Entanglement: Definition, Purification and measures

Entanglement: Definition, Purification and measures Seminar in Quantum Information processing 3683 Gili Bisker Physics Department Technion Spring 006 Gili Bisker Physics Department, Technion

Introduction Entanglement of pure and mixed states. Identifying Entanglement is easy for a pure state of two systems, but it become extremely difficult when dealing with mixed state in higher dimension. Peres necessary criterion that a non entangled mixed state must satisfy and the proof of its sufficiency in a special case given by the Horodeckis. Entanglement is a resource needed for quantum information (Teleportation), but how can we generate maximally entangles states? Quantifying. Gili Bisker Physics Department, Technion

What is Entanglement? A pure state of a pair of quantum systems is called entangled if it cannot be written as a product state of the two systems ψ A B. A classic example is the singlet state ( ). A mixed state is entangled if it cannot be written as the sum of products of density matrices of the two subsystems ρ w ρ ρ A A A B. It means that the mixed stated cannot be prepared by two observers in distant labs. Example: the state ( ) ρ = 00 00 + is not entangled since it can be locally prepared: We choose randomly 0 or. If we have 0 we prepare both systems in the state 0 and otherwise. Gili Bisker Physics Department, Technion 3

Separability Criterion for Density Matrices Asher Peres 934-005 Gili Bisker Physics Department, Technion 4

Separability Criterion for Density Matrices A given composite quantum system can be prepared by two distant observers if the density matrix of the system is separable: ρ = w ρ ρ w > 0, w = A A A A A A A We define the partial transposed: if ρ ( ρ ) ( ρ ) we denote σ mμ, nν = ρnμ, mν. = then w mμ, nν A A A mn A μν T If the state is separable then it means ( ) ρ = σ = ρ ρ T A w A A A or: ρ ρ ρ3 ρ4 ρ ρ ρ3 ρ3 ρ ρ ρ3 ρ 4 ρ ρ ρ4 ρ 4 ρ = σ = ρ ρ 3 ρ3 ρ33 ρ34 3 ρ4 ρ33 ρ34 ρ ρ 4 ρ 4 ρ43 ρ 44 3 ρ4 ρ43 ρ 44 Gili Bisker Physics Department, Technion 5

Separability Criterion for Density Matrices T The transposed matrices ( ) ( ) with a unit trace Density matrices. None of the eigenvalues of σ are negative. ρ = ρ are non negative matrices A This is a necessary condition for the original density matrix to be separable. σ ρ ρ A ( ) T = A w A A A Gili Bisker Physics Department, Technion 6

Separability Criterion for Density Matrices Example: 0 0 0 0 0 0 0 ( ) 0 0 0 0 0 x x x ρ = xs + I = + 4 0 0 4 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 + x x 0 ρ = 4 0 x + x 0 0 0 0 x After the partial transposition: x 0 0 x 0 + x 0 0 σ = 4 0 0 + x 0 x 0 0 x Gili Bisker Physics Department, Technion 7

Separability Criterion for Density Matrices The eigenvalues of σ : λ = λ = λ = ( + ) and ( ) 3 4 x λ = 3x. 4 4 If the lowest eigenvalues is negative, meaning x >, the density 3 matrix will not be separable. In this case, this criterion is also a sufficient one, meaning that if x < the density matrix is indeed separable. 3 The Horodeckis proved that this condition is also a sufficient one for systems with dimension and 3. However, for higher dimension, the partial transpose condition was shown not to be a sufficient one. Gili Bisker Physics Department, Technion 8

Separability of mixed states: necessary and sufficient (M. Horodecki, P. Horodecki and R. Horodecki) Notations: o Finite dimensional Hilbert Space H = H H. o The sets of linear hermitian operators acting on H, H will be denoted by A, A respectively. These sets are Hilbert spaces with the scalar product A, B = Tr( B A). o The space of linear maps from A to A is denoted by (, ) o A map L( A, A ) A then ( A) 0 L A A. Λ is positive if it maps positive operators from A to A, meaning, that if 0 Λ. o We say that a map Λ is completely positive if the inducted map Λ =Λ I : A M A M n n n n n Is positive for all n, where and I is the identity map. M n We will show that if a state ρ is not separable then there exists a positive map Λ such that ( Λ I ) ρ is not a positive operator. For and 3 systems, the partial transpose will do the trick. Gili Bisker Physics Department, Technion 9

