Weak Measurement and Quantum Correlation

Arun Kumar Pati

Quantum Information and Computation GroupHarish-Chandra Research Institute

Allahabad 211 019, India

Arun Kumar Pati (HRI) 1 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Outlines

Quantum WorldWeak MeasurementsQuantum correlationSuper Quantum DiscordQuantum-Classical BoundaryConclusions

Arun Kumar Pati (HRI) 2 / 24

Quantum World

Even after more than 100 years, quantum theory still continues tosurprise us....

Linear superposition: A quantum system can remainsimultaneously in all possible allowed states.Entanglement: Two quantum systems can be in a stronglycorrelated state even if they are far apart (Einstein, Schrodinger).Non-locality: Correlations in entangled state cannot be explainedby local-realistic theory (Bell).Quantum Measurement : State vector collapses in a probabilisticway 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 tosurprise us....

Linear superposition: A quantum system can remainsimultaneously in all possible allowed states.Entanglement: Two quantum systems can be in a stronglycorrelated state even if they are far apart (Einstein, Schrodinger).Non-locality: Correlations in entangled state cannot be explainedby local-realistic theory (Bell).Quantum Measurement : State vector collapses in a probabilisticway 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 tosurprise us....

Linear superposition: A quantum system can remainsimultaneously in all possible allowed states.Entanglement: Two quantum systems can be in a stronglycorrelated state even if they are far apart (Einstein, Schrodinger).Non-locality: Correlations in entangled state cannot be explainedby local-realistic theory (Bell).Quantum Measurement : State vector collapses in a probabilisticway 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 tosurprise us....

Linear superposition: A quantum system can remainsimultaneously in all possible allowed states.Entanglement: Two quantum systems can be in a stronglycorrelated state even if they are far apart (Einstein, Schrodinger).Non-locality: Correlations in entangled state cannot be explainedby local-realistic theory (Bell).Quantum Measurement : State vector collapses in a probabilisticway 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 tosurprise us....

Linear superposition: A quantum system can remainsimultaneously in all possible allowed states.Entanglement: Two quantum systems can be in a stronglycorrelated state even if they are far apart (Einstein, Schrodinger).Non-locality: Correlations in entangled state cannot be explainedby local-realistic theory (Bell).Quantum Measurement : State vector collapses in a probabilisticway 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 tosurprise us....

Linear superposition: A quantum system can remainsimultaneously in all possible allowed states.Entanglement: Two quantum systems can be in a stronglycorrelated state even if they are far apart (Einstein, Schrodinger).Non-locality: Correlations in entangled state cannot be explainedby local-realistic theory (Bell).Quantum Measurement : State vector collapses in a probabilisticway to one of the eigenstate and coherence is lost.

Arun Kumar Pati (HRI) 3 / 24

Quantum Entanglement

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

Arun Kumar Pati (HRI) 4 / 24

Quantum Entanglement

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

Arun Kumar Pati (HRI) 4 / 24

Quantum Entanglement

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

Arun Kumar Pati (HRI) 4 / 24

Quantum Information

These are resources which can be used to design quantumcomputer, quantum information processor, quantumcommunication and quantum information technology.Merging of quantum mechanics and information theory —quantuminformation science – with important developments like quantumcryptography (Ekert 1991), quantum teleportation (Bennett et al1993), remote state preparation (Pati 1999) and many more.Quantum Correlations play important role.Entanglement and beyond (Quantum Discord) for compositesystems.

Arun Kumar Pati (HRI) 5 / 24

Quantum Information

These are resources which can be used to design quantumcomputer, quantum information processor, quantumcommunication and quantum information technology.Merging of quantum mechanics and information theory —quantuminformation science – with important developments like quantumcryptography (Ekert 1991), quantum teleportation (Bennett et al1993), remote state preparation (Pati 1999) and many more.Quantum Correlations play important role.Entanglement and beyond (Quantum Discord) for compositesystems.

