled.

Arun Kumar Pati (HRI) 4 / 24 Quantum Information

These are resources which can be used to design quantum computer, quantum information processor, quantum communication and quantum information technology. Merging of quantum mechanics and information theory —quantum information science – with important developments like quantum cryptography (Ekert 1991), quantum teleportation (Bennett et al 1993), remote state preparation (Pati 1999) and many more. Quantum Correlations play important role. Entanglement and beyond (Quantum Discord) for composite systems.

Arun Kumar Pati (HRI) 5 / 24 Quantum Information

These are resources which can be used to design quantum computer, quantum information processor, quantum communication and quantum information technology. Merging of quantum mechanics and information theory —quantum information science – with important developments like quantum cryptography (Ekert 1991), quantum teleportation (Bennett et al 1993), remote state preparation (Pati 1999) and many more. Quantum Correlations play important role. Entanglement and beyond (Quantum Discord) for composite systems.

Arun Kumar Pati (HRI) 5 / 24 Quantum Information

These are resources which can be used to design quantum computer, quantum information processor, quantum communication and quantum information technology. Merging of quantum mechanics and information theory —quantum information science – with important developments like quantum cryptography (Ekert 1991), quantum teleportation (Bennett et al 1993), remote state preparation (Pati 1999) and many more. Quantum Correlations play important role. Entanglement and beyond (Quantum Discord) for composite systems.

Arun Kumar Pati (HRI) 5 / 24 Quantum Information

These are resources which can be used to design quantum computer, quantum information processor, quantum communication and quantum information technology. Merging of quantum mechanics and information theory —quantum information science – with important developments like quantum cryptography (Ekert 1991), quantum teleportation (Bennett et al 1993), remote state preparation (Pati 1999) and many more. Quantum Correlations play important role. Entanglement and beyond (Quantum Discord) for composite systems.

Arun Kumar Pati (HRI) 5 / 24 Quantum Information

These are resources which can be used to design quantum computer, quantum information processor, quantum communication and quantum information technology. Merging of quantum mechanics and information theory —quantum information science – with important developments like quantum cryptography (Ekert 1991), quantum teleportation (Bennett et al 1993), remote state preparation (Pati 1999) and many more. Quantum Correlations play important role. Entanglement and beyond (Quantum Discord) for composite systems.

Arun Kumar Pati (HRI) 5 / 24 Quantum Information

These are resources which can be used to design quantum computer, quantum information processor, quantum communication and quantum information technology. Merging of quantum mechanics and information theory —quantum information science – with important developments like quantum cryptography (Ekert 1991), quantum teleportation (Bennett et al 1993), remote state preparation (Pati 1999) and many more. Quantum Correlations play important role. Entanglement and beyond (Quantum Discord) for composite systems.

Arun Kumar Pati (HRI) 5 / 24 Quantum Measurement

von-Neumann’s model treats both the system and the measuring apparatus as quantum systems.

Measurement correlates system and apparatus states. Measurement process can be described by a Hamiltonian HT = HS + HA + Hint, where Hint = g(t) · OS ⊗ QA.

Interaction of system and apparatus realizes the transition: P |ψi ⊗ |φi → n cn|ψni ⊗ |φni

Arun Kumar Pati (HRI) 6 / 24 Quantum Measurement

von-Neumann’s model treats both the system and the measuring apparatus as quantum systems.

Measurement correlates system and apparatus states. Measurement process can be described by a Hamiltonian HT = HS + HA + Hint, where Hint = g(t) · OS ⊗ QA.

Interaction of system and apparatus realizes the transition: P |ψi ⊗ |φi → n cn|ψni ⊗ |φni

Arun Kumar Pati (HRI) 6 / 24 Quantum Measurement

von-Neumann’s model treats both the system and the measuring apparatus as quantum systems.

Measurement correlates system and apparatus states. Measurement process can be described by a Hamiltonian HT = HS + HA + Hint, where Hint = g(t) · OS ⊗ QA.

Interaction of system and apparatus realizes the transition: P |ψi ⊗ |φi → n cn|ψni ⊗ |φni

Arun Kumar Pati (HRI) 6 / 24 Quantum Measurement

von-Neumann’s model treats both the system and the measuring apparatus as quantum systems.

