Quantum capacity for serial composition of quantum channels

Recently, I have been working with quantum channel capacity for quantum-quantum channels and I was wondering if there exist some results for channel compositions.

Specifically, I have been looking for results on what happens to quantum channel capacity when a serial composition of quantum channels is considered, i.e. $$C(\mathcal{N_1}\circ\mathcal{N_2}) = ?$$ where $$\mathcal{N_1}$$ and $$\mathcal{N_2}$$ are the quantum channels in consideration.

The channels are described by their Kraus operators $$\{E_k^1\}$$ and $$\{E_j^2\}$$. Note that the serial composition channel is a quantum channel that will be described by Kraus operators $$\{E_k^1 \cdot E_j^2\}$$, i.e. all the possible products of the Kraus operators of the individual channels.

I have been looking around but have not found anything interesting yet. I would appreciate any information, partial results and especially literature on the topic.

TL;DR

Quantum capacity of $$\mathcal{N}_2\circ\mathcal{N}_1$$ can be anywhere between zero and the minimum of the quantum capacities of $$\mathcal{N}_1$$ and $$\mathcal{N}_2$$.

Background

Quantum capacity of a quantum channel $$\mathcal{N}$$ is defined as the greatest real number $$Q(\mathcal{N})$$ such that for any $$R < Q(\mathcal{N})$$ (representing a transmission rate) and any $$\delta > 0$$ (representing an acceptable error rate), there exists a quantum error correcting code with encoding operation $$\mathcal{E}$$ that maps $$m$$ qubits into $$n$$ inputs to the channel $$\mathcal{N}$$ and decoding operation $$\mathcal{D}$$ that maps $$n$$ outputs from the channel $$\mathcal{N}$$ to $$m$$ qubits which enables recovery of any $$m$$-qubit input state $$|\psi\rangle$$ with fidelity $$1-\delta$$ and such that $$\frac{m}{n} > R$$. See also discussion on page $$1$$ and figure $$1$$ on page $$4$$ in this paper.

Let $$\mathcal{N}_2\circ\mathcal{N}_1$$ denote the channel representing the application of $$\mathcal{N}_1$$ followed by the application of $$\mathcal{N}_2$$ and note that

$$0 \le Q(\mathcal{N}_2\circ\mathcal{N}_1) \le \min(Q(\mathcal{N}_1),Q(\mathcal{N}_2)).\tag1$$

The first inequality follows from the fact $$m$$ and $$n$$ in the definition above are non-negative integers and therefore $$Q(\mathcal{N})$$ is a non-negative real number for any channel $$\mathcal{N}$$. The second inequality follows from the fact that we can absorb $$\mathcal{N}_1$$ (respectively, $$\mathcal{N}_2$$) into the encoding operation $$\mathcal{E}$$ (respectively, the decoding operation $$\mathcal{D}$$) to improve $$Q(\mathcal{N}_2)$$ (respectively, $$Q(\mathcal{N}_1)$$) to at least $$Q(\mathcal{N}_2\circ\mathcal{N}_1)$$.

Two extreme cases

Can we tighten the bounds in $$(1)$$? It turns out that without additional assumptions on the channels, we can't. To show this, we will use the quantum erasure channel

$$\mathcal{R}_\epsilon(\rho) = (1 - \epsilon)\rho + \epsilon|2\rangle\langle 2|$$

to construct two examples, one saturating the left inequality in $$(1)$$ and one saturating the right inequality. We choose the erasure channel because its quantum capacity is known and is given by the formula

$$Q(\mathcal{R}_\epsilon) = \max(0, 1 - 2\epsilon),\tag2$$

see equation $$(2)$$ on page $$2$$ (and figure $$2(a)$$ on page $$4$$) in this paper. Now, the effect of composing two instances of the erasure channel is

\begin{align} \mathcal{R}_\epsilon(\mathcal{R}_\epsilon(\rho)) &= (1 - \epsilon)\mathcal{R}_\epsilon(\rho) + \epsilon\mathcal{R}_\epsilon(|2\rangle\langle 2|) \\ &= (1 - \epsilon)^2\rho + (2\epsilon - \epsilon^2)|2\rangle\langle 2| \\ &= \mathcal{R}_{2\epsilon-\epsilon^2}(\rho). \end{align}

Thus,

\begin{align} 0 &\le Q(\mathcal{R}_{1/3} \circ \mathcal{R}_{1/3}) \le \min(Q(\mathcal{R}_{1/3}), Q(\mathcal{R}_{1/3})) \\ 0 &\le Q(\mathcal{R}_{5/9}) \le \min\left(\frac13, \frac13\right) \\ 0 &\le 0 \le \frac13 \\ \end{align}

saturating the left inequality in $$(1)$$ and

\begin{align} 0 &\le Q(\mathcal{R}_0 \circ \mathcal{R}_0) \le \min(Q(\mathcal{R}_0), Q(\mathcal{R}_0)) \\ 0 &\le Q(\mathcal{R}_0) \le \min\left(1, 1\right) \\ 0 &\le 1 \le 1 \\ \end{align}

saturating the right inequality in $$(1)$$.

• Thank you for the answer. As you state that assumptions on the channels should be made for tightening the bounds, I can tell you that I am interested in the combined amplitude and phase damping channel. This channel is the serial concatenation of an amplitude damping and a dephasing channel. I do not know if you are aware of some result on this channel. You can see the definition for those channels in: ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9201447 – Josu Etxezarreta Martinez Apr 28 at 15:32
• Qubit channels with two-dimensional environment (i.e. with two Kraus operators) have a known formula for quantum capacity (see this paper). Both, the amplitude damping and phase damping channels have two Kraus operators (in their minimal representations). Unfortunately, I think your channel has three. Still, you may be able to use results of section IV in the paper to get an upper bound which may be tighter than the one above. HTH – Adam Zalcman Apr 28 at 17:08