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On some characterization of Entangled states. Indrani Chattopadhyay University of Calcutta. Pure Bipartite States. Unique Schmidt form where is the Schmidt rank, and are the Schmidt coefficients, such that
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On some characterization ofEntangled states Indrani Chattopadhyay University of Calcutta
Pure Bipartite States • Unique Schmidt form where is the Schmidt rank, and are the Schmidt coefficients, such that . • Schmidt vector is
Pure Bipartite Entanglement • Entanglement of Formation = Distillable entanglement • Entanglement is uniquely measured by von Nuemann entropy of its subsystem,
Concurrence • For any bipartite state ρ a measure of entanglement is Concurrence given be
Two locally unitarily connected states have same Schmidt vector, hence have equal amount of entanglement. Is there any LOCC, rather than local unitary operation that preserves entanglement of pure bipartite system?
Constraints • Thermo-dynamical law of Entanglement : Amount of Entanglement of a state cannot be increased by any LOCC. • Schmidt rank of a state cannot be increased by LOCC
Local Conversion of Bipartite Pure Entangled states Object : Transform one pure bipartite state to another Operation: All possible LOCC performed on each subsystem Let , have Schmidt vectors, and where , ,
Nielsen’s criteria A necessary-sufficient criteria for deterministic transformation of one pure bipartite state to another by LOCC, is : In Schmidt form, the Schmidt vector of second state should Majorize that of the first, i.e.
Comparability and amount of Entanglement • The states are said to be comparable if either or • Thermo-dynamical law of entanglement, implies • Also implies rank of is greater or equals to the rank of • In comparable class, EF is a monotonic function of concurrence.
Incomparability • Two pure bipartite states, such that neither can be deterministically transformed to the other by LOCC(i.e., both the criteria fails both way ). • We denote it as • Their cannot be any direct comparison of the amount of their entanglement. • Feature of entangled system only(bipartite level) • Existence of such pairs can be connected with the no-go theorems
Theorem:For any comparable pair ofstates of Schmidt rank d≥3, only locally unitarily connected states can have same amount of entanglement, i.e., The theorem says that if there exists any pair of distinct (i.e., when ) pure bipartite states, with , then
Class of states having same amount of Entanglement Eeveryrank 3 non-maximally entangled state, have an entanglement E(0, E(ΨMax) 1.584962501). For every value of Ein this interval there exists a class of infinite number of states, with an exact amount of entanglementE, all incomparable with each other.
Thus Equi-entangled bipartite pure statesare eitherlocally unitarily connected or not connected by deterministic LOCC, i.e., can not be deterministically transformed to either way by any local operations together with an infinite amount of classical communications. A simple consequence is the next result.
Existence of infinite number of equally-entangled states with different Schmidt vector (thus having different Concurrence) shows, even in pure bipartite system, Entanglement of Formation(EF)is not a monotonic function of concurrence • It follows from incomparability of the system.
SLOCC transformation • The transformation is possible by SLOCC if there exists invertible operators A and B such that with a non-zero probability. • Any pair of bipartite states are SLOCC convertible to each other.
Pure Tri-qubit system Any three qubit pure state can be maid into the form, Where i are all real, There are exactly 2 SLOCC inequivalent classes: GHZ and W class. W. Dur et.al., PRA, 62, 062314, 2000
SLOCC Classification of a system ~ Searching for SLOCC invariants (Entanglement Monotones) in the system
SLOCC Classification of pure 3 qubit entangled states • 1.1- 1:2 cut separable, bipartite entanglement. • 2.0- Genuine tripartite entanglement with all residual bipartite entanglement to be zero • 2.1- Genuine tripartite entanglement with only one residual bipartite entanglement is non-zero • 2.2- Genuine tripartite entanglement with only one residual bipartite entanglement is zero • 2.3- Genuine tripartite entanglement with all residual bipartite entanglement to be non-zero
3 qubit state in Generalized Schmidt form where coefficients are all real. Class 1.1 Case-1 When 3 = 5 =0, 1 ,2 ,4> 0 Concurrence of reduced density matrices of the subsystems are,
Result: There exists many states in the same subclass such that the concurrence of reduced subsystem are monotonic, i.e., where ’AC be the reduced subsystem of the state Φwith the imposed restriction • Either 1 > 1 > 2 > 2 > 4 > 4 or 4 > 4 > 2 > 2 > 1 > 1
Case 2: When 4 = 5 = 0, 1 , 2 ,3 > 0 Concurrence of the subsystems are, • Result: For every statein same subclass, monotonic nature of concurrence of reduced subsystem is observed, if Either 1 > 1 > 2 > 2 > 3 > 3 or 3 > 3 > 2 > 2 > 1 > 1
Case 3: When 1 = 3 =0, 2 ,4 , 5 > 0 Concurrence of the subsystems are, • Result: For every statein same subclass, monotonic nature of concurrence of reduced subsystem is observed, if Either 2 > 2 > 4 > 4 > 5 > 5 or 5 > 5 > 4 > 4 > 2 > 2
Class 2.0 When 2 =3 = 4 = 0 so the Key state is Concurrence of reduced bipartite subsystems of this state are,
Class 2.1 Case-13 =4=0 1, 2, 5 > 0 Concurrence of the subsystems are, • Result: For every statein same subclass, monotonic nature of concurrence of reduced subsystem is observed, if Either 2 > 2 > 1 > 1 > 5 > 5 or 5 > 5 > 1 > 1 > 2 > 2
Class 2.1 Case-2 If 2=3=0, 1 , 4,5 > 0 Concurrence of the subsystems are, • Result: For every statein same subclass, monotonic nature of concurrence of reduced subsystem is observed, if Either 1 > 1 > 5 > 5 > 4 > 4 or 4 > 4 > 5 > 5 > 1 > 1
Class 2.1 Case-3 If 2=4=0, 1, 3,5 > 0 Concurrence of the subsystems are, • Result: For every statein same subclass, monotonic nature of concurrence of reduced subsystem is observed, if Either 1 > 1 > 5 > 5 > 3 > 3 or 3 > 3 > 5 > 5 > 1 > 1
Class 2.2 Case-1If 3=0, 1, 2, 4,5>0 Concurrence of the subsystems are, • Result: There is an intrinsic monotonicity in concurrence, assuming 1>2 and 4>5 we get • For every statein same subclass, monotonicity of concurrence of both subsystems is observed, and Either 1 > 1 > 2 > 2 > 4 > 4 > 5 > 5 or 4 > 4 > 5 > 5 > 1 > 1 > 2 > 2
Class 2.2 Case-2If 4=0 and 1 , 2 , 3 ,5 > 0 Concurrence of the subsystems are, • Result: There is an intrinsic monotonicity in concurrence, assuming 1>2 and 3>5 we get • For every statein same subclass, monotonicity of concurrence of both subsystems is observed, and Either 1 > 1 > 2 > 2 > 3 > 3 > 5 > 5 or 3 > 3 > 5 > 5 > 1 > 1 > 2 > 2
Class 2.3If 2=5 =0 and 1 , 3 ,4> 0 Concurrence of the subsystems are, • Result: There is an intrinsic monotonicity in concurrence, assuming 1>3>4 we get • For every statein same subclass, monotonicity of concurrence of one subsystems is assured, if Either 1 > 1 > 3 > 3 > 4 > 4 or 4 > 4 > 3 > 3 > 1 > 1 Similarly for the cases 3>4>1 ,4>1>3 etc.
Classification of 4 qubit pure states • Verstraete et. al., PRA, 65, 052112
4 Qubit Generic state • With , Concurrence of the reduced subsystems are Monotonicity is observed under such relations