Measurement correlates system and apparatus states. Measurement process can be described by a Hamiltonian HT = HS + HA + Hint, where Hint = g(t) · OS ⊗ QA.

Interaction of system and apparatus realizes the transition: P |ψi ⊗ |φi → n cn|ψni ⊗ |φni

Arun Kumar Pati (HRI) 6 / 24 Weak Measurement

The concept of the weak measurements, for the first time, was introduced by Aharonov et al.1

Quantum state is preselected in |ψi i and allowed to interact weakly with apparatus. Measurement strength can be tuned and for “small g(t)” it is called ’weak measurement’.

With postselection in |ψf i, apparatus state is shifted by an amount equal to the weak value hAiw = hψf |A|ψi i . hψf |ψi i Weak value can lie outside the spectrum of the observable measured, unlike the expectation value of the observable.

1 Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). Arun Kumar Pati (HRI) 7 / 24 Weak Measurement

The concept of the weak measurements, for the first time, was introduced by Aharonov et al.1

Quantum state is preselected in |ψi i and allowed to interact weakly with apparatus. Measurement strength can be tuned and for “small g(t)” it is called ’weak measurement’.

With postselection in |ψf i, apparatus state is shifted by an amount equal to the weak value hAiw = hψf |A|ψi i . hψf |ψi i Weak value can lie outside the spectrum of the observable measured, unlike the expectation value of the observable.

1 Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). Arun Kumar Pati (HRI) 7 / 24 Weak Measurement

The concept of the weak measurements, for the first time, was introduced by Aharonov et al.1

Quantum state is preselected in |ψi i and allowed to interact weakly with apparatus. Measurement strength can be tuned and for “small g(t)” it is called ’weak measurement’.

With postselection in |ψf i, apparatus state is shifted by an amount equal to the weak value hAiw = hψf |A|ψi i . hψf |ψi i Weak value can lie outside the spectrum of the observable measured, unlike the expectation value of the observable.

1 Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). Arun Kumar Pati (HRI) 7 / 24 Weak Measurement

Aharonov: “Weak measurement finds what is there without disturbing it...” Weak measurement gives a handle to explore the quantum world without destroying superpositions. Weak measurement opens up a new window for understanding the weirdness of quantum theory. Weak values have found numerous applications such as direct measurement of the wave function of single photon (Nature, 2012), understanding non-locality, amplification of weak signal etc (Science, 2011).

Arun Kumar Pati (HRI) 8 / 24 Weak Measurement

Aharonov: “Weak measurement finds what is there without disturbing it...” Weak measurement gives a handle to explore the quantum world without destroying superpositions. Weak measurement opens up a new window for understanding the weirdness of quantum theory. Weak values have found numerous applications such as direct measurement of the wave function of single photon (Nature, 2012), understanding non-locality, amplification of weak signal etc (Science, 2011).

Arun Kumar Pati (HRI) 8 / 24 Weak Measurement

Aharonov: “Weak measurement finds what is there without disturbing it...” Weak measurement gives a handle to explore the quantum world without destroying superpositions. Weak measurement opens up a new window for understanding the weirdness of quantum theory. Weak values have found numerous applications such as direct measurement of the wave function of single photon (Nature, 2012), understanding non-locality, amplification of weak signal etc (Science, 2011).

Arun Kumar Pati (HRI) 8 / 24 Weak Measurements Without Postselection

The weak measurements are universal.2

Consider the measurement of projectors {Π0, Π1} on a qubit state ρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1) q 1∓tanh x P † where a(±x) = 2 and y=±x P(y) P(y) = I. These are called weak measurement operators because they do not cause complete collapse.

2 O. Oreshkov and T. A. Brun, Phys. Rev. Lett. 95, 110409 (2005). Arun Kumar Pati (HRI) 9 / 24 Weak Measurements Without Postselection

The weak measurements are universal.2

Consider the measurement of projectors {Π0, Π1} on a qubit state ρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1) q 1∓tanh x P † where a(±x) = 2 and y=±x P(y) P(y) = I. These are called weak measurement operators because they do not cause complete collapse.

