2.2 Quantum Information Theory
2.2.2 Classical Capacity of a Quantum Channel
One generalization of Shannon’s second coding theorem is to consider sending classical information through a quantum channel, Φ(ρ), as described by a completely positive, trace preserving memoryless map.
Since it is classical information we are sending and receiving, the generalization actually involves the original entropic concepts. That is, the concept of mutual information between the initial and final random variables is exactly the same as in Shannon’s second coding theorem, because the variables are classical and the overall transfer of classical information is ‘oblivious’ to the fact that a quantum channel is being used in between.
Whilst the mutual information definition is the same, its calculation is very different.
Indeed, it involves a series of mappings from classical to quantum through the channel and back to classical. Finally, to arrive at the capacity, the maximization over more choices has to be taken because there are extra levels of mappings.
Another major difference for the quantum case, is that if we allow the receiver to conduct a joint quantum measurement on multiply-received quantum states, that is, we allow the receiver to entangle the output before measuring, then the capacity can be shown to be greater than if this is not allowed. This means that the capacity is not additive in the
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2.2 Quantum Information Theory
sense discussed above for the classical capacity. Since the capacity is an asymptotic average quantity of increasing number of uses anyway, we naturally choose to allow joint quantum measurements and over larger and larger block sizes. The full capacity per use should be defined taking into account this extra strategy. Having said this, we treat entanglement in input states as a separate case (entanglement assisted capacity) and prefer to maintain that the sender is only allowed product state inputs.
So the definition of the capacity per use involves aregularization procedure,
CX→Φ→Y = lim
n→∞C(n)/n, (2.1)
whereC(n)is the classical capacity through a quantum channel where joint measurements are allowed on blocks of size n.
Now that we have laid out the setting, we introduce a quantum information entropic quantity called the Holevo χ quantity [5] based on the output states of the channel for given input states, ρi, occurring with probability pi,
χ({pi, ρi}) = S(Φ(X
i
piρi))−X
i
piS (Φ (ρi)).
This is a quantum property because it involves quantum states and channels. However, it refers to our goal of transmitting classical information through the mapping of the classical stateito an assigned quantum state, i7→ρi.
This quantity captures the size of the carrier quantum state being sent through the channel (in the first term) and the average amount of noise the channel disturbs those carriers by (in the second term).
The difference is related to the amount of classical information that can be reliably sent through the quantum channel. Indeed, Holevo [5] and Schumacher-Westmoreland [6]
(HSW) proved that the classical capacity is the maximization of the χ quantity over input pure states and their probabilities,
CX→Φ→Y = max
{pi,ρi}χ({pi, ρi}).
2 Channel Capacity Background
This is a remarkable ‘single-letter’ quantum entropic characterization of the complicated limiting regularization process Eq. (2.1). This is not to say that the capacity can be achieved by code-words of single letters, but that combining the letters and the mapped quantum states with joint measurements and taking the asymptotic limit, leads to a capacity that can be described in terms of the properties of the single letters and their associated quantum states. Indeed, the capacity depends on taking larger and larger blocks of letters, and for example capping the joint measurement size, leads to a lower capacity.
The HSW theorem became a fundamental building block for many other entropic and capacity related quantities. One extension [7] that we use in this thesis, is a weakening of the restriction that the channel must be memoryless. One of the simplest extensions to memoryless channels are calledforgetful channels. The HSW theorem applies directly to these forgetful channels. In this thesis, we calculate theχ quantity for a specific simple forgetful channel and study the effects that the strength of the correlations have on the capacity.
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Bibliography
[1] The Mathematical Theory of Communication, C. Shannon and W. Weaver (Urbana IL: University Illinois Press; 1949)
[2] Quantum theory, the Church-Turing principle and the universal quantum com- puter, D. Deutsch (Proceedings of the Royal Society of London Ser. A; A400; pg.
97–117; 1985)
[3] On the Einstein Podolsky Rosen Paradox, J. Bell, (Physics 1 (3); 195200; 1964) [4] A New Proof of the Quantum Noiseless Coding Theorem, R. Josza and B. Schu-
macher (J. Mod. Opt.; 41 23439; 1994)
[5] The Capacity of Quantum Communication Channel with General Signal States, A.S. Holevo (IEEE Trans. Inform. Theory; 44 26972; arXiv:quant-ph/9611023;
1998)
[6] Sending Classical Information via Noisy Quantum Channels, B. Schumacher and M.D. Westmoreland (Phys. Rev. A; 56 1318; 1997)
[7] Quantum Channels with Memory, D. Kretschmann and R.F. Werner, Phys. Rev.
A, 72(6):62323, 2005 (arXiv:quant-ph/0502106)
Chapter 3
Classical Capacity of a Qubit Depolarizing Channel with Memory
Published:
The Classical Capacity of a Qubit Depolarizing Channel with Memory,
J. Wouters, I. Akhalwaya, M. Fannes, F. Petruccione (Phys. Rev. A 79, 042303, 2009)
Abstract
The classical product state capacity of a noisy quantum channel with memory is inves- tigated. A forgetful noise-memory channel is constructed by Markov switching between two depolarizing channels which introduces non-Markovian noise correlations between successive channel uses. The computation of the capacity is reduced to an entropy computation for a function of a Markov process. A reformulation in terms of algebraic measures then enables its calculation. The effects of the hidden-Markovian memory on the capacity are explored. An increase in noise-correlations is found to increase the capacity.
3 Classical Capacity of a Qubit Depolarizing Channel with Memory