Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 323. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.

Also available via SciRate.

]]>]]>“This is probably what it felt like to be a British foreign service officer after World War II, when you realize, no, the sun actually does set on your empire,” said the mid-level officer. “America is over. And being part of that, when it’s happening for no reason, is traumatic.”

I think the basic cause of the trouble is the following:

The application of mathematics to biological evolution is rooted, historically, in statistics rather than in dynamics. Consequently, a lot of model-building starts with tools that belong, essentially, to descriptive statistics (e.g., linear regression). This is fine, but then people turn around and discuss those models in language that implies they have constructed a dynamical system. This makes life quite difficult for the student trying to learn the subject by reading papers! The problem is not the algebra, but the assumptions. And that always makes for a thorny situation.

]]>You can punch a neo-Nazi who might fall onto the switch lever, which might divert the train away from millions of people, or you can do nothing. Choose.

From Stephanie Zvan.

]]>The SIC problem gives us the opportunity to travel all throughout mathematics, because, while the definition looks pretty small, the question is bigger on the inside.

For a taste of why this is so, try here:

- M. Appleby, S. Flammia, G. McConnell and J. Yard, “SICs and Algebraic Number Theory,” arXiv:1701.05200 [quant-ph] (2017).

We share the concerns expressed by many APS members about recent U.S. government actions that will harm the open environment that is essential for a successful global scientific enterprise. The recent executive order regarding immigration, and in particular, its implementation, would reduce participation of international scientists and students in U.S. research, industry, education, and conference activities, and sends a chilling message to scientists internationally.

The American Chemical Society had already spoken up:

While ACS understands the administration has communicated that the intent of the order is to prevent terrorists from entering the country, it feels that the order itself is overly broad in its reach, unfairly targets individuals from a handful of nations, ignores established mechanisms designed to achieve the ends sought by the order, and sets potential precedent for future executive orders.

The Society notes that reliable media sources are reporting that the executive order and its implementation have caused tremendous confusion both in the U.S. and worldwide, where individuals with valid green cards have been detained or otherwise prevented from completing previously approved travel to the U.S. Adding to the confusion are orders from several federal judges directing the administration to temporarily halt the travel ban.

And the American Mathematical Society:

For many years, mathematical sciences in the USA have profited enormously from unfettered contact with colleagues from all over the world. The United States has been a destination of choice for international students who wish to study mathematics; the US annually hosts hundreds of conferences attracting global participation. Our nation’s position of leadership in mathematics depends critically upon open scientific borders. By threatening these borders, the Executive Order will do irreparable damage to the mathematical enterprise of the United States.

We urge our colleagues to support efforts to maintain the international collegiality, openness, and exchange that strengthens the vitality of the mathematics community, to the benefit of everyone.

And the International Astronomical Union:

The IAU considers that mobility restrictions imposed by any country, similar to the ones recently included in the US executive order, run counter to its mission, which is inspired by the principles of the International Council for Science (ICSU) on the Freedom in the Conduct of Science. Such restrictions can have a direct impact on the astronomical communities of countries at both ends of the ban, as well as astronomy as a whole.

And the Association for Computing Machinery:

The open exchange of ideas and the freedom of thought and expression are central to the aims and goals of ACM. ACM supports the statute of International Council for Science in that the free and responsible practice of science is fundamental to scientific advancement and human and environmental well-being. Such practice, in all its aspects, requires freedom of movement, association, expression and communication for scientists. All individuals are entitled to participate in any ACM activity.

ACM urges the lifting of the visa suspension at or before the 90-day deadline so as not to curtail the studies or contributions of scientists and researchers.

And my *alma mater* MIT, in the person of its president, Rafael Reif:

The Executive Order on Friday appeared to me a stunning violation of our deepest American values, the values of a nation of immigrants: fairness, equality, openness, generosity, courage. The Statue of Liberty is the “Mother of Exiles”; how can we slam the door on desperate refugees? Religious liberty is a founding American value; how can our government discriminate against people of any religion? In a nation made rich by immigrants, why would we signal to the world that we no longer welcome new talent? In a nation of laws, how can we reject students and others who have established legal rights to be here? And if we accept this injustice, where will it end? Which group will be singled out for suspicion tomorrow?

