I have been reading about the Quantum Channel Capacity and it seems to be an open problem to find such capacity in general. Quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to a receiver.

Known results on the field are the Hashing bound, which is a lower bound on such quantum capacity and which is given by the LSD (Lloyd-Shor-Devetak) theorem; or the HSW (Holevo-Schumacher-Westmoreland) theorem for classical capacity over quantum channels.

I was wondering if there have been any advances in what a general expression for the quantum capacity since the release of those theorems. A glimpse on the advance in such field is enough for me, or references to papers where such task is developed.


Let's recap a bit:

In classical information theory, the analogous formula is the Shannon noisy channel coding theorem (https://en.wikipedia.org/wiki/Noisy-channel_coding_theorem). It's charming, because it is basically just a very simple optimization of the mutual information.

The quantum channel capacity is that it is given by

$$ \lim\limits_{n\to\infty} \frac{1}{n}Q(T^{\otimes n}) $$

where $T$ is the quantum channel in question and $Q$ is the coherent information.

Now let's try to answer your question:

Obviously, we'd want a formula that doesn't depend on $n$ just like in the classical case. The problem is: It's known that such an expression cannot exist (see https://arxiv.org/abs/quant-ph/9706061). It gets worse: You could hope that there is a maximal $n$ after which you at least know that the capacity is zero. But that's false (published recently: https://www.nature.com/articles/ncomms7739.pdf?origin=ppub).

In addition, if you use two different channels, their capacity can both be zero while the capacity when used together is larger than zero (see https://arxiv.org/abs/0807.4935), which makes it even more difficult to imagine simple formulas for channels.

This implies that the best we can hope for is formulas for particular channels or particular subclasses of channels. There are a few results scattered throughout the literature. For instance, capacities of certain Gaussian bosonic channels are known (https://arxiv.org/pdf/quant-ph/0606132.pdf).

However, please note that the quantum capacity is only one of many capacities defined and it's not necessarily the most interesting one. A different capacity which is often discussed in the literature is the classical capacity of a quantum channel (how much classical information can I send over a quantum channel?). Let me point you to a recent review providing a lot of pointers to the literature: https://arxiv.org/pdf/1801.02019.pdf

  • $\begingroup$ Thanks for the answer, I will go through the literature you sent to see if I can get some more insight about the field! $\endgroup$ – Josu Etxezarreta Martinez Jul 2 '18 at 8:04

Known results on the field are the Hashing bound, which is a lower bound on such quantum capacity and which is given by the LSD (Lloyd-Shor-Devetak) theorem; or the HSW (Holevo-Schumacher-Westmoreland) theorem for classical capacity over quantum channels.

Those theorems are not very new, I was able to find some newer work.

A quick list of those theories and their dates:

"... quantitative bounds are put to the computational power of an ‘ultimate laptop’ with a mass of one kilogram confined to a volume of one litre. ... Applying this result to a one kilogram computer with energy $E = mc^2 = 8.9874 × 10^{16}$ joules show that our ultimate laptop can perform a maximum of $5.4258 × 10^{50}$ operations per second.

There's no suggestion that is a reasonable and obtainable upper bound, it's simply a max limit.

Dr Shor's papers should be consulted directly to derive his capacity formulas.

"... the quantum capacity $Q(\mathcal{N})$ of a quantum channel $\mathcal{N}$

$$\qquad\qquad\qquad Q( \mathcal{N} ) = \tilde{Q} ( \mathcal{N} ) = E( \mathcal{N} ) = \lim\limits_{l \to \infty} \frac{1}{l} \max\limits_{\rho \in \mathcal{H}^{\otimes l}_\mathcal{P}} \; I_c(\rho, \mathcal{N}^{\otimes l}). \qquad\qquad\qquad (53)$$

The Holevo-Schumacher-Westmoreland theorem is arguably over 50 years old, certainly 45.