Arun Kumar Pati (HRI) 5 / 24

Quantum Information

These are resources which can be used to design quantumcomputer, quantum information processor, quantumcommunication and quantum information technology.Merging of quantum mechanics and information theory —quantuminformation science – with important developments like quantumcryptography (Ekert 1991), quantum teleportation (Bennett et al1993), remote state preparation (Pati 1999) and many more.Quantum Correlations play important role.Entanglement and beyond (Quantum Discord) for compositesystems.

Arun Kumar Pati (HRI) 5 / 24

Quantum Information

These are resources which can be used to design quantumcomputer, quantum information processor, quantumcommunication and quantum information technology.Merging of quantum mechanics and information theory —quantuminformation science – with important developments like quantumcryptography (Ekert 1991), quantum teleportation (Bennett et al1993), remote state preparation (Pati 1999) and many more.Quantum Correlations play important role.Entanglement and beyond (Quantum Discord) for compositesystems.

Arun Kumar Pati (HRI) 5 / 24

Quantum Information

These are resources which can be used to design quantumcomputer, quantum information processor, quantumcommunication and quantum information technology.Merging of quantum mechanics and information theory —quantuminformation science – with important developments like quantumcryptography (Ekert 1991), quantum teleportation (Bennett et al1993), remote state preparation (Pati 1999) and many more.Quantum Correlations play important role.Entanglement and beyond (Quantum Discord) for compositesystems.

Arun Kumar Pati (HRI) 5 / 24

Quantum Information

These are resources which can be used to design quantumcomputer, quantum information processor, quantumcommunication and quantum information technology.Merging of quantum mechanics and information theory —quantuminformation science – with important developments like quantumcryptography (Ekert 1991), quantum teleportation (Bennett et al1993), remote state preparation (Pati 1999) and many more.Quantum Correlations play important role.Entanglement and beyond (Quantum Discord) for compositesystems.

Arun Kumar Pati (HRI) 5 / 24

Quantum Measurement

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

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

Interaction of system and apparatus realizes the transition:|ψ〉 ⊗ |φ〉 →

∑n cn|ψn〉 ⊗ |φn〉

Arun Kumar Pati (HRI) 6 / 24

Quantum Measurement

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

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

Interaction of system and apparatus realizes the transition:|ψ〉 ⊗ |φ〉 →

∑n cn|ψn〉 ⊗ |φn〉

Arun Kumar Pati (HRI) 6 / 24

Quantum Measurement

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

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

Interaction of system and apparatus realizes the transition:|ψ〉 ⊗ |φ〉 →

∑n cn|ψn〉 ⊗ |φn〉

Arun Kumar Pati (HRI) 6 / 24

Quantum Measurement

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

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

Interaction of system and apparatus realizes the transition:|ψ〉 ⊗ |φ〉 →

∑n cn|ψn〉 ⊗ |φn〉

Arun Kumar Pati (HRI) 6 / 24

Weak Measurement

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

Quantum state is preselected in |ψi〉 and allowed to interactweakly with apparatus.Measurement strength can be tuned and for “small g(t)” it iscalled ’weak measurement’.With postselection in |ψf 〉, apparatus state is shifted by an amountequal to the weak value 〈A〉w = 〈ψf |A|ψi 〉

〈ψf |ψi 〉.

Weak value can lie outside the spectrum of the observablemeasured, 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, wasintroduced by Aharonov et al.1

Quantum state is preselected in |ψi〉 and allowed to interactweakly with apparatus.Measurement strength can be tuned and for “small g(t)” it iscalled ’weak measurement’.With postselection in |ψf 〉, apparatus state is shifted by an amountequal to the weak value 〈A〉w = 〈ψf |A|ψi 〉

〈ψf |ψi 〉.

Weak value can lie outside the spectrum of the observablemeasured, 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, wasintroduced by Aharonov et al.1

Quantum state is preselected in |ψi〉 and allowed to interactweakly with apparatus.Measurement strength can be tuned and for “small g(t)” it iscalled ’weak measurement’.With postselection in |ψf 〉, apparatus state is shifted by an amountequal to the weak value 〈A〉w = 〈ψf |A|ψi 〉

〈ψf |ψi 〉.