On Sunday, many members of our campus community joined a protest in Boston to make plain their rejection of these policies and their support for our Muslim friends and colleagues. As an immigrant and the child of refugees, I join them, with deep feeling, in believing that the policies announced Friday tear at the very fabric of our society.

And the chancellor of my own current academic residence, UMass Boston:

These executive orders are very concerning, but as we have seen with the latest developments—court orders in New York and here in Boston blocking the implementation of the ban on immigration from seven particular countries—the policies they may usher in are far from settled.

I’m writing you today to make sure that you are aware that we are following these issues carefully and to remind you of our stand as a public research university—to protect and advocate for the most vulnerable to ensure all people have access to higher education. That means that the safety and academic success of each and every student is our number one priority.

As I stated in a communication to the campus community last month, this university supports the Deferred Action for Childhood Arrivals (DACA) program and our undocumented immigrant students. I have signed a letter along with other higher education leaders calling for DACA’s preservation, enhancement, and expansion. We strongly uphold and advocate for the rights of international students and faculty and staff with legal status to be here to be allowed entry to the country to continue their education, research, and other work they do here. We will continue to hold firm on these positions.

**EDIT TO ADD:** A little while after posting this, I noticed this post by Terry Tao, which says in part as follows:

]]>This is already affecting upcoming or ongoing mathematical conferences or programs in the US, with many international speakers (including those from countries not directly affected by the order) now cancelling their visit, either in protest or in concern about their ability to freely enter and leave the country. Even some conferences outside the US are affected, as some mathematicians currently in the US with a valid visa or even permanent residency are uncertain if they could ever return back to their place of work if they left the country to attend a meeting. In the slightly longer term, it is likely that the ability of elite US institutions to attract the best students and faculty will be seriously impacted. Again, the losses would be strongest regarding candidates that were nationals of the countries affected by the current executive order, but I fear that many other mathematicians from other countries would now be much more concerned about entering and living in the US than they would have previously.

It is still possible for this sort of long-term damage to the mathematical community (both within the US and abroad) to be reversed or at least contained, but at present there is a real risk of the damage becoming permanent. To prevent this, it seems insufficient for me for the current order to be rescinded, as desirable as that would be; some further legislative or judicial action would be needed to begin restoring enough trust in the stability of the US immigration and visa system that the international travel that is so necessary to modern mathematical research becomes “just” a bureaucratic headache again.

An academic type like me has a hard time responding to accusations of “identity politics” or “political correctness,” not because the accusations have any intellectual merit, but because the real message isn’t the words on the page. People like me, we see a thing wrapped up in the form of a scholarly argument, and we try to respond with footnotes and appendices. But the clauses and locutions are just dances around the real issue, the fundamental point that was expressed most clearly by the Twitter account @ProBirdRights:

]]>I am feel uncomfortable when we are not about me?

I’m really not feeling that good about our ability to handle the next epidemic that comes our way.

And on a personal note, I’m a queer scientist who has published on biological evolution and the need for financial regulation. So, you can sod off with your cheery hot takes about America becoming Great Again through space exploration, or whatever the Quisling line is this week. Stuff your white dick back in your pants and sit your ass down while the adults work, m’kay?

]]>We went out in the way a bad joke would have predicted. We lost against our own racism and sexism, our endemic illnesses whose symptoms were intensified by corrupt law enforcement and institutionally rotten mass media. Undone at the final hour by a bizarre codicil in a slaveowners’ constitution. Undone, pushed over the edge—but the edge was too close all along. When it really mattered, we proved ourselves incompetent: not able to handle our civil responsibilities, indeed, in a sense, not ready for adulthood. In the name of national glory, we have voted ourselves a government of the worst. And now a generation will grow up ignorant, poor and sick, if they get the chance to grow up at all. Many of the things we will lose will be things we can never regain, from international respect to endangered species to the lives of our loved ones.

Many good people will keep up the good fight and stir up, as John Lewis says, the good trouble.

The abyss has opened before us.

Whether the future we make for ourselves will have anything to commend it now depends upon our ability to stare into that abyss and make it blink.