  • The paper "Optimal signal ensembles" (Dec 31 1999, last updated Apr 4, 2018), by Benjamin Schumacher and Michael D. Westmoreland, on page 2, reads:

"Holevo$^{[1]}$ proved (as Gordon$^{[2]}$ and Levitin$^{[3]}$ had previously conjectured) that the mutual information between the input and output of this channel, regardless of Bob’s choice of decoding observable, can never be greater than $\chi$, where:

$$\qquad\qquad\qquad\qquad\qquad \chi = S(\rho) - \sum_k p_kS(\rho_k) \qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)$$

where $S(ρ) = −\text{Tr}\, \rho \, \text{log} \, \rho$ is the von Neumann entropy of the density operator $\rho$.

More recently, it has been shown by Holevo$^{[4]}$ and by Schumacher and Westmoreland$^{[5]}$ that the Holevo bound is asymptotically achievable."

[1] A. S. Kholevo, Probl. Peredachi Inf. 9, 3 (1973) [Probl. Inf. Transm. (USSR) 9, 110 (1973)].

[2] J. P. Gordon, in Quantum Electronics and Coherent Light, Proceedings of the International School of Physics “Enrico Fermi,” Course XXXI, edited by P. A. Miles (Academic, New York, 1964), pp. 156-181.

[3] L. B. Levitin, “On the quantum measure of the amount of information,” in Proceedings of the IV National Conference on Information Theory, Tashkent, 1969, pp. 111–115 (in Russian); “Information Theory for Quantum Systems,” in Information, Complexity, and Control in Quantum Physics, edited by A. Blaqui`ere, S. Diner, and G. Lochak (Springer, Vienna, 1987).

[4] A. S. Holevo, IEEE Trans. Inform. Theory 44, 269 (1998).

[5] B. Schumacher and M. Westmoreland, Phys. Rev. A 51, 2738 (1997).

I was wondering if there have been any advances in what a general expression for the quantum capacity [is] since the release of those theorems. A glimpse on the advance in such field is enough for me or references to papers where such task is developed.

Some of the more recent work is:

  • "From Classical to Quantum Shannon Theory" (7 Jun 2011, last revised 22 Mar 2016) by Mark M. Wilde

    The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and post-millenium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quantum information theory (Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution (Part III), and we develop many of the tools necessary for understanding information transmission or compression (Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory.

  • "A Mini-Introduction To Information Theory" (30 May 2018, last revised 3 Jun 2018) by Edward Witten - Nicholas Teague's review and presentation.

    This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case.

Noboru Watanabe lists a number of papers on ReseachGate about key attacks and quantum communication. In particular, see:

  • "Quantum Entropy and Its Applications to Quantum Communication and Statistical Physics" May 2010 in Entropy 12(5), by Masanori Ohya and Noboru Watanabe

    ... We will review the mathematical aspects of quantum entropy (entropies) and discuss some applications to quantum communication, statistical physics. All topics taken here are somehow related to the quantum entropy that the present authors have been studied. ...

  • "Quantum probability from classical signal theory" Nov 2011, Article in International Journal of Quantum Information 9(supp01), by Masanori Ohya, Andrei Khrennikov and Noboru Watanabe

    We present quantum mechanics (QM) as theory of special classical random signals. On one hand, this approach provides a possibility to go beyond conventional QM: to create a finer description of micro-processes than given by the QM-formalism. In fact, we present a model with hidden variables of the wave-type. On the other hand, our approach establishes coupling between quantum and classical information theories. We recall that quantum information theory has already been used for description of the entropy of Gaussian input signals for noisy channels. The entropy of a classical random input was invented as the entropy of the quantum density operator corresponding to the covariance operator of the input process. In this paper, we proceed the other way around: we apply classical signal theory to create a measurement model which reproduces quantum probabilities.

  • "On Treatment of Communication Processes by Quantum Entropies" Mar 2012, Noboru Watanabe

    In information communication theory, one can study the efficiency of information transmission of the communication processes by using the Shannon's type inequalities by means of the entropy and the mutual entropy. In this paper, we discuss about entropy type inequalities for treating the classical Gaussian communication processes consistently.