Weak value can lie outside the spectrum of the observablemeasured, 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 disturbingit...”

Weak measurement gives a handle to explore the quantum worldwithout destroying superpositions.Weak measurement opens up a new window for understandingthe weirdness of quantum theory.Weak values have found numerous applications such as directmeasurement 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 disturbingit...”

Weak measurement gives a handle to explore the quantum worldwithout destroying superpositions.Weak measurement opens up a new window for understandingthe weirdness of quantum theory.Weak values have found numerous applications such as directmeasurement 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 disturbingit...”

Weak measurement gives a handle to explore the quantum worldwithout destroying superpositions.Weak measurement opens up a new window for understandingthe weirdness of quantum theory.Weak values have found numerous applications such as directmeasurement 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 stateρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1)

where a(±x) =√

1∓tanh x2 and

∑y=±x P(y)†P(y) = I.

These are called weak measurement operators because they donot 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 stateρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1)

where a(±x) =√

1∓tanh x2 and

∑y=±x P(y)†P(y) = I.

These are called weak measurement operators because they donot 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 stateρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1)

where a(±x) =√

1∓tanh x2 and

∑y=±x P(y)†P(y) = I.

These are called weak measurement operators because they donot 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 stateρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1)

where a(±x) =√

1∓tanh x2 and

∑y=±x P(y)†P(y) = I.

These are called weak measurement operators because they donot 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 stateρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1)

where a(±x) =√

1∓tanh x2 and

∑y=±x P(y)†P(y) = I.

These are called weak measurement operators because they donot 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 stateρ. It can be modeled using following operators,

P(±x) = a(±x)Π0 + a(∓x)Π1, (1)

where a(±x) =√

1∓tanh x2 and

∑y=±x P(y)†P(y) = I.

These are called weak measurement operators because they donot 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|0〉+ β|0〉) = αa(x)|0〉+ βa(−x)|1〉For small x , the distance between the initial state and state afterthe measurement is close to zero, i.e., action of P(±x) does notalter 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|0〉+ β|0〉) = αa(x)|0〉+ βa(−x)|1〉For small x , the distance between the initial state and state afterthe measurement is close to zero, i.e., action of P(±x) does notalter 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|0〉+ β|0〉) = αa(x)|0〉+ βa(−x)|1〉For small x , the distance between the initial state and state afterthe measurement is close to zero, i.e., action of P(±x) does notalter the state of the system much.

Arun Kumar Pati (HRI) 10 / 24

Features of Weak Measurement

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

But weak measurements act very gently, thereby, destroying onlya little amount of correlation between the subsystems of acomposite system.

This comes at the cost of inferring the state of the systemambiguously.

Arun Kumar Pati (HRI) 11 / 24

Features of Weak Measurement

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

But weak measurements act very gently, thereby, destroying onlya little amount of correlation between the subsystems of acomposite system.

This comes at the cost of inferring the state of the systemambiguously.

Arun Kumar Pati (HRI) 11 / 24

Features of Weak Measurement

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

But weak measurements act very gently, thereby, destroying onlya little amount of correlation between the subsystems of acomposite system.

This comes at the cost of inferring the state of the systemambiguously.

Arun Kumar Pati (HRI) 11 / 24

Features of Weak Measurement

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

But weak measurements act very gently, thereby, destroying onlya little amount of correlation between the subsystems of acomposite system.

This comes at the cost of inferring the state of the systemambiguously.

Arun Kumar Pati (HRI) 11 / 24

Quantum Discord

Given a composite state ρAB the mutual informationI(A : B) = S(A) + S(B)− S(AB) contains total correlation.Measurement on subsystem tries to extract information. Mutualinformation of postmeasured state is the classical correlation 3

J(A : B)

Difference between total correlation I(A : B) and classicalcorrelation J(A : B) is quantum discord 4

DB(A : B) = I(A : B)− J(A : B)

= minΠi

∑i

piS(ρA|ΠBi

)− S(A|B), (2)

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 informationI(A : B) = S(A) + S(B)− S(AB) contains total correlation.Measurement on subsystem tries to extract information. Mutualinformation of postmeasured state is the classical correlation 3