]]>For example, here’s Jan-Åke Larsson getting credit for work by Giacomo D’Ariano:

And here’s Chris picking up 38 bonus points for research on Mutually Unbiased Bases—a topic not far from my own heart!—research done, that is, by Ingemar Bengtsson:

Note that again, the publication appeared in *AIP Conference Proceedings.*

I’d like to know how far and wide the rot goes, but that’s the kind of question I’d need to crowdsource!

**EDIT TO ADD (later in the day):** The problem is not confined to articles *AIP Conference Proceedings.* It also hits articles published in *Physical Review A,* one of the flagship journals of physics. Here, for example, is Adán Cabello getting credit for a paper by Klimov and Muños:

He is also credited with a paper by Caball**ar**, Ocampo and Galapon, also in *Physical Review A.*

Of course, there may be multiple underlying software problems at work.

**EDIT TO ADD (slightly later still):** Paul Pukite writes in via Twitter to say,

yesterday I caught an entry in an SPIE conf misattributed to me. I had a cite in the same issue so thought it a 1 time mistake. But today it also removed cites to individual chapters in my book, which were redundant. Looks like crawling a new algorithm.

It looks like Google Scholar is also crediting Pukite with “Article#: INSPEC Accession Number,” by a certain “P. Options” (who may be a collaborator of a certain “P. Login,” active some years ago).

**EDIT TO ADD (21 January):** Marcus Appleby wrote in with an example where Google Scholar attributes a solo-author paper of his to seventy people:

I think that in this case, Google Scholar is taking the author list from *everyone who published in the first twenty-five papers* on the Quantum Physics arXiv during December 2004. No, really: take a look!

Let’s say you tell your students that arm folding is a genetic trait, with the allele for right forearm on top (R) being dominant to left forearm on top (L). Results from a large number of studies show that about 11 percent of your students will be R children of two L parents; if they understand the genetics lesson correctly, they will think that either they were secretly adopted, or Mom was fooling around and Dad isn’t their biological father. More of your students will reach this conclusion with each bogus genetic trait that you add to the lesson. I don’t think this is a good way to teach genetics.

Via PZ Myers, who is teaching genetics this semester and has an interest in getting it right.

]]>My answer: They don’t. But not for the reason that most physicists who bother with such speculations probably think.

As a university graduate *and* veteran of many science-blog comment threads, I’ve seen my share of arguments about the mind-body problem, the Hard Problem of Consciousness and all that. The older I get, the more I feel that the way these arguments set up the problems automatically forbids a solution, and then passes this failure off as profundity. One defines *Matter* as something entirely unlike *Mind,* and then one argues about how the latter can or cannot arise out of the former. After enough rounds of this, any elegant formulation of words that appears to provide a solution immediately becomes suspect: If it *appears* to make the problem more tractable, then it *must* be treating Matter and Mind *insufficiently differently.*

When I was younger, I embraced the idea that *reality*—you know, all the stuff out there and in here that is, like, real and all—is at bottom a *mathematical entity* of some kind. What can I say? I also listened to Foreigner and thought that racism in America was on the wane. Now that my day job is actually doing physics, as opposed to reading breathless books about it, I find that this attitude, the instinctive identification of the physical and the mathematical, is entirely optional. It says more about your own temperament than pretty much anything else, nothing in particular compels it, and you can cheerfully do research without it. Countervailing influences, some of which I also carried with me since childhood, eventually led me to find it a rather empty business!

My colleague Marcus Appleby wrote an essay about this general topic a while back, an essay called (straightforwardly enough) “Mind and Matter” (arXiv:1305.7381). It prompted me to formulate a slogan, which goes like this:

*Cognitive science led me to reject the Cartesian idea of “Mind.” Quantum physics makes it legitimate to reject the Cartesian idea of “Matter.” Confusion reigns when we discard one pole but not the other!*

Above and beyond that, I have grown to doubt that the imagery of “mindless mathematical laws” is sustainable, once we confront the daily practice and the philosophy of mathematics. To get a tiny taste of this, consider the following puzzle, posed by John Baez:

You might say the standard natural numbers are those of the form 1 + ··· + 1, where we add 1 to itself some finite number of times. But what does ‘finite number’ mean here? It means a standard natural number! So this is circular.

So, conceivably, the concept of ‘standard’ natural number, and the concept of ‘standard’ model of Peano arithmetic, are more subjective than most mathematicians think. Perhaps some of my ‘standard’ natural numbers are nonstandard for you!