  • "An Entropy Based Treatment of Gaussian Communication Process for General Quantum Systems" Sept 2013, by Noboru Watanabe

    The quantum entropy introduced by von Neumann around 1932 describes the amount of information of the quantum state itself. It was extended by Ohya for C * -systems before Conne-Narnhoffer-Thirring (CNT) entropy. The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras and it was subsequently extended by H. Araki [Publ. Res. Inst. Math. Sci. Kyoto Univ. 11, No. 3, 809–833 (1975/76; Zbl 0326.46031); ibid. 13, No. 1, 173–192 (1977/78; Zbl 0374.46055)] and A. Uhlmann [Comm. Math. Phys. 54, No. 1, 21–32 (1977; Zbl 0358.46026)] for general von Neumann algebras and * -algebras, respectively. By introducing a new notion, the so-called compound state, M. Ohya, IEEE Trans. Inf. Theory 29, 770–774 (1983; Zbl 0511.94007)] succeeded to construct the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review Ohya’s S-mixing entropy and the quantum mutual entropy for general quantum systems. Based on a concept of structure equivalent, we apply the general framework of quantum communication to the Gaussian communication processes.

  • "Entropy type complexity of quantum processes" Nov 2014 in Physica Scripta T163(T163), by Noboru Watanabe

    von Neumann entropy represents the amount of information in the quantum state, and this was extended by Ohya for general quantum systems 10. Umegaki first defined the quantum relative entropy for σ -finite von Neumann algebras, which was extended by Araki, and Uhlmann, for general von Neumann algebras and *-algebras, respectively. In 1983 Ohya introduced the quantum mutual entropy by using compound states; this describes the amount of information correctly transmitted through the quantum channel, which was also extended by Ohya for general quantum systems. In this paper, we briefly explain Ohyaʼs S-mixing entropy and the quantum mutual entropy for general quantum systems. By using structure equivalent class, we will introduce entropy type functionals based on quantum information theory to improve treatment for the Gaussian communication process.

Now back to papers by other authors.

Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. By constructing an upperbound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed "teleportation stretching", we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we determine the fundamental rate-loss trade-off affecting any protocol of quantum key distribution. Our findings set the ultimate limits of point-to-point quantum communications and provide the most precise and general benchmarks for quantum repeaters.

  • "Quantum Information - Chapter 10. Quantum Shannon Theory" (Jan 2018), by John Preskill (.PDF)

    This is the 10th and final chapter of my book Quantum Information, based on the course I have been teaching at Caltech since 1997. An early version of this chapter (originally Chapter 5) has been available on the course website since 1998, but this version is substantially revised and expanded. The level of detail is uneven, as I’ve aimed to provide a gentle introduction, but I’ve also tried to avoid statements that are incorrect or obscure. Generally speaking, I chose to include topics that are both useful to know and relatively easy to explain; I had to leave out a lot of good stuff, but on the other hand the chapter is already quite long. My version of Quantum Shannon Theory is no substitute for the more careful treatment in Wilde’s book 1, but it may be more suitable for beginners.

I'll add a few more links if OP Josu Etxezarreta Martinez or anyone else runs out of reading, otherwise I'll review this in several months to ensure it remains correct and current.

For Math.SE / English.SE: "I have been reading about the Quantum Channel Capacity and it seems to be an open problem to find such capacity in general." [See also: Channel Capacity and Transmission rate of a channel].

In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (no solution for it is known).

Let's focus on quantum computing, leave that to Math.SE / English.SE, and limit QC.SE to answering QC questions. There are some open problems within some of the conjectures about channel capacity.

  • $\begingroup$ Thanks for the detailed answer and the links, I will go through them. However, I have to disagree with your opinion that questions about Quantum Information Theory should be left to the Math.SE site. This questions are really important for the development of quantum computers and the communications in between them, so I think that considering them in QC.SE is the most logical thing to do. $\endgroup$ – Josu Etxezarreta Martinez Jul 2 '18 at 8:12
  • $\begingroup$ @JosuEtxezarretaMartinez - You are most welcome. You should re-read the part you disagree with. I appreciate that English is not your first language. The portion at the end says that the definition of an "open problem" is not best suited for here, where we want to focus on quantum computing. Don't let the people whom don't understand and the drive-by downvoters discourage you from our site we have a number of truly exceptional experts from around the world with first rate abilities. $\endgroup$ – Rob Jul 2 '18 at 11:06

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