Separability of mixed states: necessary and sufficient An hermitian operator A that acts on H = H H is an witness if it fulfills:. A is not a positive operator.. Tr ( Aσ ) 0 for every separable state σ. + + Example: For qubits, the operator W = I Φ Φ, where ( ) + Φ = +, is an witness: 00 Φ + Φ + = Φ + Φ + Φ + Φ + = < 0, so this operator is not a positive one.. For any pair of states, a = α 0 + β and b = γ 0 + δ, we have:. W ( I ) + + * * * * ( ) ( )( ) abw, ab, = ab, I Φ Φ ab, = αγ + βδ αγ+ βδ = * * * * ( ( ) ) ( ) * * * * ( ) ( ) = αγ + Re αγβ δ + βδ = α + β αγ Re αγβ δ βδ = = αδ + βγ Re αγβ δ αδ + βγ αγβ δ = αδ βγ 0 Gili Bisker Physics Department, Technion 0

Separability of mixed states: necessary and sufficient Lemma: For any inseparable state ρ A A there exists an Tr Aρ <. witness such that ( ) 0 (The proof uses tools from functional analysis like the Hahn Banach's theorem). Theorem : A state ρ A A is separable if and only if for every positive map Λ : A A the operator ( I Λ ) ρ is positive. (Again, the proof uses tools from functional analysis). Gili Bisker Physics Department, Technion

Separability of mixed states: necessary and sufficient 3 Theorem : A state ρ in or is separable if and only if its partial transposition is a positive operator. T Proof: if ρ is separable then its partial transpose ρ is also positive. T For the converse statement, we assume that ρ is positive. Every positive map Λ: A A, where H = H = or 3 CP CP H =, H = is of the form Λ =Λ +Λ T, where Λ CP is i complete positive and T is the transposition. CP CP Since Λ is complete positive, the map Λ = I Λ is positive, and i i i T if ρ is positive then CP CP CP CP T T I Λ ρ = I Λ +Λ T ρ = I Λ ρ+ I Λ ρ =Λ ρ+λ ρ ( ) ( ) ( ) ( ) ( ) Is also positive, and by Theorem, ρ is separable. Gili Bisker Physics Department, Technion

Concentrating Partial Entanglement by Local Operations (Bennett, Bernstein, Popescu, Schumacher) If Alice and Bob are supplied with n pairs of particles in identical partly entangled pure states, they can concentrate their into smaller number of maximally entangle pairs (like the singlet), by local actions of each observer. The of partly entangle pure state ( A, B) parameterize by its entropy of : H = Tr ρ A logρa = Tr ρblog ρb Where: ρ ( ) = Tr ( ){ Ψ ( A, B AB BA )} { } { } The yield of singlets approaches nh ( n) Ο log. Ψ can be Gili Bisker Physics Department, Technion 3

Concentrating Partial Entanglement by Local Operations Initially, Alice and Bob share n partly entangled particles with initial entropy of nh. They can perform only local operations and communicate classically. We can write the initial state as: ( ) n Ψ AB, = cosθ 0 + sinθ 0 AB AB The entropy of of one particle: ρ = cos θ 0 0 + sin θ 0 0 { } i= A AB AB H = Tr ρ Alog ρa = cos θlogcos θ + sin θlogsin θ The total entropy Alice and Bob share is nh. Gili Bisker Physics Department, Technion 4

Concentrating Partial Entanglement by Local Operations If they have only one particle: ( ) Ψ AB, = α 0 + β 0, where AB AB we assume α < β, Alice can perform the following measurement: β M 0 = λ 0 0 + α M M + M M = M = U M M 0 0 0 0 Whenever she measures μ = 0, the state will be: μ= 0 M 0 Ψ λβ Ψ = 0 + 0 p p normalization 0 + 0 0 0 ( ) ( ) AB AB AB AB Gili Bisker Physics Department, Technion 5

Concentrating Partial Entanglement by Local Operations The probability for getting a maximally entangled state is p0 = λβ. In order to increase this probability we would like to increase λ. But we must keep the operator M 0 M 0 positive: * β β β 0 0 * 0 M0 M0 = λ α α = λ α 0 0 0 And since we have M M M0 M0 β α p0 + =, then λ λ = λβ α α β So we cannot increase λ as much as we want. Gili Bisker Physics Department, Technion 6

Concentrating Partial Entanglement by Local Operations For two particles we have cosθ 0 + sinθ 0 cosθ 0 + sinθ 0 = ( ) ( ) cos θ 0 0 sinθcosθ 0 0 0 0 sin θ 0 0 AB AB AB AB AB AB AB AB + + + = cos θ 00 + sinθcosθ 0 0 + 0 0 + sin θ 00 AA BB AA BB AA BB AA BB After appropriate measurement done by Alice, they can be left with the state ( 0 0 + 0 0 AA BB AA BB) With probability sin θ cos θ. After a CNOT performed locally by Alice and Bob: ( ) ( ) 0 + 0 = 0 + 0 AA BB AA BB AB AB AB They have one particle in a maximally entangled state. Gili Bisker Physics Department, Technion 7

