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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.
Arun Kumar Pati (HRI) 20 / 24
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.
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
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