J(A : B)

Difference between total correlation I(A : B) and classicalcorrelation J(A : B) is quantum discord 4

DB(A : B) = I(A : B)− J(A : B)

= minΠi

∑i

piS(ρA|ΠBi

)− S(A|B), (2)

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 informationI(A : B) = S(A) + S(B)− S(AB) contains total correlation.Measurement on subsystem tries to extract information. Mutualinformation of postmeasured state is the classical correlation 3

J(A : B)

Difference between total correlation I(A : B) and classicalcorrelation J(A : B) is quantum discord 4

DB(A : B) = I(A : B)− J(A : B)

= minΠi

∑i

piS(ρA|ΠBi

)− S(A|B), (2)

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 informationI(A : B) = S(A) + S(B)− S(AB) contains total correlation.Measurement on subsystem tries to extract information. Mutualinformation of postmeasured state is the classical correlation 3

J(A : B)

Difference between total correlation I(A : B) and classicalcorrelation J(A : B) is quantum discord 4

DB(A : B) = I(A : B)− J(A : B)

= minΠi

∑i

piS(ρA|ΠBi

)− S(A|B), (2)

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 thatcannot be extracted by doing measurement on one of thesubsystem.It depends on the observer who performs measurement onsubsystem.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 inspeed-up in computation and many information processing tasks.

We also know that weak measurements can maintainquantumness.

Can weak measurements reveal more quantumness about thestate?Indeed, weak measurements can reveal more. Quantum discordwith 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 inspeed-up in computation and many information processing tasks.

We also know that weak measurements can maintainquantumness.

Can weak measurements reveal more quantumness about thestate?Indeed, weak measurements can reveal more. Quantum discordwith 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 inspeed-up in computation and many information processing tasks.

We also know that weak measurements can maintainquantumness.

Can weak measurements reveal more quantumness about thestate?Indeed, weak measurements can reveal more. Quantum discordwith 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 inspeed-up in computation and many information processing tasks.

We also know that weak measurements can maintainquantumness.

Can weak measurements reveal more quantumness about thestate?Indeed, weak measurements can reveal more. Quantum discordwith 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 inspeed-up in computation and many information processing tasks.

We also know that weak measurements can maintainquantumness.

Can weak measurements reveal more quantumness about thestate?Indeed, weak measurements can reveal more. Quantum discordwith 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 MeasurementsSuper Quantum Discord

The SQD is defined as

DBw (A,B) := min

{PB(x)}Sw (ρA|{PB(x)})− S(A|B), (3)

where

Sw (ρA|{PB(x)}) =∑

{y=x ,−x}

p(y)S(ρA|PB(y)), (4)

with

ρA|PB(±x) =TrB[(I ⊗ PB(±x))ρAB(I ⊗ PB(±x))][

p(±x) = TrAB{(I ⊗ PB(±x))ρAB(I ⊗ PB(±x))}] . (5)

Arun Kumar Pati (HRI) 15 / 24

Weak MeasurementsSuper Quantum Discord

The SQD is defined as

DBw (A,B) := min

{PB(x)}Sw (ρA|{PB(x)})− S(A|B), (3)

where

Sw (ρA|{PB(x)}) =∑

{y=x ,−x}

p(y)S(ρA|PB(y)), (4)

with

ρA|PB(±x) =TrB[(I ⊗ PB(±x))ρAB(I ⊗ PB(±x))][

p(±x) = TrAB{(I ⊗ PB(±x))ρAB(I ⊗ PB(±x))}] . (5)

Arun Kumar Pati (HRI) 15 / 24

Weak MeasurementsSuper Quantum Discord

The SQD is defined as

DBw (A,B) := min

{PB(x)}Sw (ρA|{PB(x)})− S(A|B), (3)

where

Sw (ρA|{PB(x)}) =∑

{y=x ,−x}

p(y)S(ρA|PB(y)), (4)

with

ρA|PB(±x) =TrB[(I ⊗ PB(±x))ρAB(I ⊗ PB(±x))][

p(±x) = TrAB{(I ⊗ PB(±x))ρAB(I ⊗ PB(±x))}] . (5)

Arun Kumar Pati (HRI) 15 / 24

Weak MeasurementsSuper Quantum Discord: Properties

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

For pure maximally entangled state |Ψ〉 = 1√2

(|0〉|1〉 − |1〉|0〉,normal discord D(A : B) = 1.For weak measurement at x = 0.2, super discordDw (A : B) = 1.4689, which is greater than the entanglemententropy.