Concentrating Partial Entanglement by Local Operations Now, let us examine a more general case, with n pairs: ( ) n Ψ AB, = cosθ 0 + sinθ 0 AB AB i= The state has n terms with n + distinct coefficients: n n n cos θ, cos θsin θ,,sin θ Alice performs a measurement, projecting the initial state into one of n + orthogonal subspaces corresponding to the power k = 0,, n of sinθ in the coefficients, obtaining a result k with probability: n ( n k cos ) ( sin k pk = θ θ ) k Alice then tells Bob which outcome she obtained. Gili Bisker Physics Department, Technion 8

Concentrating Partial Entanglement by Local Operations Alice and Bob will be left with a state ψ k which is a maximally entangled state in a subsystem with n k dimensions, because we can change the base and get the state: n n 00 + + + AB AB k k When n, the probability is maximal for AB k = sin θn. Gili Bisker Physics Department, Technion 9

Concentrating Partial Entanglement by Local Operations If we had m identical maximally entangled states ( 00 ) +, AB AB the product has m orthogonal elements with the same coefficient. Alice and Bob have the state: m n n 00 + + + AB AB k k We'll Define H, such that the above n k equivalent to nh maximally entangled pairs: n nh = k AB orthogonal states is Gili Bisker Physics Department, Technion 0

Concentrating Partial Entanglement by Local Operations If we take the most likely case, where k = sin θn then: n n! nh = log log sin θn = = ( sin θn)!( n sin θn)! n! = log n sin θ log sin θ + cos θlogcos θ ( sin θn)!( cos θn)! H = sin θ logsin θ + cos θlogcos θ, which is the entropy of of the initial pure state. n non maximally entangled nh maximally entangled. Gili Bisker Physics Department, Technion

Purification of Noisy Entanglement (Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters) Two observers, who share entangled states, can purify them by local operations and measurements, using classical communication, and sacrificing some of the pairs to increase the purity of the rest. The initial state is: + + + + W = Fψ ψ + F ψ ψ + φ φ + φ φ F ( 3 ){ } Where ψ ± = ( ± ) and ( ) 0 0 φ ± = 00 ±. The purity of W F can be express by the Fidelity: F = ψ W F ψ The yield of pure states is shown to be positive if F >. Gili Bisker Physics Department, Technion

Purification of Noisy Entanglement First, Bob performs a unilateral σ y rotation. The Unilateral Pauli rotation I σ i maps the bell states onto one other, leaving no state unchanged: ± ± σ maps ψ φ x ± σ maps ψ φ y ± ± σ maps ψ ψ and φ φ z The new state is: F φ + φ + + F φ φ + ψ + ψ + + ψ ψ ( 3 ){ } Gili Bisker Physics Department, Technion 3

Purification of Noisy Entanglement The next step is BXOR which is the quantum XOR operating on two pairs. The quantum XOR operates on two qubits, flipping the second (target) spin iff the first (source) spin is up. BXOR Source Target Alice 3 Flips 3 iff is up Bob 4 Flips 4 iff is up For example, we take ψ as the source and φ as the target. BXOR ψ φ BXOR ( 0 0 ) ( 00 ) ( 000 0 000 0 ) ( 00 00 00 00 ) ( 0 ( 0 0 ) 0 ( 0 0 )) ( 0 0 ) ( 0 0 ) + ψ ψ = = 34 = BXOR + = = + = = + = = + = Gili Bisker Physics Department, Technion 4

Purification of Noisy Entanglement Summarizing, omitting overall phase: Before After BXOR Source Target Source Target φ ± φ + same same ψ ± φ + same ψ + ψ ± ψ + same φ + φ ± ψ + same same φ ± φ φ same ψ ± φ ψ ψ ψ ± ψ ψ φ φ ± ψ φ same Gili Bisker Physics Department, Technion 5