2 O. Oreshkov and T. A. Brun, Phys. Rev. Lett. 95, 110409 (2005). Arun Kumar Pati (HRI) 9 / 24 Weak Measurements Without Postselection

The weak measurements are universal.2

Consider the measurement of projectors {Π0, Π1} on a qubit state ρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1) q 1∓tanh x P † where a(±x) = 2 and y=±x P(y) P(y) = I. These are called weak measurement operators because they do not cause complete collapse.

2 O. Oreshkov and T. A. Brun, Phys. Rev. Lett. 95, 110409 (2005). Arun Kumar Pati (HRI) 9 / 24 Weak Measurements Without Postselection

The weak measurements are universal.2

Consider the measurement of projectors {Π0, Π1} on a qubit state ρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1) q 1∓tanh x P † where a(±x) = 2 and y=±x P(y) P(y) = I. These are called weak measurement operators because they do not cause complete collapse.

2 O. Oreshkov and T. A. Brun, Phys. Rev. Lett. 95, 110409 (2005). Arun Kumar Pati (HRI) 9 / 24 Weak Measurements Without Postselection

The weak measurements are universal.2

Consider the measurement of projectors {Π0, Π1} on a qubit state ρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1) q 1∓tanh x P † where a(±x) = 2 and y=±x P(y) P(y) = I. These are called weak measurement operators because they do not cause complete collapse.

2 O. Oreshkov and T. A. Brun, Phys. Rev. Lett. 95, 110409 (2005). Arun Kumar Pati (HRI) 9 / 24 Weak Measurements Without Postselection

The weak measurements are universal.2

Consider the measurement of projectors {Π0, Π1} on a qubit state ρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1) q 1∓tanh x P † where a(±x) = 2 and y=±x P(y) P(y) = I. These are called weak measurement operators because they do not cause complete collapse.

2 O. Oreshkov and T. A. Brun, Phys. Rev. Lett. 95, 110409 (2005). Arun Kumar Pati (HRI) 9 / 24 Weak Measurements

P(x) and P(−x) constitute a valid measurement operator. Application of P(x) on a qubit: P(x)(α|0i + β|0i) = αa(x)|0i + βa(−x)|1i For small x, the distance between the initial state and state after the measurement is close to zero, i.e., action of P(±x) does not alter the state of the system much.

Arun Kumar Pati (HRI) 10 / 24 Weak Measurements

P(x) and P(−x) constitute a valid measurement operator. Application of P(x) on a qubit: P(x)(α|0i + β|0i) = αa(x)|0i + βa(−x)|1i For small x, the distance between the initial state and state after the measurement is close to zero, i.e., action of P(±x) does not alter the state of the system much.

Arun Kumar Pati (HRI) 10 / 24 Weak Measurements

P(x) and P(−x) constitute a valid measurement operator. Application of P(x) on a qubit: P(x)(α|0i + β|0i) = αa(x)|0i + βa(−x)|1i For small x, the distance between the initial state and state after the measurement is close to zero, i.e., action of P(±x) does not alter the state of the system much.

Arun Kumar Pati (HRI) 10 / 24 Features of Weak Measurement

The local projective measurements on apparatus destroy the quantum correlation in the composite state of system and apparatus and make it classical.

But weak measurements act very gently, thereby, destroying only a little amount of correlation between the subsystems of a composite system.

This comes at the cost of inferring the state of the system ambiguously.

Arun Kumar Pati (HRI) 11 / 24 Features of Weak Measurement

The local projective measurements on apparatus destroy the quantum correlation in the composite state of system and apparatus and make it classical.

But weak measurements act very gently, thereby, destroying only a little amount of correlation between the subsystems of a composite system.

This comes at the cost of inferring the state of the system ambiguously.

Arun Kumar Pati (HRI) 11 / 24 Features of Weak Measurement

The local projective measurements on apparatus destroy the quantum correlation in the composite state of system and apparatus and make it classical.

But weak measurements act very gently, thereby, destroying only a little amount of correlation between the subsystems of a composite system.

This comes at the cost of inferring the state of the system ambiguously.

Arun Kumar Pati (HRI) 11 / 24 Features of Weak Measurement

The local projective measurements on apparatus destroy the quantum correlation in the composite state of system and apparatus and make it classical.

But weak measurements act very gently, thereby, destroying only a little amount of correlation between the subsystems of a composite system.

This comes at the cost of inferring the state of the system ambiguously.

