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The Golden-Thompson inequality holds not only for the trace but in fact also for all unitarily invariant norms, for example the Schatten norms. That is, it holds that

$$\| e^{B+C} \|_p \leq \| e^{B} e^C \|_p$$

for any $p \geq 1$. (See, for example, Theorem 9.3.7 in ``R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer Verlag, New York, 1997'') If we apply this to $p=\infty$ we get something similar to what you want, albeit including a maximization:

$$ \mathrm{tr}( A e^{B+C} ) \leq \max_{A' : \| A' \|_1 = 1} \mathrm{tr} \big( A' e^{B} e^{C} \big) .$$

This statement holds for any $0 \leq A \leq 1$ with $\mathrm{tr}(A) = 1$.

The Golden-Thompson inequality holds not only for the trace but in fact also for all unitarily invariant norms, for example the Schatten norms. That is, it holds that

$$\| e^{B+C} \|_p \leq \| e^{B} e^C \|_p$$

for any $p \geq 1$. (See, for example, Theorem 9.3.7 in ``R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer Verlag, New York, 1997'') If we apply this to $p=\infty$ we get something similar to what you want, albeit including a maximization:

$$ \mathrm{tr}( A e^{B+C} ) \leq \max_{A' : \| A' \|_1 = 1} \mathrm{tr} \big( A' e^{B} e^{C} \big) .$$

This statement holds for any $0 \leq A \leq 1$ with $\mathrm{tr}(A) = 1$, or in fact any $A$ with $\| A\|_1 \leq 1$.

The Golden-Thompson inequality holds not only for the trace but in fact also for all unitarily invariant norms, for example the Schatten norms. That is, it holds that

$$\| e^{B+C} \|_p \leq \| e^{B} e^C \|_p$$

for any $p \geq 1$. (See, for example, Theorem 9.3.7 in ``R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer Verlag, New York, 1997'') If we apply this to $p=\infty$ we get something similar to what you want, albeit including a maximization:

$$ \mathrm{tr}( A e^{B+C} ) \leq \max_{A' : \| A' \|_1 = 1} \mathrm{tr} \big( A' e^{B} e^{C} \big) .$$

This statement holds for any $0 \leq A \leq 1$.

The Golden-Thompson inequality holds not only for the trace but in fact also for all unitarily invariant norms, for example the Schatten norms. That is, it holds that

$$\| e^{B+C} \|_p \leq \| e^{B} e^C \|_p$$

for any $p \geq 1$. (See, for example, Theorem 9.3.7 in ``R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer Verlag, New York, 1997'') If we apply this to $p=\infty$ we get something similar to what you want, albeit including a maximization:

$$ \mathrm{tr}( A e^{B+C} ) \leq \max_{A' : \| A' \|_1 = 1} \mathrm{tr} \big( A' e^{B} e^{C} \big) .$$

This statement holds for any $0 \leq A \leq 1$ with $\mathrm{tr}(A) = 1$.

The Golden-Thompson inequality holds not only for the trace but in fact also for all unitarily invariant norms, for example the Schatten norms. That is, it holds that

$$\| e^{B+C} \|_p \leq \| e^{B} e^C \|_p$$

for any $p \geq 1$. (See, for example, Theorem 9.3.7 in ``R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer Verlag, New York, 1997'') If we apply this to $p=\infty$ we get something similar to what you want, albeit including a maximization:

$$ \mathrm{tr}( A e^{B+C} ) \leq \max_{A' : \| A' \|_1 = 1} \left| \mathrm{tr} \big( A' e^{B} e^{C} \big) \right|.$$

The Golden-Thompson inequality holds not only for the trace but in fact also for all unitarily invariant norms, for example the Schatten norms. That is, it holds that

$$\| e^{B+C} \|_p \leq \| e^{B} e^C \|_p$$

for any $p \geq 1$. (See, for example, Theorem 9.3.7 in ``R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer Verlag, New York, 1997'') If we apply this to $p=\infty$ we get something similar to what you want, albeit including a maximization:

$$ \mathrm{tr}( A e^{B+C} ) \leq \max_{A' : \| A' \|_1 = 1} \mathrm{tr} \big( A' e^{B} e^{C} \big) .$$

This statement holds for any $0 \leq A \leq 1$.