After the BXOR: BXOR Purification of Noisy Entanglement + + + + ( F φ φ + 3( F) φ φ + 3( F) ψ ψ + 3( F) ψ ψ ) + + + + F φ φ + ( F) φ φ + ( F) ψ ψ + ( F) ψ ψ ( 3 3 3 ) + + + + + + + + φ φ φ φ + 3 ( F ) F φ φ ψ ψ ( F) ( F) ( F) ( F) F( F) ( F) ( F) ( F) F φ φ ( F) F FF + + + + + 3 3 φ φ φ φ + 3 3 φ φ ψ ψ + + + + + 3 φ φ φ φ + 3 3 φ φ ψ ψ + 3 φ φ + 3 φ φ ψ ψ = + + + + + + + + + 3( F) 3( F) ψ ψ φ φ + F 3( F) ψ ψ ψ ψ + + + + + 3( F) 3( F) ψ ψ ψ ψ + 3( F) 3( F) ψ ψ φ φ + + + + + 3( F) 3( F) ψ ψ φ φ + F 3( F) ψ ψ ψ ψ + 3( F) 3( F) ψ ψ ψ ψ + 3( F) 3( F) ψ ψ φ φ Gili Bisker Physics Department, Technion 6

Purification of Noisy Entanglement The target pair is locally measured in the z direction, if the spins + + come out parallel (which mean that the target is in the φ φ or the φ φ state) the unmeasured source pair is kept, otherwise, it is discarded. If the target pair's z spins parallel, the new source state is: + + ( F + 9 ( F) ) φ φ 5 F + ( ) ( ) 3F F + 9 F + 3 F F φ φ + ψ ψ + ψ ψ + + ( )( ) After unilateral σ y rotation, we get: ( F + 9 ( F) ) ψ ψ 5 F + ( ) ( ) 3F F + 9 F + + + + + 3 F F ψ ψ + φ φ + φ φ ( )( ) Gili Bisker Physics Department, Technion 7

Purification of Noisy Entanglement The new Fidelity is: F F = F + F F + F + 9 ( F) ( ) ( ) 5 3 9 F'(F) 0.5 F` F 0 0 0.5 F Gili Bisker Physics Department, Technion 8

Introduction to Entanglement Measures Goal: quantifying and finding magnitudes which behave monotonically under local transformation. A typical framework would be two observers who can communicate classically, sharing a composite system in an entangled state, on which they can only operate locally (LOCC): Gili Bisker Physics Department, Technion 9

Basic Properties of Entanglement Separable state contains no. A state of many parties A,B,C is said to be separable if it can be written in the form i i i ρabc... = piρ A ρ B ρ C... pi = i i These states can be created by LOCC: Alice samples from the distribution p i, informs all the others of the outcome and then each party X locally prepares ρ. i X All non separable states are entangled. Any non separable state ρ can enhance the teleportation power of some other state σ (L. Masanes 006) Gili Bisker Physics Department, Technion 30

Basic Properties of Entanglement The of states does not increase under LOCC transformations. LOCC can only create separable so LOCC cannot create from a non entangled state. If quantum state ρ can be transformed to σ using LOCC, Then ρ is at least as entangled as σ. Entanglement does not change under local unitary transformation. This one follows from the previous one, since local unitary transformations are invertible. Gili Bisker Physics Department, Technion 3

Basic Properties of Entanglement There are maximally entangled states. In two party systems a maximally entangled state is: 00 + + + d ; d ψ + d = d Any pure or mixed state can be prepared from such a state with certainty using only LOCC operations. (By the way, an equivalent statement does not exist in multi particle systems, so there is a difficulty in establishing a theory of multi particle ). Gili Bisker Physics Department, Technion 3

Local Manipulation of Quantum States For bi-partite system any pure state can be created using LOCC from a maximally entangle state, which is a state that is unitary equivalent to ψ + = ( 00 + ). We'll consider a general states written in the Schmidt decomposition form: φ = α 00 + β Gili Bisker Physics Department, Technion 33

Local Manipulation of Quantum States Let us define the operators: A α A 0 ( 0 0 + β ) I ( α 0 + β 0 ) ( 0 + 0 ) They satisfy: And: So: A A A + A A = I I 0 0 ψ = φ A ψ = φ + + 0 φ φ = A ψ ψ A + A ψ ψ A + + + + 0 0 Gili Bisker Physics Department, Technion 34

Local Manipulation of Quantum States Let us now add an ancilla 0 to Alice's system: ( ) ψ + + 0 00 0 A AB A AB Alice applies to her system the unitary transformation: 00 α 00 + β We get: 0 α 0 + β 0 (( α 00 + β ) 0 + ( α 0 + β 0 ) ) = A B A B 0 ( α 00 + β ) + ( α 0 + β 0 ) ( ) A AB AB A AB AB Alice measures her ancilla, if the outcome is 0, then they are left with φ. If the outcome is, Bob performs σ x on his particle, and they are left with φ. Gili Bisker Physics Department, Technion 35