Arun Kumar Pati (HRI) 16 / 24

Weak MeasurementsSuper Quantum Discord: Properties

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

For pure maximally entangled state |Ψ〉 = 1√2

(|0〉|1〉 − |1〉|0〉,normal discord D(A : B) = 1.For weak measurement at x = 0.2, super discordDw (A : B) = 1.4689, which is greater than the entanglemententropy.

Arun Kumar Pati (HRI) 16 / 24

Super Quantum Discord for Werner State

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

0 0.2 0.4 0.6 0.8 1z

0

0.5

1

1.5

2

Dis

cord

z=1/3 Super DiscordNormal Discord

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

Arun Kumar Pati (HRI) 17 / 24

Super Quantum DiscordApplications

The necessary and sufficient conditions for vanishing of SQD isfound and there, some application of SQD to optimal statediscrimination is shown.6

The vanishing of super discord only for product states supportsthe evidences where total correlations behave as if it wereexclusively 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 DiscordApplications

The necessary and sufficient conditions for vanishing of SQD isfound and there, some application of SQD to optimal statediscrimination is shown.6

The vanishing of super discord only for product states supportsthe evidences where total correlations behave as if it wereexclusively 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 differencebetween 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 correlationbecomes zero.

The extra quantum correlation is revealed only with weakmeasurement.

Arun Kumar Pati (HRI) 19 / 24

Extra Quantum Correlation

The extra quantum correlation is defined as the differencebetween 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 correlationbecomes zero.

The extra quantum correlation is revealed only with weakmeasurement.

Arun Kumar Pati (HRI) 19 / 24

Extra Quantum Correlation

The extra quantum correlation is defined as the differencebetween 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 correlationbecomes zero.

The extra quantum correlation is revealed only with weakmeasurement.

Arun Kumar Pati (HRI) 19 / 24

Quantum-Classical Boundary

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

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Q

Total Correlation

Decreasing x

Q

x = 0

Q = Quantum correlationJ = Classical correlation

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Conclusions

Weak measurements can reveal more quantum correlation.

Super Quantum Discord is a monotonically decreasing function ofthe measurement strength.

Weak measurements can be used to capture the extra quantumcorrelation.

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Conclusions

Weak measurements can reveal more quantum correlation.

Super Quantum Discord is a monotonically decreasing function ofthe measurement strength.

Weak measurements can be used to capture the extra quantumcorrelation.

Arun Kumar Pati (HRI) 22 / 24

Conclusions

Weak measurements can reveal more quantum correlation.

Super Quantum Discord is a monotonically decreasing function ofthe measurement strength.

Weak measurements can be used to capture the extra quantumcorrelation.

Arun Kumar Pati (HRI) 22 / 24

Weak measurements have found numerous applications startingfrom the precision quantum measurements to foundationalquestions of quantum mechanics.

Super Discord can be used to harness quantumness of acomposite state.

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

Arun Kumar Pati (HRI) 23 / 24

Weak measurements have found numerous applications startingfrom the precision quantum measurements to foundationalquestions of quantum mechanics.

Super Discord can be used to harness quantumness of acomposite state.

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

Arun Kumar Pati (HRI) 23 / 24

Weak measurements have found numerous applications startingfrom the precision quantum measurements to foundationalquestions of quantum mechanics.

Super Discord can be used to harness quantumness of acomposite state.

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

Arun Kumar Pati (HRI) 23 / 24

In the darkness if not one, but thousand lamps can lighten a path.....

THANK YOUArun Kumar Pati (HRI) 24 / 24