One can remove the ambiguity in the informal definition of “natural number,” but not in a unique way. The resulting multiplicity is irrelevant for practical purposes—ah, but tonight, we’re concerned with *impractical purposes!* We can shove this befuddlement under the rug and try to move on to defining the real numbers, only to find that a similar difficulty arises there. There is a stupendous diversity of viable candidates for the position of “the real numbers.” Technically, they are known as *real closed fields.* They are equivalent to the extent that if you can prove a mathematical statement involving the numbers 0 and 1, the operations of addition and multiplication, and the less-than-or-equal comparison in one such arena, you can prove it in any other. But two different real closed fields do not have to be isomorphic: There might not be a one-to-one mapping between them that respects the structure of arithmetic.

Mathematics is that domain of life in which “two minds can know one thing,” as best as that is possible—but you have to work painfully hard to subtract the *minds* from that image, and the result might leave you with nothing at all.

What does this have to do with physics? Perhaps nothing, except for the mental entertainment and recuperative value obtained by taking a break *from* physics! The mathematics that finds application in physics is peculiarly robust against changes in the underpinnings—to how we choose to define the real numbers, for example. On the other hand, if we’re being honest, this might be an artifact of what kinds of problems the physicists have chosen to work on! And we have to admit the vexing hassles that even our best physical theories run into because we assume that spacetime is a continuum—i.e., because we habitually think that events happen along a line of real numbers.

Another thing I’ve lost my taste for with the passing years is the blog comment section. So, I don’t have a fantastically strong urge to engage in the FQXI essay contest officially. Instead, I’ll close with a pointer to a manifesto I posted recently, and in particular to the wild speculations in the latter sections.

]]>]]>A curious thing: in the four classes so far, the number of students attending has been, respectively, 19, 17, 15, 13. Assuming that the arithmetic progression continues, our final class will have $-1$ student. Some of Joachim’s colleagues have expressed an interest in coming along to see what $-1$ student looks like. This presents problems of a philosophical type.

\[ ax^2 + bx + c = 0 \] with the quadratic formula,

\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a},\] and neglecting the fact that if the expression under the square root sign equals

This sort of glitch is usually not worth a lot of breath, though I do tend to write in when I notice them, to keep down the overall confusingness of the scientific literature. In this case, however, there’s a surprise bonus. The extra solutions you pick up turn out to have a very interesting structure to them, and they include mathematical objects that were already interesting for other reasons. So, I wrote a little note explaining this. In order to make it self-contained, I had to lay down a bit of background, and with one thing and another, the little note became more substantial. *Too* substantial, I learned: The journal that published the original paper wouldn’t take it as a Comment on that paper, because it said too many new things! Eventually, after a little more work, it found a home:

- B. C. Stacey, “SIC-POVMs and Compatibility among Quantum States,”
*Mathematics***4,**2 (2016): 36 arXiv:1404.3774 [quant-ph].

The number of citations that Google Scholar lists for this paper (one officially published in a journal, mind) fluctuates between 5 and 6. I think it wavers on whether to include a paper by Szymusiak and Słomczyński (*Phys. Rev. A* **94,** 012122 = arXiv:1512.01735 [quant-ph]). Also, if you compare against the NASA ADS results, it turns out that Google Scholar is missing other citations, too, including a journal-published item by Bellomo *et al.* (*Int. J. Quant. Info.* **13,** 2 (2015), 1550015 = arXiv:1504.02077 [quant-ph]).

As I said in 2014, this would be a rather petty thing to care about, *if* people didn’t rely on these metrics to make decisions! And, as it happens, all the problems I noted then are still true now.

This round of depression isn’t just worse than my previous episodes: it’s different. My symptoms, the things that help, the things that make it worse — they’re different. I’ve spent the last four years learning how to manage depression, and now, at least to some extent, I need to start all over again.

It’s different because the world is genuinely terrible. That’s not the depression talking: that’s a reasonable, evidence-based assessment of reality. You know the joke, “just because you’re paranoid doesn’t mean they’re not out to get you?” Well, just because you’re depressed doesn’t mean the world’s not terrible. And just because you’re anxious doesn’t mean the world’s not terrifying.

