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Characterization of local quantum processes by Local Quantum Uncertainty

Indrani Chattopadhyay Department of Applied Mathematics, University of Calcutta

Quantum Disentanglement

• This is a local quantum process.
• It is defined on composite system consists of two or more

subsystems, so that resulting state is separable.The disentangling machine(DM) is then defined as

• It assumed to preserve the local properties of the state, by

preserving the state of each subsystem.

d DM

ρ ρ e   

       

d A e A B d B e B A

ρ Tr ρ Tr ρ ρ Tr ρ Tr ρ    

• An exact universal disentangling process may defined

in two ways. (i) The process may defined to convert every entangled input states to some product states. (ii) Disentangling process, that transforms any entangled input state to some separable states .

• B

A AB

    

) ( ) ( ), ( ) ( ;         

B B A A i i B i A i AB AB

Tr Tr Tr Tr        



No-Disentanglement Theorem

• Exact universal disentangling machine

doesn’t exist in either process.

• Hence disentangling process is defined to

be either state-dependent exact disentanglement or it can be universal inexact disentanglement.

Results on state-dependent exact disentanglement

• Any set of perfectly distinguishable states can be

disentangled.

• Any set of states with identical reduced density

matrices can be disentangled.

• Any set of maximally entangled states can be

disentangled.

Inexact Universal Disentanglement

• An inexact universal disentangling machine can

disentangle any entangled state, and for which the local system are related by the reduction factors ηA and ηB as

• Where the reduction factors are independent of

the initial entangled state and

   

B B B B d A A A A A d B

I Tr I Tr                   2 1 2 1        

1 ,  

B A η

η

Universal Disentangling Machine

• Suppose two parties A and B share an

entangled state of two qubit system given by |ψ = α |00+ β |11, where |α|2 + |β|2= 1.

• A disentangling machine will then assumed to

be some unitary operation acting on any one subsystem with some Machine state or two Local unitary operation acting of the two subsystems.

Local Cloning induces Disentanglement

• Bandyopadhyay et.al. proposed that entanglement of

bipartite system HAB, can be reduced by introducing local isopropic cloner to any subsystem (Say A).

• The process spilt the entanglement of the joint

system between two joint system(HA’B and HA’’B ) each having a less amount of entanglement.

    M

b M a M b U M b M a M b U

Á A Á A Á A Á A Á A Á A Á A Á A

             

                       

1 1 1 1 1 1 1

Symmetric Optimal Universal Machine

• If reduction factors are chosen to be equal, i.e.

, the final state remain entangled for all initial state, if . Fidelity of cloner is related with the reduction factor by

• It is possible to disentangle arbitrary pure two-qubit

entangled state, by applying universal isotropic cloner whose Fidelity in one subsystem.

• If isotropic local cloning machine is applied on both
• f the subsystem, then the any pure bipartite state

entangled states can be disentangled, if the common reduction factor

    

B A

3 1  

3 2  F

 

   1

2 1

F

           3 2 3 1 i.e., 3 1 F 

Asymmetric Optimal Universal Machine

• When the disentangling machine is allowed to
• perate locally on both the subsystems, and

reduction factors are not bound to be equal(asymmetric case) then it is shown that for

• ptimal disentanglement process the reduction

factors ηA and ηB satisfy the following relation:

3 1 

B Aη

η

Disentanglement resulting from Decoherence Process in Open System Dynamics

• Decoherence process is the destruction of

quantum interference.

• Disentanglement and Decoherence

phenomena are shown to be connected by Dodd et.al. for open quantum dynamics. All possible initial state of the two particle system become separable after a finite time, under the evolution process that produce decoherence of both particles.

Disentangling Capacity of a Joint Unitary

In evolution of pure bipartite system, the Entangling E↑(UAB) and Disentangling (E↓(UAB)) Capacity of a joint unitary UAB are U’ is the extension of U in the extended Hilbert space(H’

AB=H’ AaBb) of bipartite system, introducing

ancillary spaces to both subsystem, | be an arbitrary state of H’. For any 22 unitary E↑(U) =E↓(U) whereas from 23 dimension the two capacities are not always equal.

 

   

 

 

   

 

         

   

U E E U E E U E U E sup sup

Discord

• A measure of non-classicality of bipartite correlation.
• Consider a composite quantum system
• The total correlation of a density matrix AB of the

composite system is characterized by the quantum mutual information

• I(ρ) = H(ρA) + H(ρB) − H(ρ) , (1)
• where H(.) is the von Neumann entropy function.
• ρA and ρB are local subsystems of parties A and

B respectively. where H(.) is the von Neumann entropy function.

