Continuity of the quantum conditional entropy

The quantum conditional entropy is given by

\[S(A|B)_\rho := S(\rho^{AB}) - S(\rho^{B})\]

where $S$ is the von Neumann entropy,

\[S(\rho) := - \text{tr}(\rho \log \rho)\]

and $\rho$ is a positive semidefinite trace-1 matrix (density matrix).

Here, $\rho^{AB}$ represents an operator on the tensor product of two finite-dimensional Hilbert spaces $A$ and $B$ (with dimensions $d_A$ and $d_B$ respectively), so we can regard $\rho_{AB}$ as a matrix on the vector space $\mathbb{C}^{d_Ad_B}$. Moreover, $\rho^B$ denotes the partial trace of $\rho^{AB}$ over the system $A$, so $\rho^B$ is a matrix on $\mathbb{C}^{d_B}$.

One question is how much can $S(A|B)_\rho$ vary between two density matrices $\rho$ and $\sigma$ as a function of the trace-distance $\text{trdist}(\rho, \sigma) := \|\rho-\sigma\|_1 = \frac{1}{2} \text{tr}\left(\sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)}\right)$ (that is, half of the nuclear norm). Here the trace distance is meaningful as it is the quantum analog to the total variation distance, and has an interpretation in terms of the maximal possible probability to distinguish between $\rho$ and $\sigma$ by measurement.

The Alicki-Fannes-Winter (AFW) bound (Winter 2015, Lemma 2) states that if $\rho$ and $\sigma$ are density matrices, then $\text{trdist}(\rho, \sigma) \leq \varepsilon \leq 1$, then

\[| S(A|B)_\rho - S(A|B)_\sigma| \leq 2 \varepsilon \log d_A + (1 + \varepsilon) h \left(\frac{\varepsilon}{1+\varepsilon}\right)\]

where $h(x) = -x\log x - (1-x)\log(1-x)$ is the binary entropy.

We can illustrate this bound by computing

\[ \max_{\rho} S(A|B)_\rho - S(A|B)_\sigma\]

for a fixed state $\sigma$, and comparing to the AFW bound.

We will choose $d_A=d_B=2$, and $\sigma$ as the maximally entangled state:

\[\sigma = \frac{1}{2}\begin{pmatrix}1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1\\\end{pmatrix}\]

math

First, we can formulate the conditional entropy in terms of the relative entropy using the relationship

\[S(A|B)_\rho = - D(\rho^{AB} \| I^A \otimes \rho^B)\]

math

where $D$ is the quantum relative entropy and $I^A$ is the $d_A$-dimensional identity matrix. Thus:

using Convex
using LinearAlgebra: I

function quantum_conditional_entropy(ρ_AB, d_A, d_B)
    ρ_B = partialtrace(ρ_AB, 1, [d_A, d_B])
    return -quantum_relative_entropy(ρ_AB, kron(I(d_A), ρ_B))
end

quantum_conditional_entropy (generic function with 1 method)

Now we setup the problem data:

ϵ = 0.1
d_A = d_B = 2
σ_AB = 0.5 * [
    1 0 0 1
    0 0 0 0
    0 0 0 0
    1 0 0 1
]

4×4 Matrix{Float64}:
 0.5  0.0  0.0  0.5
 0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0
 0.5  0.0  0.0  0.5

And we build and solve problem itself

using SCS

ρ_AB = HermitianSemidefinite(d_A * d_B)
add_constraint!(ρ_AB, tr(ρ_AB) == 1)

problem = maximize(
    quantum_conditional_entropy(ρ_AB, d_A, d_B),
    0.5 * nuclearnorm(ρ_AB - σ_AB) ≤ ϵ,
)

solve!(problem, SCS.Optimizer; silent = false)

Problem statistics
  problem is DCP         : true
  number of variables    : 1 (32 scalar elements)
  number of constraints  : 3 (67 scalar elements)
  number of coefficients : 31
  number of atoms        : 29

Solution summary
  termination status : OPTIMAL
  primal status      : FEASIBLE_POINT
  dual status        : FEASIBLE_POINT
  objective value    : -0.2582

Expression graph
  maximize
   └─ Convex.NegateAtom (concave; negative)
      └─ quantum_relative_entropy (convex; positive)
         ├─ 4×4 complex variable (id: 351…737)
         └─ vcat (affine; complex)
            ├─ …
            └─ …
  subject to
   ├─ ≤ constraint (convex)
   │  └─ + (convex; real)
   │     ├─ * (convex; positive)
   │     │  ├─ …
   │     │  └─ …
   │     └─ [-0.1;;]
   ├─ PSD constraint (convex)
   │  └─ vcat (affine; real)
   │     ├─ hcat (affine; real)
   │     │  ├─ …
   │     │  └─ …
   │     └─ hcat (affine; real)
   │        ├─ …
   │        └─ …
   ├─ == constraint (affine)
   │  └─ + (affine; complex)
   │     ├─ sum (affine; complex)
   │     │  └─ …
   │     └─ [-1;;]
   ⋮

We can then check the observed difference in relative entropies:

difference = evaluate(
    quantum_conditional_entropy(ρ_AB, d_A, d_B) -
    quantum_conditional_entropy(σ_AB, d_A, d_B),
)

0.4349438715678587

We can compare to the bound:

h(x) = -x * log(x) - (1 - x) * log(1 - x)
bound = 2 * ϵ * log(d_A) + (1 + ϵ) * h(ϵ / (1 + ϵ))

0.473729143196151

In fact, in this case we know the maximizer is given by

ρ_max = σ_AB * (1 - ϵ) + ϵ * (I(d_A * d_B) - σ_AB) / (d_A * d_B - 1)

4×4 Matrix{Float64}:
 0.466667  0.0        0.0        0.433333
 0.0       0.0333333  0.0        0.0
 0.0       0.0        0.0333333  0.0
 0.433333  0.0        0.0        0.466667

We can check that ρ_AB obtained the right value:

norm(evaluate(ρ_AB) - ρ_max)

1.158261803578682e-7

Here we see a result within the expected tolerances of SCS.

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