If you have two systems A and B, an important question is how much can you  tell about system A knowing about system B. Obviously if they’re correlated then you can learn a lot, and if they’re not correlated, then you learn very little. Mathematically this is measured by the mutual information:

Where measures the entropy. The entropy of a random variable is essentially how random it is (measured in the number of bits you need to specify it). Its formal definition is

Of course with two systems, things get a little bit more complicated. You could assign a value for all possible values of and , which we call . Now imagine if I don’t know anything about system A or B. What is the probability distribution for system A? Well it’s just,

and similarly

If I come along and start measuring system , I’ll actually learn things about A. In fact if I measure a particular b then I’ll have a new probability distribution for A,

That’s Bayes’ rule. The is just the new normalization.

Returning to the definition of mutual information, there’s two parts. The first, is just the number of bits you need to specify an A, if you know nothing about B. The second term, is given by

which is the average number of bits needed to specify , after you’ve measured . If there’s no correlation, then, you need the same number of bits to specify A no matter what you measure from B, and the mutual information is zero. But if there’s some correlation, then you need fewer bits to specify A after you know B, and the mutual information is greater than zero.

With a little bit of algebraic manipulation we get an equivalent version (at least classically):

The problem which this paper addresses is how do we generalize mutual information to a quantum mechanical setting. There seems to be two obvious ways to do it. The first is to just replace the Shannon entropies with Von Neumann entropies:

where these entropies are now taken according to given density matrices,

That seems perfectly reasonable. The problem is when you pick a particular measurement basis, things become a little bit difficult. What does P(A|B) mean? Is it B measured in the ideal basis for determining A? Is it measured in some other basis? If we pick a particular basis then we can evaluate things like what is. We define these things like this:

Which is the density matrix you’d expect after has been measured. Now becomes

and we can define

Of course, picking a given measurement basis isn’t picking the perfect basis, so these two measures of mutual information are (generally) different. The difference is known as the discord:

This is a measure of how much coherence you lose when making the measurements. If you choose a perfectly diagonal basis to measure in, then you won’t lose any coherence. But if you choose a basis which makes far from diagonal, then discord will be high. If you think of a diagonal as classical, and a state with lots of coherences as quantum, then this is a measure of exactly how quantum your system is. It is very interesting that even systems which aren’t entangled have discord.

Title: Quantum discord: A measure of quantum correlations
Authors: Ollivier and Zurek
Reference: PRL, 88, 017901

Typically, quantum error correction codes treat both types of error the same. If an error correction code can correct up to three bit flips, it can correct up to three phase flips as well. More recently researchers have been considering what happens when one type of error is far more prevalent than the other. For example, in real quantum computers often bits will not flip, but they will mostly just lose phase information (equivalent to having a phase flip). Ideally we would like error correction codes which can correct more phase flips than they do bit flips. In this paper, the authors describe a whole set of error correction codes which do just that.

In fact, the error correction codes they describe can be tailored (“adapted”) to the specific ratio of bit flips and phase flips that are experienced as the experiment progresses. They give a relatively simple way to adapt the code given different error rates which are being experienced.

Title: Adaptively correcting quantum errors with entanglement
Authors: Fujiwara, Hsieh
Reference: 
arXiv:1104.5004v1 [quant-ph]

Quantum mechanics normally manifests itself at very cold temperatures, near to absolute zero. So far there is only one system, NV centers in diamond, in which quantum coherence (of an individual qubit) and control has been demonstrated at anything much higher. For the NV center, even room temperatures is okay. But for other systems, any quantum coherence generally dies off way too fast to be measured, and certainly too fast to control.

Your only option if you want to measure (and control) this type of coherence is to be fast. And by fast I mean on the timescale of femto-seconds. That’s 1×10-15 s. A computer operating at 1GHz is operating approximately 1 million times slower than that. So these things are fast, very fast.

In a recent paper, Hildner, Brinks and van Hulst did control molecules on this timescale. Amazingly they observed the quantum coherence of just a single molecule. Taking a single terrylenediimide (TDI) molecule, they excited it with two 75 fs pulses. In between the two pulses they waited, differing amounts of time from 0 fs to 600 fs. During this time the molecule loses its quantum mechanical nature – it decoheres. They were able to monitor that process, watching the molecule go from largely coherent to a completely classical mixed state. In doing so (and repeating the experiment) they were able to quantify exactly how fast a molecule lost its phase information. They measured times from 25 fs to 110 fs with most molecules being around 60 fs.

Title: Femtosecond Coherence and Quantum Control of Single Molecules at Room Temperature

Authors: Hildner, Brinks, van Hulst

Reference: Nature, 7, 172-178, 2011

Overall the Hamiltonian becomes,
.

In the ideal case:

They study the dynamics of this chain experimentally and analytically. In particular they consider multiple qubit coherences and find situations which the system deviates from the simplest nearest neighbor (NN) tight binding approximation.