• ρA and ρB are local subsystems of parties A and

B respectively.

B B A A B A AB

d H d H H H H     ) dim( , ) dim( ,

• A generalization of the classical conditional

entropy is H(ρB|A), where ρB|A is the state of the subsystem B given a measurement on subsystem

• A. By optimizing over all possible measurements

in A, we get an alternative version of mutual information as

• , (2)
• Where is the state of B
• conditioned on outcome k of the measurement

performed on subsystem A and {Ek} represents the set of positive operator valued measure(POVM) elements.



        

    k κ κ κ

ρ ρ ρ ρ Q

Β Ε Β Α

Η min Η )

) ( ) ( ρ I E Tr ρ I E Tr ρ

B k B k A k B

  

• Then the discrepancy between the two

measures of information defined above in equations (1) and (2) will be termed as Quantum Discord : DA(ρ) = I(ρ) − QA(ρ) (3)

• States having highly mixed in this sense,

though not have much entanglement, but may used as resource for performing some information theoretic tasks exponentially faster than any classical algorithm.

• Even separable states having this resource

are shown to be powerful than classical system.

• The discord is always non-negative.
• The value of this measure reaches zero for

classically correlated states.

• Discord is not a symmetric quantity DA(ρ) and

DB(ρ) denotes the left discord and right discord

• f ρ .
• If DA(ρ) = DB(ρ) = 0 , then the state ρ is said to

be completely classically correlated.

Classical Quantum States

• The states of a quantum system with zero value of quantum

discord, are known as Classical-Quantum states.

• A state ρ has zero-discord if and only if there exist a von

Neumann measurement such that, (4)

• The zero-discord state is of the form
• Where is some orthonormal basis set, ρk are the

quantum states of subsystem B and pk are non-negative numbers such that kpk = 1.

• The set of zero-discord states is not convex.

 

k

k k k

;    

   

ρ I ρ I

k B k B k

  



Π Π

k k k k k AB

p      

 

k

ψ

• We consider the singular value decomposition of  as

diag[c1, c2, . . . ]. Singular value decomposition defines new basis in local Hilbert-Schmidt spaces

,

• The state ρ in the new basis is of the form

where L is the rank of correlation matrix R (i.e., the number of non-zero eigenvalues cn).

• The necessary and sufficient condition (4) becomes
• This is equivalent to :





' ' ' n n nn n

A U S





' ' ' m m mm n

B W F





 

L n n n n

F S c ρ

1

L 1,2, n ; Π Π   



n k k n k

S S

 

n k, ; Π ,  

k n

S

• This means that the set of operators {Sn} have

common eigenbasis defined by the set of projectors {k}. Therefore, the set {k} exists if and only if:

• By checking a maximum of L(L − 1)/2 number of these

commutators, one may identify the zero discord states, where L = rank(R) ≤ min{dA

2 , dB 2 }.

• Now zero-discord state ρ is a sum of dA product
• perators. This bounds the rank of the correlation

tensor to L ≤ dA.

• Thus, the rank of the correlation tensor is itself the

discord witness: If L > dA, the state has a non-zero discord.

 

L , 1,2, n m, ,    

n m S

S

Geometric Discord

• Geometric discord of a quantum state is defined as

the minimum distance from the set(0) of states with zero quantum discord

where ||.||2 is the Hilbert Schmidt distance.

• Geometric discord can also be defined as

minimization is taken over all local von Neumann measurement {X} on party X.

 

2 2

1

min :   



 

  G

D

 

 

 

2 2 Π

Π min : ρ ρ ρ D

X X G

X

 

Geometric Discord of Two-Qubit system

• Any state of two-qubit system can be expressed as
• where =(1,2,3)t ; i are generators of SU(2).
• Let T={tij} and is maximum eigenvalue of the

matrix

• Geometric Discord of two-qubit system can be

described as

             



j i j i ij t t AB

λ λ t λ y I I λ x I I ρ

, 2 2 2 2

4 1

max

k

t t

xx ΤΤ 

 

 

max 2 2

4 1 k T x ρ D

AB A G

  

Coherence Vector Representation of Bipartite System

• In coherent vector representation a bipartite state

AB of composite system with and , can be expressed as where coherent vectors for reduced density matrix A and B are

B A

H H 

 

B B

d H  dim

 

A A

d H  dim

   

 

  

       

           

1 1 1 1 1 1 1 1

2 2 2 2

4 1 2 1 2 1 1

A B j i B i A i

d i d j B A ij d i B A i A d i B A i B B A B A AB

K y d x d d d     

       

B A AB t B A AB t

tr y y tr x x             , , , , ,

2 1 2 1

• The generators of SU(dA) and SU(dB) are denoted as

and respectively. IA and IB are identity vectors of subsystem A and B. Also, a matrix with elements are required for this representation. The triplet define a tensor as,

• This tensor is used to characterize the correlations
• f the bipartite state AB .

j

B

λ

i

A



 

j i

B A AB ij

λ λ ρ tr K  

 

K y x , ,

t t

xx y KK

2

Λ  

Condition for  being product state

• A bipartite state AB is a product state, if

and only if the criterion tensor =0 or rank()=0.