Local Manipulation of Quantum States These two procedures are equivalent: if we want to perform a measurement described by the measurement operators M : We add an ancilla with an orthogonal basis m corresponding to the possible measurement outcomes. If the state of the ancilla at the beginning is 0, then we define an operator U on the product ψ 0 by: U ψ 0 = M m m ψ m. If we do a projective measurement of the ancilla P m = I m m, the outcome m occurs with probability: p m = ψ 0U PU ψ 0 = ψ M M ψ ( ) m m m And the state after the measurement will be: M m ψ m ψ M M ψ m m m Gili Bisker Physics Department, Technion 36

Local Manipulation of Quantum States Let ρ H A H B, where H A = HB = H. A B Let φ = φ φ H H be a product state, then there is a LOCC operation Λ such that ( ρ ) φ φ A B Λ = : { } d A B B d A A B A A B A B B ( ρ) ( φ j ) ( φ i ) ρ( i φ ) ( j φ j= i= ) d A B A B A B A B ( φ φ i j ) ρ( i j φ φ ) φ Tr( ρ) φ φ φ Λ = = = = i, j= Gili Bisker Physics Department, Technion 37

Local Manipulation of Quantum States We want to transform ψ to ψ, (general pure state of two parties) using LOCC. We can write the state in their Schmidt decomposition: n ψ = α i i ψ = α i i i A B i A B i= i= Where the αi, α i are real and given in a decreasing order: α > α > > αn α > α > > α It has been shown (M.A Nielsen, 999) that the task is possible if α, meaning, for every and only if the { } i α are majorized by { } i l l l < n we have α i α = = i n n and α i α = = = i. i i { 3} { } For example, consider the distribution:,, and,,, n i n i 555 5555 Gili Bisker Physics Department, Technion 38

Local Manipulation of Quantum States Consequences: There are pairs of states that neither of them can be converted into the other with certainty. { 5} { 43} For example, consider the distribution:,,, and,, Given a pure state d Ψ +, where 8888 888 d Ψ + is a maximally entangled d Λ Ψ = σ, for state, there is a LOCC operation Λ such that ( + ) any state H A H B σ, because the distribution ( d ) by any probability distribution on {,...,d }. d i= is majorized Gili Bisker Physics Department, Technion 39

Definition of Entanglement Measures and Entanglement Monotones Entanglement monotone E ( ρ ) needs to satisfy to following:. ( ) numbers: ρ E ( ρ) E ρ is a mapping from density matrices into positive real +.. For normalization, we set E( ) log 3. E ( ρ ) = 0 If the state is separable. ψ + = d. 4. E ( ρ ) Doesn't increase under LOCC. d Gili Bisker Physics Department, Technion 40

Definition of Entanglement Measures and Entanglement Monotones Frequently, some may also impose additional requirements: For pure states the measure reduces to the entropy of E ψ ψ = H ψ ψ. : ( ) ( ) Convexity (mixing of state does not increase ): E p ρ pe ρ ( i i) i ( i) i Additivity: because there are measures that do not n satisfy the condition E( σ ) = ne( σ ) or E( σ ρ) = E( σ) + E( ρ), we define an asymptotic version: n E ( σ ) E ( σ ) lim n n Which automatically satisfies additivity. i Gili Bisker Physics Department, Technion 4

Definition of Entanglement Measures and Entanglement Monotones Continuity: any monotone that is additive on pure state and "asymptotically continuous" must equal H ( ρ ) on all pure states. Asymptotically continuous: for two sequences of states ρ, σ such that ρn σ n 0, the following property holds: E( ρn) E( σ n) 0 + log dim H ( ) n H n n n Gili Bisker Physics Department, Technion 4

Survey of Entanglement Monotones We discuss in this section a variety of bipartite monotones (meaning they cannot increase under LOCC). Gili Bisker Physics Department, Technion 43

Entanglement Cost n m Say we want to transform ρ σ, for some large integers n and m. The larger ratio r = m n we may achieve would give an indication about the relative The cost E ( ) C ρ quantifies the maximal possible rate r at which one can convert blocks of two qubits maximally entangled states into many copies of ρ. We denote a general LOCC operation by Ψ, we can define the cost as: The E ( ) { { } } rn n Ψ n ( ρ + + ) ρ ( ψ ψ ) EC = inf r: lim inf Tr Ψ = 0 C ρ measures how many maximally entangles states are required to create copies of ρ by LOCC alone. Gili Bisker Physics Department, Technion 44

