Ph219C/CS219C
Due: Thursday 14 April 2022
1.1 Positivity of quantum relative entropy
a) Show that ln x ≤ x−1 for all positive real x, with equality iff x = 1.
b) The (classical) relative entropy of a probability distribution {p(x)} relative to {q(x)} is defined as
D(p∥q)≡p(x)(logp(x)−logq(x)) . (1) x
with equality iff the probability distributions are identical. Hint:
D(p ∥ q) ≥ 0 , (2) Apply the inequality from (a) to ln (q(x)/p(x)).
c) The quantum relative entropy of the density operator ρ with re- spect to σ is defined as
D(ρ∥σ)=trρ(logρ−logσ) . (3) Let {pi} denote the eigenvalues of ρ and {qa} denote the eigen-
values of σ. Show that
D(ρ∥σ)=pi logpi −Dialogqa , (4)
where Dia is a doubly stochastic matrix. Express Dia in terms of the eigenstates of ρ and σ. (A matrix is doubly stochastic if its entries are nonnegative real numbers, where each row and each column sums to one.)
d) Show that if Dia is doubly stochastic, then (for each i)
log Diaqa ≥Dialogqa , (5) aa
with equality only if Dia = 1 for some a.
D(ρ ∥ σ) ≥ D(p ∥ r) , (6)
f) ShowthatD(ρ∥σ)≥0,withequalityiffρ=σ. 1.2 Properties of Von Neumann entropy
a) Use nonnegativity of quantum relative entropy to prove the subad- ditivity of Von Neumann entropy
H(ρAB) ≤ H(ρA) + H(ρB), (7) with equality iff ρAB = ρA ⊗ ρB. Hint: Consider the relative
entropyofρAB andρA⊗ρB.
b) Use subadditivity to prove the concavity of the Von Neumann en-
e) Show that
where ri = a Diaqa.
Hint: Consider
H( pxρx) ≥ pxH(ρx) . (8) xx
ρAB=px(ρx)A⊗(|x⟩⟨x|)B , (9) x
where the states {|x⟩B} are mutually orthogonal. c) Use the condition
H(ρAB) = H(ρA) + H(ρB) iff ρAB = ρA ⊗ ρB (10) to show that, if all px’s are nonzero,
H pxρx = pxH(ρx) (11)
xx iff all the ρx’s are identical.
1.3 Monotonicity of quantum relative entropy
Quantum relative entropy has a property called monotonicity:
D(ρA∥σA) ≤ D(ρAB∥σAB); (12)
The relative entropy of two density operators on a system AB cannot be less than the induced relative entropy on the subsystem A.
Programming Help
a) Use monotonicity of quantum relative entropy to prove the strong subadditivity property of Von Neumann entropy. Hint: On a tripartite system ABC, consider the relative entropy of ρABC and ρA ⊗ρBC.
b) Use monotonicity of quantum relative entropy to show that the action of a quantum channel N cannot increase relative entropy:
D(N (ρ)∥N (σ) ≤ D(ρ∥σ), (13) Hint: Recall that any quantum channel has an isometric dilation.
1.4 Separability and majorization
The hallmark of entanglement is that in an entangled state the whole is less random than its parts. But in a separable state the correlations are essentially classical and so are expected to adhere to the classical principle that the parts are less disordered than the whole. The ob- jective of this problem is to make this expectation precise by showing that if the bipartite (mixed) state ρAB is separable, then
λ(ρAB) ≺ λ(ρA) , λ(ρAB) ≺ λ(ρB) . (14) Here λ(ρ) denotes the vector of eigenvalues of ρ, and ≺ denotes ma-
jorization.
A separable state can be realized as an ensemble of pure product states,
so that if ρAB is separable, it may be expressed as
ρAB = pa |ψa⟩⟨ψa| ⊗ |φa⟩⟨φa| . (15)
We can also diagonalize ρAB, expressing it as
ρAB = rj|ej⟩⟨ej| , (16) j
where {|ej⟩} denotes an orthonormal basis for AB; then by the HJW theorem, there is a unitary matrix V such that
√r|e⟩=V √p|ψ⟩⊗|φ⟩. (17) jj jaaaa
Also note that ρA can be diagonalized, so that
ρA = pa|ψa⟩⟨ψa| = sμ|fμ⟩⟨fμ| ; (18) aμ
4 here {|fμ⟩} denotes an orthonormal basis for A, and by the HJW
theorem, there is a unitary matrix U such that
√p|ψ⟩=U √s|f⟩. (19)
Now show that there is a doubly stochastic matrix D such that
rj =Djμsμ . (20)
That is, you must check that the entries of Djμ are real and non- negative, and that j Djμ = 1 = μ Djμ. Thus we conclude that λ(ρAB) ≺ λ(ρA). Just by interchanging A and B, the same argument also shows that λ(ρAB) ≺ λ(ρB).
Remark: Note that it follows from the Schur concavity of Shannon entropy that, if ρAB is separable, then the von Neumann entropy has the properties H(AB) ≥ H(A) and H(AB) ≥ H(B). Thus, for separable states, conditional entropy is nonnegative: H(A|B) = H(AB) − H(B) ≥ 0 and H(B|A) = H(AB) − H(A) ≥ 0. In contrast, if the state of AB is an entangled pure state, then H(AB) = 0 and H(B|A) = H(A|B) < 0.
1.5 The first law of Von Neumann entropy
We’ll use S(ρ) = −tr (ρ ln ρ) to denote the entropy of a density oper- ator when using natural logarithms instead of logarithms with base 2. As in §10.2.6, a d × d density matrix can be expressed as
ρ= −K, (21)
where K is a d × d Hermitian matrix called the modular Hamiltonian associated with ρ. (Under this definition of K, we have the freedom to shift K by a multiple of the identity operator without changing ρ.) We assume that ρ has full rank; that is, it has d positive eigenvalues. We will see that when ρ changes slightly, the first-order change in S(ρ) can be related to the change in the expectation value of K.
a) SupposeA(λ)isaboundedHermitianoperatorsmoothlyparametrized by the real number λ. Show that
d n dA n−1
dλ (trA ) = n tr dλ A . (22)
Code Help
Do not assume that dA/dλ commutes with A. b) Suppose the density operator is perturbed slightly:
ρ→ρ′ =ρ+δρ. (23) Since ρ and ρ′ are both normalized density operators, we have
tr (δρ) = 0. Show that
S(ρ′) − S(ρ) = tr ρ′K − tr (ρK) + O (δρ)2 ; (24)
δS = δ⟨K⟩ (25)
to first order in the small change in ρ. This statement generalizes the first law of thermodynamics; for the case of a thermal density operator with K = H/T (where H is the Hamiltonian and T is the temperature), it becomes the more familiar statement
δE = δ⟨H⟩ = TδS. (26)
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