CONDITION FOR  BEING ZERO-DISCORD STATE

• A bipartite state AB is a state of zero

discord, if and only if the criterion tensor  satisfies .

 

1 Λ  

A

d rank

Local Quantum Uncertainty

• For a bipartite quantum state ρAB, Girolami et.al. give

us the concept of local quantum uncertainty(LQU). It is defined as

• The minimization is performed over all non-

degenerate spectrum Λ(characterized as local maximally informative observable)

 

A AB K A

K ρ I U , min

Λ

Λ 

I K K

A 



Λ Λ

LQU as Measure of Bipartite Quantumness

• The LQU quantifies the minimum amount of

uncertainty in a quantum state. Non-zero value of this quantity for a bipartite state ρAB indicates the non-existence of any quantum certain observable for ρAB.

• This quantity vanishes for all zero discord state w.r.t.

measurement on party A.

• LQU is invariant under local unitary operation.
• It reduces to entanglement monotone for pure state.

For pure bipartite states LQU reduces to linear entropy of reduced subsystems.

From the analytical formula for local quantum uncertainty it has been shown that the whole class of O⊗O invariant states (including Werner and Isotropic Class) in n ⊗ n systems, possess quantum correlation.

Effect of Universal Isotropic Disentanglement Process on Geometric Discord of Two-Qubit State

             



j i B A ij B t A B A t B A AB

j i

λ λ K λ y I I λ x I I ρ

,

4 1

• For a two qubit state
• Geometric Discord is

where K is the matrix, present in coherent vector representation of the state and kmax is the largest eigenvalue of the matrix

t t

KK xx 

 

 

max 2 2

4 1 k K x ρ D

AB A G

  

• Under an universal isotropic disentanglement

process initial state AB changes to the final state where A , B are the reduction factors.

• The Geometric Discord of the final state is
• where kmax is maximum eigenvalue of the matrix

                



j i B A ij B A B t B A B A t A B A AB

j i

λ λ K η η λ y η λ x η ρ

,

Ι Ι Ι Ι 4 1

 

 

max 2 2 2 2 2

4 1 k K η η x η ρ D

B A A AB A G

    

t B A t A

KK η η xx η

2 2 2



Optimal Disentanglement

• It is known that for the case of optimal

asymmetric disentanglement process

• The criterion tensor  for AB is given by

 

Λ Λ

2 2 2 2 2 B A t t B A

η η xx y KK η η    

3 1 

B Aη

η

• Final state AB is a product state if and only if

intial state AB is a product state.

• AB is a zero discord state if and only if AB is a

zero discord state.

• The above two findings lead us to conclude

that Optimal Disentanglement Process is a local process that preserve exactly zero discord state.

• All the non-classical correlations that have

zero value for only zero discord states in two qubit systems, quantum disentanglement is the only local process that preserves their

• structures. For example:
• Quantum Discord,
• Geometric Discord,
• Relative Entropy of discord,
• Local Quantum Uncertainty

Co-authors of the paper are Ajay Sen & Debasis Sarkar

References

• T. Mor et.al., PRA 60, 4341, (1999).
• T. Mor, Phys. Rev. Lett. 83 (1999) 1451;
• D. Terno, Phys. Rev. A 59 (1999) 3320
• S. Ghosh et.al., Phy. Rev. A, 61, 052301 (2000).
• P.J.Dodd & J.J.Halliwell, Phys Rev A 69, 052105 (2004)
• N. Linden et.al., PRL 103, 030501, (2009)
• S. Bandyopadhyay et.al, Phys.Lett.A 258, 205,1999
• H. Ollivier & W. C. Zurek, PRL, 88, 017901 (2001)
• L. Henderson, and V. Vedral, J. Phys. A 34, 6899 (2001)
• J. Oppenheim et al., Phys. Rev. Lett. 89, 180402 (2002)
• Davide Girolami et.al., Phys. Rev. Lett. 110, 240402 (2013)