The Distillable Entanglement We can ask about the reverse process: at what rate may we obtain maximally entangled states (of two qubits) from an input supply of states of the form ρ? (distillation or concentration) The distillable E ( ) D ρ provide us the rate at which noisy mixed state ρ may be converted into singlet state by LOCC: ( ) { ( ) ( ) } rn n + + ED ρ = sup r: lim inf Tr Ψ ρ ψ ψ = 0 n Ψ Two of the Horodeckis showed (Phys. Rev. Lett. 84, 000) that any measure E with the properties: Normalized E ψ + = log d, Monotonic under LOCC, Continuous and ( d ) n Additive ( E( σ ) ne( σ ) = ) satisfies E ( ρ ) E( ρ) E ( ρ) D < <. C Gili Bisker Physics Department, Technion 45

For pure states E ( ρ ) E ( ρ ) Entropy of Entanglement D = C and they are equal to the entropy of, which is defined by: H ψ ψ S tr ψ ψ = S tr ψ ψ ( ) ( ) ( ) Where S is the von Neumann entropy S( ρ ) tr{ ρ ρ} A B = log. For a general mixed state σ, S( traσ ) may not equal S( trbσ ) Now we can say: Given a large number N of copies of ψ ψ, we can distill N H( ψ ψ ) singlet states and then create from those singlets M NH( ψ ψ ) H( ψ ψ ) copies of ψ ψ. In the infinite limit H( ) H( ) conversion rate from ψ ψ to ψ ψ ψ ψ is the optimal ψ ψ Gili Bisker Physics Department, Technion 46

Entanglement of Formation The Entanglement of formation E ( ) F ρ of a mixed state ρ is the least expected of any ensemble of pure states realizing ρ : E ρ = min ph ψ ψ : ρ = p ψ ψ { } F( ) i ( i i ) i i i i i Where H( ψ ψ ) = S( ρa ) = S( ρb) = Tr{ ρalogρa} = Tr{ ρblogρb} And ρ = Tr { ψ ψ }, ρ = Tr { ψ ψ } A B B A In order for Alice and Bob to create the system ρ without transferring quantum states between them, they must already share E ρ. the amount of equivalent of ( ) F Gili Bisker Physics Department, Technion 47

A closed formula for E ( ) Entanglement of Formation We define ρ = ( σ y σ y) ρ ( σ y σ y) F. We denote C ( ρ ) max{ 0, λ λ λ λ } ρ is known for bi-partite qubit systems: = 3 4, where the λ i are the square roots of the eigenvalues of the matrix ρρ, in decreasing order, and + C ( ρ ) EF ( ρ ) = s Where s( x) xlog x ( x) log ( x) =. The asymptotic version defined as: E ( ρ ) E ( ) lim F F ρ = n n E ρ = E ρ. is equal the cost ( ) ( ) F C Gili Bisker Physics Department, Technion 48

Entanglement of Formation Example for maximally entangled state: 0 0 0 0 0 0 ρ = s s = ( 0 0 )( 0 0 ) = 0 0 0 0 0 0 0 0 0 i 0 i 0 σ y σ y = i 0 i 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = ( y y) ( y y) = = = = 0 0 0 0 0 0 0 0 0 0 0 0 ρ σ σ ρ σ σ ρρ ρ ρ eigenvalues { ρρ} { } C ( ρ ) =,0,0,0 = + C ( ρ ) EF ( ρ ) = s = s = log log = log = Gili Bisker Physics Department, Technion 49

Entanglement of Formation Example for a separable state: 0 0 0 0 0 0 0 ρ = 0 0 0 0 0 0 0 0 0 0 0 i 0 i 0 σ y σ y = = i 0 i 0 0 0 0 0 0 0 0 0 0 0 0 ρ = ( σ y σ y) ρ ( σ y σ y) = 0 0 0 0 0 0 0 ρρ = 0 eigenvalues C { ρρ} = { 0,0,0,0} ( ρ ) = 0 + C ( ρ ) EF ( ρ ) = E( C) = s = s() = log log = 0 Gili Bisker Physics Department, Technion 50

Entanglement of Formation Examples for Inseparable state: 8 0 0 0 3 0 8 4 0 ρ = 3 0 4 8 0 0 0 0 8 0 0 0 0 3 3 0 ρσ ( y σ y) ρ ( σ y σ y) = 64 0 3 3 0 0 0 0 eigenvalue C ( ρ ) 5 ( ρρ ) = {,,, } 64 64 64 64 5 8 8 8 8 4 = = + C ( ρ ) EF ( ρ ) = s = s( 0.984) = 0.78 Gili Bisker Physics Department, Technion 5

Negativity and Logarithmic Negativity The Negativity, N ( ρ ), is an monotone that attempts to quantify the negativity in the spectrum of the partial transposed matrix. We will define it as: T ρ N ( ρ ) = Where X = Tr X X is the trace norm. The trace norm of any Hermitian operator A is equal to the sum of the absolute values of the eigenvalues of A. For density matrices: = Tr = Tr =. ρ ρ ρ ρ Gili Bisker Physics Department, Technion 5