Greta’s project of blogging once a weekday this month is one of the incentives that’s got me posting more often, though I’m not keeping up with her rate.

]]>- “Information and structure in complex systems,”
*Plektix*(24 October 2014).

Ben’s blog post is a capsule introduction to this article that he and I wrote with Yaneer Bar-Yam:

- “An Information-Theoretic Formalism for Multiscale Structure in Complex Systems,” arXiv:1409.4708 [cond-mat.stat-mech].

I also cover this topic, as well as a fair bit of background on how to relate *probability* and *information,* in my PhD thesis:

*Multiscale Structure in Eco-Evolutionary Dynamics,*arXiv:1509.02958 [q-bio.PE].

In this post, I’ll carry the ideas laid out in these sources a little bit farther in a particular direction.

Imagine a system that is made up of a large number of pieces. A theme one encounters in many areas of science is that such a system is simpler to understand when the pieces are independent of one another, each one doing its own thing. In thermodynamics and statistical physics, for example, we spend a lot of time studying gases, and the easiest conceptual model of a gas to work with is an *ideal gas,* in which the atoms sail past each other without interacting. On the other hand, a system can also be simple to understand if all the pieces are so tightly correlated or so strongly bound together that they essentially move as a unit. Then, instead of having to understand lots of little things, we just have to understand one big thing, because knowing what any one part is doing tells us most all of what we need to know about the whole.

A similar high-level intuition is helpful in pure mathematics, as Terry Tao once explained. In mathematics, one finds *structured* collections, where the relations among the individual elements have a high degree of predictability, and *random or pseudo-random* collections, where the status of one element is not informative about the status of another. Moreover, one also encounters *hybrid* collections, which partake in features of both.

The general approach of the references I bullet-pointed above is that we can use *information theory* to make these heuristic discussions quantitative. One conclusion is that we should not aim to quantify a system’s “complicatedness” by a single number. Instead, it is more illuminating to devise *curves* that indicate *how much* structure is present *at different scales.*

We start by saying that a system $\mathcal{S}$ is composed of pieces, or components. We’ll denote the set of components of the system $\mathcal{S}$ by $S$. It is important to distinguish the two, because we could take the same components and arrange them in a different way to create a different system. We express the arrangement or the patterning of the components by defining an **information function.** For each subset $T \subset S$, the information function $H$ assigns a nonnegative number $H(T)$, which expresses how much information is necessary to say what all the pieces in $T$ are doing. One can actually prove quite a lot from the starting point that if $H$ is to be an information function, it had better satisfy a few basic axioms.

First, as we already said, $H(T) \geq 0$ for any $T \subset S$.

Second, if $T \subset V \subset S$, then $H(T) \leq H(V)$. We can call this property **monotonicity**.

Third, if we have two subsets $T \subset S$ and $V \subset S$, not necessarily nested in one another or anything like that, the total information assigned to their union, $H(T \cup V)$, must be limited. Information pertinent to the components in $T$ can be pertinent to the components in $V$. For one reason, $T$ and $V$ might have some components in common. And even those components that are not shared between the two sets might be correlated in some way such that the amount of information necessary to describe the whole lot is reduced. So, we require that

\[

H(T \cup V) \leq H(T) + H(V) – H(T \cap V),\] which we call **strong subadditivity**. Given two components $s$ and $t$ within $S$, the **shared information** expresses the difference between what is required to describe them separately versus describing them together:

\[

I(s;t) = H(\{s\}) + H(\{t\}) – H(\{s,t\}).\] It follows from strong subadditivity that the shared information is always nonnegative.

A **descriptor** of a system is an entity outside that system which tells us about it in some way. Mathematically speaking, we take the system $\mathcal{S}$ and augment it with a new component that we can call $d$, to form a new system whose component set is $S \cup \{d\}$. The information function of the augmented system reduces to that of the original system $\mathcal{S}$ when applied to subsets of $S$. The values of the augmented system’s information function on subsets that include the descriptor $d$ tell us how $d$ shares information with the original components of $\mathcal{S}$.

