Quantum Lecture 7 The Holevo bound Typical sequences and subspaces - PDF document

Quantum Lecture 7 • The Holevo bound • Typical sequences and subspaces • Compression Mikael Skoglund, Quantum Info 1/15 The Holevo Bound Assume a discrete random variable X ∈ X with pmf p ( x ) is embedded on a set of states ρ x , as the ensemble { p ( x ) , ρ x } A measurement described by { M n } N n =1 is performed, resulting in Y ∈ { 1 , . . . , N } The Holevo bound states that � I ( X ; Y ) ≤ S ( ρ ) − p ( x ) S ( ρ x ) x ∈X over all possible { M n } , and with � ρ = p ( x ) ρ x x ∈X Mikael Skoglund, Quantum Info 2/15

The entity � χ ( p ( x ) , ρ x ) = S ( ρ ) − p ( x ) S ( ρ x ) x ∈X is the Holevo information of the ensemble { p ( x ) , ρ x } Note that the joint entropy of the classical-quantum state � σ = p ( x ) | e ( x ) �� e ( x ) | ⊗ ρ x x ∈X (where { e ( x ) } is a basis) is H ( p ) + � x ∈X p ( x ) S ( ρ x ) , hence χ ( p ( x ) , ρ x ) = H ( p ) + S ( ρ ) − S ( σ ) = mutual information between the classical and the quantum state Mikael Skoglund, Quantum Info 3/15 Fano’s Inequality For discrete random variables, consider X = variable of interest Y = observed variable ˆ X = f ( Y ) estimate of X based on Y With P e = Pr( ˆ X � = X ) and h ( x ) = − x log x − (1 − x ) log(1 − x ) , we have Fano’s inequality h ( P e ) + P e log( |X| − 1) ≥ H ( X | Y ) Hence, in the quantum setting: For any measurement that tries to conclude X as ˆ X from ρ , � h ( P e ) + P e log( |X| − 1) ≥ H ( X ) − S ( ρ ) + p ( x ) S ( ρ x ) x ∈X Mikael Skoglund, Quantum Info 4/15

Typical Sequences For a sequence x n = ( x 1 , . . . , x n ) with letters in X and a pmf p ( x ) on X , let T ( x n ) = − 1 � log p ( x i ) n i For fixed n and ε > 0 , let = { x n : | T ( x n ) − H ( p ) | ≤ ε } T ( n ) ε be the set of ε -typical sequences (of length n , given p ) Mikael Skoglund, Quantum Info 5/15 By the (weak) LLN, if X n ∼ � i p ( x i ) then for any ε > 0 there is an N such that for all n > N Pr( X n ∈ T ( n ) ) > 1 − ε ε We also have |T ( n ) | ≤ 2 n ( H ( p )+ ε ) ε and there is an N such that for n ≥ N |T ( n ) | ≥ (1 − ε )2 n ( H ( p ) − ε ) ε Mikael Skoglund, Quantum Info 6/15

Compression We can enumerate all elements of T ( n ) using numbers from ε [1 : M n ] with M n ≥ ⌈ 2 n ( H ( p )+ ε ) ⌉ Assume X n ∼ � i p ( x i ) Compression code: Observe X n = x n ; if x n ∈ T ( n ) then produce ε i ∈ [1 : M n ] corresponding to x n ; if x n / ∈ T ( n ) then declare error ε For any ε > 0 , there is an N such that for all n > N , Pr( error ) ≤ ε as long as n log M n ≥ H ( p ) + ε + 1 1 n Mikael Skoglund, Quantum Info 7/15 On the other hand, from Fano’s inequality Pr( error )log M n + 1 n ≥ H ( p ) − 1 n log M n n Hence, for large n , choosing n − 1 log M n slightly bigger than H ( p ) is the best compression we can accomplish Mikael Skoglund, Quantum Info 8/15