Negativity and Logarithmic Negativity Another monotone is the logarithmic negativity: T EN ( ρ ) = log ρ E ρ ρ = E ρ + E ρ : Which is an additive one ( ) ( ) ( ) E N ( ) log ( ) N N N T T ρ ρ ρ ρ log ρ ρ = = = ( ) ( ) ( ) log Tr ( ) ( ( ) ) log ( ) T T T T Tr ρ ρ ρ ρ ρ ρ ρ ρ = log = = T T T T = logtr ρ ρ ρ ρ = Tr ρ ρtr ρ ρ = T T ( ) N ( ) N ( ) = log Tr ρ ρ + log Tr ρ ρ = E ρ + E ρ T If the state is separable then ρ = and the negativity and the logarithmic negativity vanish. Gili Bisker Physics Department, Technion 53

Negativity and Logarithmic Negativity Consider the example we used for Peres criterion: T The eigenvalues of ( x) ρ = xs + I 4 λ λ λ x 3 4 ρ : = = = ( + ) and ( ) λ = 3x. 4 4 If the lowest eigenvalues is negative, meaning x >, the density 3 matrix will not be separable: T 3 3 ρ = ( + x) ( 3x) = + x > + = 4 4 If x < the density matrix is indeed separable: 3 T 3 ρ = ( + x) + ( 3x) = 4 4 Gili Bisker Physics Department, Technion 54

Relative Entropy of Entanglement The quantum relative entropy is S( ρ σ) Tr{ ρlog ρ σ log ρ} = and it measures the distinguish-ability between quantum states. The relative entropy of with respect to a set X will be defined as: X E ρ = inf S ρσ R ( ) ( ) σ X The set X can be taken as the set of separable states, or any other set of states, depending upon what you are regarding as "worthless" states. Gili Bisker Physics Department, Technion 55

Relative Entropy of Entanglement We can also define this measure using another distance measure to quantify how far a particular state is from a chosen set of disentangled states: Gili Bisker Physics Department, Technion 56

Entanglement Witness Monotone An hermitian operator A is defined as an witness if: ρ SEP Tr ( Aρ ) 0 and ρ such that Tr ( Aρ) < 0 So A acts as a linear hyperplane separating some entangled states from the convex set of separable ones. Gili Bisker Physics Department, Technion 57

Entanglement Witness Monotone The amount of violation of an Entanglement Witness is a measure of the non Separability of a given state: E A = max 0, Tr Aρ wit { } ( ) ( ) We can choose sets of witnesses and define a monotone as the minimal violation over all witnesses of the set. Gili Bisker Physics Department, Technion 58

Entanglement Measure (G. Gour and R. W. Spekkens) We define the "Entanglement of Assistance" (EoA): Suppose that Alice and Bob have a bipartite system in the mixed state ρ AB and Charlie holds a purification of this state. The tripartite system held by Alice, Bob and Charlie is in a pure state ρ = Tr Ψ Ψ. Ψ, such that { } ABC AB C ABC Gili Bisker Physics Department, Technion 59

Entanglement Measure The EoA quantifies the maximum of pure state (quantified by the entropy of H ) that Alice and Bob can extract from ρ AB by Charlie performing a measurement on his system and reporting the outcome to Alice and Bob: EAst ( ρab ) = max ph i ( ψi AB ) { p, ψ } i i AB Where H ( ψ i ) S ( ρa, i) Tr ( ρa, ilog ρa, i), ρ AB A, i TrB( ψi ψ AB i ) The maximization is over all the ensembles { p, ψ } such that: i = = =. ρ = ψ ψ p AB i i i AB i i i AB Gili Bisker Physics Department, Technion 60

Entanglement Measure The EoA is a measure of for tripartite states and for it to be an monotone; it must be non increasing under LOCC operations between all three parties. "Entanglement of Collaboration" (EoC): Same as the EoA, only now, the parties can use arbitrary tripartite LOCC operation. Here Alice and Bob are allowed to assist Charlie in assisting them. If there exists a state for which the EoC is greater then the EoA, then the EoA is not an monotone, because in this state it would be possible to increase the EoA by an LOCC operation (Alice and Bob communicating with Charlie). Gili Bisker Physics Department, Technion 6