Arun Kumar Pati (HRI) 11 / 24 Quantum Discord

Given a composite state ρAB the mutual information I(A : B) = S(A) + S(B) − S(AB) contains total correlation. Measurement on subsystem tries to extract information. Mutual information of postmeasured state is the classical correlation 3 J(A : B) Difference between total correlation I(A : B) and classical correlation J(A : B) is quantum discord 4

DB(A : B) = I(A : B) − J(A : B) X = min pi S(ρA|ΠB ) − S(A|B), (2) Πi i i

where S(A|B) = S(ρAB) − S(ρB). 3 L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). 4 H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). Arun Kumar Pati (HRI) 12 / 24 Quantum Discord

Given a composite state ρAB the mutual information I(A : B) = S(A) + S(B) − S(AB) contains total correlation. Measurement on subsystem tries to extract information. Mutual information of postmeasured state is the classical correlation 3 J(A : B) Difference between total correlation I(A : B) and classical correlation J(A : B) is quantum discord 4

DB(A : B) = I(A : B) − J(A : B) X = min pi S(ρA|ΠB ) − S(A|B), (2) Πi i i

where S(A|B) = S(ρAB) − S(ρB). 3 L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). 4 H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). Arun Kumar Pati (HRI) 12 / 24 Quantum Discord

Given a composite state ρAB the mutual information I(A : B) = S(A) + S(B) − S(AB) contains total correlation. Measurement on subsystem tries to extract information. Mutual information of postmeasured state is the classical correlation 3 J(A : B) Difference between total correlation I(A : B) and classical correlation J(A : B) is quantum discord 4

DB(A : B) = I(A : B) − J(A : B) X = min pi S(ρA|ΠB ) − S(A|B), (2) Πi i i

where S(A|B) = S(ρAB) − S(ρB). 3 L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). 4 H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). Arun Kumar Pati (HRI) 12 / 24 Quantum Discord

Given a composite state ρAB the mutual information I(A : B) = S(A) + S(B) − S(AB) contains total correlation. Measurement on subsystem tries to extract information. Mutual information of postmeasured state is the classical correlation 3 J(A : B) Difference between total correlation I(A : B) and classical correlation J(A : B) is quantum discord 4

DB(A : B) = I(A : B) − J(A : B) X = min pi S(ρA|ΠB ) − S(A|B), (2) Πi i i

where S(A|B) = S(ρAB) − S(ρB). 3 L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). 4 H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). Arun Kumar Pati (HRI) 12 / 24 Quantum discord represents the amount of information that cannot be extracted by doing measurement on one of the subsystem. It depends on the observer who performs measurement on subsystem. For pure entangled state discord is entanglement entropy.

Arun Kumar Pati (HRI) 13 / 24 Super Quantum Discord

We know quantum correlations can play important roles in speed-up in computation and many information processing tasks.

We also know that weak measurements can maintain quantumness.

Can weak measurements reveal more quantumness about the state? Indeed, weak measurements can reveal more. Quantum discord with weak measurement –Super Quantum Discord (SQD).5

5 U. Singh and A. K. Pati, quant-ph/1211.0939 (2012). Arun Kumar Pati (HRI) 14 / 24 Super Quantum Discord

We know quantum correlations can play important roles in speed-up in computation and many information processing tasks.

We also know that weak measurements can maintain quantumness.

Can weak measurements reveal more quantumness about the state? Indeed, weak measurements can reveal more. Quantum discord with weak measurement –Super Quantum Discord (SQD).5

5 U. Singh and A. K. Pati, quant-ph/1211.0939 (2012). Arun Kumar Pati (HRI) 14 / 24 Super Quantum Discord

We know quantum correlations can play important roles in speed-up in computation and many information processing tasks.

We also know that weak measurements can maintain quantumness.

Can weak measurements reveal more quantumness about the state? Indeed, weak measurements can reveal more. Quantum discord with weak measurement –Super Quantum Discord (SQD).5

5 U. Singh and A. K. Pati, quant-ph/1211.0939 (2012). Arun Kumar Pati (HRI) 14 / 24 Super Quantum Discord

We know quantum correlations can play important roles in speed-up in computation and many information processing tasks.

We also know that weak measurements can maintain quantumness.

