Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$ where $(s_1,s_2,\dots)$ is the sequence of eigenvalues of the operator $|T|=(T^*T)^{1/2})$ written in any order.

Let $H_1$ and $H_2$ be Hilbert spaces of dimention $n$ and $m$ respectively. Let $H_1\otimes_2 H_2$ denote their Hilbert-space tensor product (hence, an inner product space of dimension $nm$). Consider the map $$\varphi:S_1(H_1)\hat{\otimes}S_1(H_2)\to S_1(H_1\otimes_2 H_2),$$ which is defined by $$\varphi( A \otimes B) (\xi_1\otimes\xi_2) = (A\xi_1 )\otimes (B\xi_2) \quad(\xi_1\in H_1, \xi_2\in H_2) $$

Is it the case that $$\Vert\varphi^{-1}\Vert=\min\lbrace{m, n}\rbrace ?$$ Here, $\hat{\otimes}$ denotes the projective tensor product of Banach spaces.

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