The **utility** of a descriptor $d$ is

\[ U(d) = \sum_{s \in S} I(d;s). \] Given the basic axioms of information functions that we listed above, we can define an **optimal descriptor** as the one which has the largest possible utility, given its own amount of information. That is, if we invest an amount of information $y$ in describing the system $\mathcal{S}$, then an optimal descriptor has $H(d) = y$, and it relates to $\mathcal{S}$ in such a way that $U(d)$ is as large as the basic axioms of information functions allow. This defines a linear programming problem whose solution is the **optimal utility**, and so the theory of linear programming lets us prove helpful results about how the optimal utility varies as a function of $y$. Taking the derivative of the optimal utility gives the **marginal utility of information**, or MUI.

We prove a fair bit about the MUI in the paper where we introduced it. For systems of the ideal-gas type, where information applies to one component at a time, the MUI is a low and flat function: Investing one bit of description buys one unit of utility, until the whole system is described. On the other hand, if all the components are bound together and “move as one,” then investing a small amount of information buys us utility on a large scale, because that small amount applies across the board. In this case, the MUI starts high and falls sharply.

When the construction of a system is specified in detail, it is sometimes possible to make a finer degree of analysis, which reveals features that a first application of a structure index can overlook. For example, take the two systems defined by James and Crutchfield (2016). These are three-component systems composed of random variables whose joint states are chosen by picking a row at random from a table, with uniform probability. When systems are defined as collections of random variables with some joint probability distribution, we can use the **Shannon information** (a.k.a., Shannon entropy, Shannon index) as our information function $H$. The **dyadic** and **triadic** systems are defined respectively by the tables

\[

D = \left(\begin{array}{ccc}

0 & 0 & 0 \\

0 & 2 & 1 \\

1 & 0 & 2 \\

1 & 2 & 3 \\

2 & 1 & 0 \\

2 & 3 & 1 \\

3 & 1 & 2 \\

3 & 3 & 3

\end{array}\right);

\ T =

\left(\begin{array}{ccc}

0 & 0 & 0 \\

1 & 1 & 1 \\

0 & 2 & 2 \\

1 & 3 & 3 \\

2 & 0 & 2 \\

3 & 1 & 3 \\

2 & 2 & 0 \\

3 & 3 & 1

\end{array}\right).

\]

If we compute the MUI for these two systems in the manner described above, we find that the MUI for the dyadic system is the same as for the triadic. In fact, the information functions for the two systems, computed according to the Shannon scheme, agree for all subsets $U$. However, we can detect a difference between their structures by *augmenting* them. Let the three components of the dyadic system be written $d_1$, $d_2$ and $d_3$, and introduce a new variable $\Delta_{12}$ which takes the value 1 when the state of $d_1$ and $d_2$ are the same, and is 0 otherwise. This new **ancilla** variable is determined completely by the original system, and is sensitive to the particular values taken by the original system components $d_1$, $d_2$ and $d_3$. We can define two other ancillae in the same way, $\Delta_{13}$ and $\Delta_{23}$. Carrying out this construction for the dyadic example system, we find that the three ancillae form a completely correlated block system (that is biased towards the joint state 000).

In contrast, for the triadic example, defining three ancillae in the same way, we find that they form a **parity-bit system** (with odd parity). The ancillary system has four possible joint states, all of which are equally probable, and so it has two bits of information overall. For each possible joint state of the ancillary system, the original system can be in one of two joint states, with equal probability. Therefore, we see that the three bits of information necessary to specify the state of the original system break down into a pair of bits for describing the ancillae, plus one more bit of additional detail.

The logical extreme of this approach is to define an ancilla for each possible value of each component of a system. I like to call this “exploding” the original system. Applying the our theory to an exploded system can reveal new details of organization, at the price of increasing the number of components one must consider. For example, consider a two-component system defined by picking two successive characters at random out of a large corpus of English text. The shared information between the two components quantifies how much knowing the value of the first character helps us predict the value of the second. However, knowing that the first character is a $Q$ is a stronger constraint than knowing that it is, say, a $T$, because fewer characters can follow a $Q$. We can express this in our theory by exploding the two-component system, defining new components that represent the events of the first character being a $Q$ and the first character being a $T$. This is the same basic idea as that of the *partial information decomposition*; however, our axioms do not force the user to introduce a distinction between “input” and “output” variables as the PID framework does.