Can weak measurements reveal more quantumness about the state? Indeed, weak measurements can reveal more. Quantum discord with weak measurement –Super Quantum Discord (SQD).5

5 U. Singh and A. K. Pati, quant-ph/1211.0939 (2012). Arun Kumar Pati (HRI) 14 / 24 Super Quantum Discord

We know quantum correlations can play important roles in speed-up in computation and many information processing tasks.

We also know that weak measurements can maintain quantumness.

Can weak measurements reveal more quantumness about the state? Indeed, weak measurements can reveal more. Quantum discord with weak measurement –Super Quantum Discord (SQD).5

5 U. Singh and A. K. Pati, quant-ph/1211.0939 (2012). Arun Kumar Pati (HRI) 14 / 24 Weak Measurements Super Quantum Discord

The SQD is defined as

B Dw (A, B) := min Sw (ρA|{PB(x)}) − S(A|B), (3) {PB(x)}

where X Sw (ρA|{PB(x)}) = p(y)S(ρA|PB(y)), (4) {y=x,−x}

with

B B TrB[(I ⊗ P (±x))ρAB(I ⊗ P (±x))] ρ B = . (5) A|P (±x)  B B  p(±x) = TrAB{(I ⊗ P (±x))ρAB(I ⊗ P (±x))}

Arun Kumar Pati (HRI) 15 / 24 Weak Measurements Super Quantum Discord

The SQD is defined as

B Dw (A, B) := min Sw (ρA|{PB(x)}) − S(A|B), (3) {PB(x)}

where X Sw (ρA|{PB(x)}) = p(y)S(ρA|PB(y)), (4) {y=x,−x}

with

B B TrB[(I ⊗ P (±x))ρAB(I ⊗ P (±x))] ρ B = . (5) A|P (±x)  B B  p(±x) = TrAB{(I ⊗ P (±x))ρAB(I ⊗ P (±x))}

Arun Kumar Pati (HRI) 15 / 24 Weak Measurements Super Quantum Discord

The SQD is defined as

B Dw (A, B) := min Sw (ρA|{PB(x)}) − S(A|B), (3) {PB(x)}

where X Sw (ρA|{PB(x)}) = p(y)S(ρA|PB(y)), (4) {y=x,−x}

with

B B TrB[(I ⊗ P (±x))ρAB(I ⊗ P (±x))] ρ B = . (5) A|P (±x)  B B  p(±x) = TrAB{(I ⊗ P (±x))ρAB(I ⊗ P (±x))}

Arun Kumar Pati (HRI) 15 / 24 Weak Measurements Super Quantum Discord: Properties

Theorem : Given a bipartite state ρAB, the super quantum discord (SQD) revealed by the weak measurement is always greater than or equal to the normal quantum discord with the strong measurement, i.e., Dw (A : B) ≥ D(A : B).

For pure maximally entangled state |Ψi = √1 (|0i|1i − |1i|0i, 2 normal discord D(A : B) = 1. For weak measurement at x = 0.2, super discord Dw (A : B) = 1.4689, which is greater than the entanglement entropy.

Arun Kumar Pati (HRI) 16 / 24 Weak Measurements Super Quantum Discord: Properties

Theorem : Given a bipartite state ρAB, the super quantum discord (SQD) revealed by the weak measurement is always greater than or equal to the normal quantum discord with the strong measurement, i.e., Dw (A : B) ≥ D(A : B).

For pure maximally entangled state |Ψi = √1 (|0i|1i − |1i|0i, 2 normal discord D(A : B) = 1. For weak measurement at x = 0.2, super discord Dw (A : B) = 1.4689, which is greater than the entanglement entropy.

Arun Kumar Pati (HRI) 16 / 24 Super Quantum Discord for Werner State

(1−z) SQD for a mixture of pure and random state ρ = z|ΨihΨ| + 4 I.

2 Super Discord z=1/3 Normal Discord 1.5

1 Discord

0.5

0 0 0.2 0.4 0.6 0.8 1 z

Figure: The super and the normal discords as a function of z for the Werner state at x = 0.2.

Arun Kumar Pati (HRI) 17 / 24 Super Quantum Discord Applications

The necessary and sufficient conditions for vanishing of SQD is found and there, some application of SQD to optimal state discrimination is shown.6

The vanishing of super discord only for product states supports the evidences where total correlations behave as if it were exclusively quantum 7.