Authors: Wenxian Zhang, Paola Cappellaro, Natania Antler, Brian Pepper, David G. Cory, Viatcheslav V. Dobrovitski, Chandrasekhar Ramanathan, Lorenza Viola

Reference: Arxiv 0906.2434

Date: June, 2009.

Tags: quantum, one dimension, NMR, chain

It is possible to simulate pretty much any quantum system which involves at most two qubit interactions. This is normally done with the Trotter expansion

What other Hamiltonians and expansions are possible?

Not that pontiff (Photo: Giampaolo Macorig)

Dave Bacon, the quantum pontiff, is at it again. This time he has come up with a remarkably simple scheme for implementing gates adiabatically. How to do this is described in his new paper: Adiabatic Gate Teleportation.

The idea is something like teleportation. You have two Hamiltonians:
, and
.

Prepare your systems 2 and 3 in the ground state (which is a Bell state) of . Put whatever state you like in system 1. Now adiabatically (ie. slowly) change the Hamiltonian from to . The state which was initially in system 1 has been teleported to state 3. Work out the maths. It works. Great. So we can move states around adiabatically. So how do we perform gates on them?

To perform a gate, we simply perform a unitary operation on the initial Hamiltonian:

Nothing changes about the spectrum of the adiabatic passage because a unitary doesn’t change the diagonalization. But when you do the adiabatic passage, the state is not only teleported but comes out with the gate applied to it.

Two qubit gates work similarly… Simple and straightforward. Yet another interesting way to do quantum computation.

Reference: Bacon and Flammia, Adiabatic Gate Teleportation, 2009.

Link: http://arxiv.org/abs/0905.0901

Tags: physics, quantum computation, bacon, adiabatic gate, teleportation

So now how about quantum mechanical channels? Well, if you want to send classical information (bits) down a quantum channel, the best you can do is the Holevo capacity, which is given by

The capacity of the channel is simply the largest (for any input state) Holevo capacity.

So here is the critical question of this post: If I have two quantum channels, can I just add their Holevo capacities to get the total capacity? Is ? The answer, as it turns out, is NO. This comes as a bit of a shock, because it was widely assumed to be true, despite the fact there was no proof yet. In a recent paper in Nature Physics, Matt Hastings found a counter-example.

Reference: M. B. Hastings. Superadditivity of communication capacity using entangled inputs. Nat Phys, 5(4):255-257, April 2009.

Arxiv: http://arxiv.org/abs/0809.3972

Tags: quantuminformation, channel capacity, physics, science

So how does it all work? The key to this paper seems to be that they were able to control the coupling between their two transmon qubits. By bringing them into resonance with the microwave cavity – near to an avoided crossing with energy levels which were not populated by the experiment – they were able to get a nonlinear coupling between their two qubits. This allowed them to create a controlled phase gate between the two qubits, and to demonstrate the two algorithms. The coupling between the qubits originally was approximately 1.2 MHz which they were able to ramp up to 160 MHz by this method.

The decoherence times in this experiment, which have always been a killer for superconductors in the past, were approximately 1μs some 3 orders of magnitude above where they were before this effort began! These long decoherence times allows them to get approximately 10 gates algorithms to run (in 100ns).

This was truly a remarkable experiment, combining all the ingredients- single qubit rotations, two qubit interactions, measurement (with 90% fidelity) and initialization. Can they keep pushing down the decoherence rates? Can they scale this system up? There seems to be no fundamental reasons stopping them moving forward. Exciting times.

Paper: Demonstration of Two-Qubit Algorithms with a Superconducting Quantum Processor.

Authors: L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, R. J. Schoelkopf

System: Superconductors (transmon qubits)

Algorithms: Grover, Deutsch

Qubits: 2

Reference: quant-ph/0903.2030

Then along came Grover. In 1996 he pointed out that in quantum mechanics, if you start off with a equal superposition of N different states then you don’t need N/2 operations to find the right (or the “marked”) one. You only need around √N operations!

And with that he cemented his place in history.

Type of Problem: Oracle based search

Complexity: O(√N) (compared to classical equivalent of O(N)).

References:

In 2007 a team from Berkley observed what looks like wavelike behaviour in a photosynthesis complex – or to be more specific in the Fenna-Matthews-Olson (FMO) which is the long antenna, wirelike, complex connecting together the light harvesting chlorosome with the reaction center. Admittedly the experiment wasn’t done at room temperature (it was done at 77K or -196C). And the oscillations they observed were on a timescale of less than one picosecond (a maximum of 660fs). But when they looked at the electronic spectroscopy, they did see wavelike behaviour.

They speculate that their system is essentially performing a single quantum computation, and the operation is analogous to Grover’s algorithm, which they speculate makes the whole operation more efficient at transporting energy. How this works exactly, they don’t say.

Reference: Gregory S. Engel, Tessa R. Calhoun, Elizabeth L. Read, Tae-Kyu Ahn, Tomas
Mancal, Yuan-Chung Cheng, Robert E. Blankenship, and Graham R. Fleming.
Evidence for wavelike energy transfer through quantum coherence in
photosynthetic systems.
Nature, 446(7137):782-786, April 2007.