Preservation of Entanglement For discrete random variables X and Y with join pmf p ( x, y ) , the mutual information I ( X ; Y ) measures the degree of mutual dependence, or (nonlinear) correlation In quantum systems, two states are dependent on each-other if they are entangled Consider a mixed state ρ in H with purification | ψ � in H ⊗ R , i.e. ρ = Tr R | ψ �� ψ | for some space R R can model the unknown environment; if we had access to both H and R then we would be considering the pure state | ψ �� ψ | The system H is entangled with the environment R , as characterized by the entangled state | ψ � ∈ H ⊗ R Mikael Skoglund, Quantum Info 9/15 Assume E is applied to ρ in H , resulting in the state σ in H ⊗ R . Then, the entanglement fidelity of ( ρ, E ) is defined as F ( ρ, E ) = � ψ | σ | ψ � F ( ρ, E ) does not depend on R , 0 ≤ F ( ρ, E ) ≤ 1 We can easily verify that F ( ρ, E ) = ( F ( | ψ �� ψ | , σ )) 2 where F ( | ψ �� ψ | , σ ) is the regular (static) fidelity between the pure � ρ 1 / 2 σρ 1 / 2 ) state | ψ �� ψ | and σ (remember F ( ρ, σ ) = Tr F ( ρ, E ) measures how well entanglement is preserved by E Let { E i } be the operation elements of E , then we also have � | Tr( ρE i ) | 2 F ( ρ, E ) = i Mikael Skoglund, Quantum Info 10/15

Typical Subspaces Any density operator ρ associated with a system H has an eigen-decomposition ρ = � i λ i | x i �� x i | Since � i λ i = 1 , we can interpret this representation for ρ as an information source; | x i � is emitted with probability p ( x i ) = λ i Let ρ n = ρ ⊗ · · · ⊗ ρ , | x n � = | x i 1 · · · x i n � = | x i 1 � ⊗ · · · ⊗ | x i n � and H n = H ⊗ · · · ⊗ H ( n times) The states ρ n and | x n � correspond to “using the information source” ( ρ, H ) a number of n independent times With T ( | x n � ) = − n − 1 � n m =1 log p ( x i m ) let T ( n ) = {| x n � : | T ( | x n � ) − S ( ρ ) | ≤ ε } ε and define the typical subspace S ( n ) = span T ( n ) = span {| x n � : | x n � ∈ T ( n ) } ε ε ε Mikael Skoglund, Quantum Info 11/15 denote the projection operator from H n to S ( n ) Let P ( n ) ε ε For any ε > 0 there is an N such that for n > N Tr( P ( n ) ρ n ) ≥ 1 − ε ε Furthermore, for any n and ε Tr P ( n ) ≤ 2 n ( S ( ρ )+ ε )) ε and for any ε > 0 there is an N such that for n > N Tr P ( n ) ≥ (1 − ε )2 n ( S ( ρ ) − ε )) ε Mikael Skoglund, Quantum Info 12/15

Compression C n maps states in H n to states in a space G n of dimension D n D n maps states in G n back to states in H n Assume | ψ � is a purification of ρ n in H n ⊗ R , and let E n = D n ◦ C n Let σ n be the resulting state in H n ⊗ R The corresponding entanglement fidelity is F ( ρ n , E n ) = � ψ | σ n | ψ � Mikael Skoglund, Quantum Info 13/15 A compression scheme Select G n ⊃ S ( n ) ⇒ Tr P ( n ) ≤ D n ε ε Set C n = P ( n ) and D n = I (identity) ε Then for any ε > 0 there is an N such that for n > N ) | 2 ≥ | 1 − ε | 2 ≥ 1 − 2 ε F ( ρ n , E n ) ≥ | Tr( ρ n P ( n ) ε It also holds that Tr P ( n ) ≤ 2 n ( S ( ρ )+ ε )) ε Thus F ( ρ n , E n ) > 1 − 2 ε as long as 1 n log D n > S ( ρ ) + ε Mikael Skoglund, Quantum Info 14/15

Converse: It can be shown that, if 1 lim n log D n < S ( ρ ) n →∞ then F ( ρ n , S n ) → 0 for any projector S n If H is d -dimensional, H n is d n -dimensional; i.e. it takes n log d qubits to describe a state in H n Then the best compression we can have is from log d qubits to S ( ρ ) ( ≤ log d ) qubits, per use of the source ( ρ, H ) , with preserved entanglement F ( ρ n , S n ) → 1 1 − ( F ( ρ, σ )) 2 we could also � � Since 1 − F ( ρ, σ ) ≤ V ( ρ, σ ) ≤ use V ( ρ, σ ) = 2 − 1 Tr | ρ − σ | as fidelity metric, F ( ρ, σ ) → 1 ⇐ ⇒ V ( ρ, σ ) → 0 Mikael Skoglund, Quantum Info 15/15