Entanglement Measure A demonstration for which this is the case, in a tripartite system of dimensions 8 4 : ( 00 + + + 33 ) 0 + ( i 00 + i 33 ) + Φ = ABC 4 ( 40 + 5 + 6 + 73 ) + + ( i 40 + 5 + i 6 + 73 ) Where ± = ( 0 ± i ). Alice performs a measurement with two possible outcomes: 0 0 + + + 3 3 and 4 4 + 5 5 + 6 6 + 7 7 and she sends the outcome to Charlie. If the first outcome occurred, Charlie performs a measurement in the { 0, } basis and if the second outcome occurred, he measures in the { +, } basis. In each case, he leaves Alice and Bob with a maximally entangled 8 4 state. Gili Bisker Physics Department, Technion 6

. Entanglement Measure If no communication from Alice and Bob to Charlie is allowed, then Charlie cannot always create a maximally entangled state for Alice and Bob. First note that the initial state can be written as follows: Φ = u0 0 + u ABC AB C AB C Where: ( 0 + z 4 ) 0 + ( + 5 3 ) + u0 = AB c 0 0, k k = k= 4 ( + z 6 ) + ( 3 + 7 ) 3 u = c k = ( i 0 + z 4 ) 0 + + ( i + z 6 ) 3 3 3 AB k= 0, k 4 z + i. And ( ) The reduced density matrix is: ρ = Tr Φ Φ = u u + u u AB C ABC 0 AB 0 AB Gili Bisker Physics Department, Technion 63

Entanglement Measure Charlie cannot even create a maximally entangled state with some probability. Given the Hughston-Jozsa-Wootters theorem, it is enough to show that no convex decomposition of ρ AB contains a maximally entangled states. (HJW gave A complete classification of quantum ensembles having a given density matrix) Any decomposition of ρ AB is proportional to 0,, AB AB (why?) xu + yu x y So it is enough to show that any state of that form is not a maximally entangled state of a 8 4 system. Gili Bisker Physics Department, Technion 64

Entanglement Measure If xu0 + yu AB is maximally entangled then all of its Schmidt AB coefficient must be equal. The coefficient are: ( x iy x y ) ( x y x ) ( x iy x y ) ( x y x ) λ = + + + 6 λ = + + 6 λ = + + 3 6 λ = + 4 6 Thus, λ = λ = λ 3 = λ 4 if and only if x= y = 0. Any state of the form xu0 + yu AB cannot by maximally AB entangled. Entanglement of Assistance is not an monotone. Gili Bisker Physics Department, Technion 65

Summary We saw some properties of, ways to increase it and ways to quantify it. The idea of monotone. This is just the tip of the iceberg!!! Gili Bisker Physics Department, Technion 66

Reference. A. Peres. "Separable Criterion for Density Matrices", Physical Review Letters, volume 77, August 996.. M. Horodecki, P. Horodecki and R. Horodecki. "Separability of mixed states: necessary and sufficient " Physics Letters A, Volume 3, November 996. 3. J. E. Avron and Netanel H. Lindner, Private conversations. 4. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters. "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels", Physical Review Letters, volume 76, January 996. 5. C.H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher. "Concentrating Partial Entanglement by local Operations", Physical Review A, volume 53, April 996. 6. C. H. Bennett, D. P. DiVincenzo, J.A. Smolin and W. K. Wootters. "Mixed State Entanglement and Quantum Error Correction", Physical Review A, volume 54, November 996. 7. M. B. Plenio and S. Virmani. "An introduction to Entanglement Measures" arxiv: quant-ph/050463, v, February 006. 8. V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight. "Quantifying Entanglement". Physical Review Letters, volume 78, March 997. 9. G. Vidal, "Entanglement Monotones", Journal of Modern Optics, Volume 47, February 000. 0. M. A. Nielsen and I. L. Chang, "Quantum Computation and Quantum Information", Cambridge, 00.. G. Vidal and R. F. Werner. "Computable measure of Entanglement", Physical Review A, volume 65, February 00.. W. K. Wootters. "Entanglement of Formation of an Arbitrary State of Two Qubits", Physical Review Letters, volume 80, March 998. 3. M. J. Donald, M. Horodecki and O. Rudolph. "The Uniqueness Theorem For Entanglement Measures", Journal of Mathematical Physics, volume 43, September 00. 4. Dieter Heiss "Fundamentals of Quantum Information: Quantum Computation, Communication, Decoherence and All That", Springer 00. 5. Lecture Notes for the course "Quantum information" given by Dr. Benni Reznik, Tel Aviv University, March 005. 6. G. Gour and R. W. Spekkens. "Entanglement of Assistance is not an monotone" arxiv: quantph/0539, December 005. 7. G. Gour, Private conversations. Gili Bisker Physics Department, Technion 67

The End Gili Bisker Physics Department, Technion 68