6 B. Li, L. Chen and H. Fan, quant-ph/1301.7500 (2013). 7 C. H. Bennett et al, Phys. Rev. A 83, 012312 (2012) Arun Kumar Pati (HRI) 18 / 24 Super Quantum Discord Applications

The necessary and sufficient conditions for vanishing of SQD is found and there, some application of SQD to optimal state discrimination is shown.6

The vanishing of super discord only for product states supports the evidences where total correlations behave as if it were exclusively quantum 7.

6 B. Li, L. Chen and H. Fan, quant-ph/1301.7500 (2013). 7 C. H. Bennett et al, Phys. Rev. A 83, 012312 (2012) Arun Kumar Pati (HRI) 18 / 24 Extra Quantum Correlation

The extra quantum correlation is defined as the difference between the SQD and the normal discord in the bipartite state, i.e., ∆(ρAB) = Dw (ρAB) − Ds(ρAB).

In the strong measurement limit the extra quantum correlation becomes zero.

The extra quantum correlation is revealed only with weak measurement.

Arun Kumar Pati (HRI) 19 / 24 Extra Quantum Correlation

The extra quantum correlation is defined as the difference between the SQD and the normal discord in the bipartite state, i.e., ∆(ρAB) = Dw (ρAB) − Ds(ρAB).

In the strong measurement limit the extra quantum correlation becomes zero.

The extra quantum correlation is revealed only with weak measurement.

Arun Kumar Pati (HRI) 19 / 24 Extra Quantum Correlation

The extra quantum correlation is defined as the difference between the SQD and the normal discord in the bipartite state, i.e., ∆(ρAB) = Dw (ρAB) − Ds(ρAB).

In the strong measurement limit the extra quantum correlation becomes zero.

The extra quantum correlation is revealed only with weak measurement.

Arun Kumar Pati (HRI) 19 / 24 Quantum-Classical Boundary

Our results shows that quantum-classical boundary depends on the measurement strength. If we perform strong measurement, we make it classical. Quantum correlation (the inaccessible information) depends on the measurement strength and on the observer. By weakly measuring a system, it can reveal more quantum correlation. Normal discord is residual quantumness that remains inaccessible for a local observer.

Arun Kumar Pati (HRI) 20 / 24 Total Correlation

Decreasing x x = 0 Q Q

Q = Quantum correlation J = Classical correlation

Arun Kumar Pati (HRI) 21 / 24 Conclusions

Weak measurements can reveal more quantum correlation.

Super Quantum Discord is a monotonically decreasing function of the measurement strength.

Weak measurements can be used to capture the extra quantum correlation.

Arun Kumar Pati (HRI) 22 / 24 Conclusions

Weak measurements can reveal more quantum correlation.

Super Quantum Discord is a monotonically decreasing function of the measurement strength.

Weak measurements can be used to capture the extra quantum correlation.

Arun Kumar Pati (HRI) 22 / 24 Conclusions

Weak measurements can reveal more quantum correlation.

Super Quantum Discord is a monotonically decreasing function of the measurement strength.

Weak measurements can be used to capture the extra quantum correlation.

Arun Kumar Pati (HRI) 22 / 24 Weak measurements have found numerous applications starting from the precision quantum measurements to foundational questions of quantum mechanics.

Super Discord can be used to harness quantumness of a composite state.

In future, it can be a useful resource for quantum information processing tasks.

Arun Kumar Pati (HRI) 23 / 24 Weak measurements have found numerous applications starting from the precision quantum measurements to foundational questions of quantum mechanics.

Super Discord can be used to harness quantumness of a composite state.

In future, it can be a useful resource for quantum information processing tasks.

Arun Kumar Pati (HRI) 23 / 24 Weak measurements have found numerous applications starting from the precision quantum measurements to foundational questions of quantum mechanics.

Super Discord can be used to harness quantumness of a composite state.

In future, it can be a useful resource for quantum information processing tasks.

Arun Kumar Pati (HRI) 23 / 24 In the darkness if not one, but thousand lamps can lighten a path.....

THANK YOU

Arun Kumar Pati (HRI) 24 / 24