Separability-Preserving Operators in Entanglement Theory

June 14th, 2011

One of the key concepts in quantum information theory is the difference between separable states and entangled states. A pure quantum state (that is, a unit vector) v ∈ CnCn is said to be separable if it can be written as v = a ⊗ b for some a,b ∈ Cn; otherwise v is called entangled. In this post we will investigate what operators preserve the set of separable pure states, as well as what operators entangle all separable pure states.

Separable Pure State Preservers and Entangling Gates

In the design of quantum algorithms, entangling gates play a very important role. Entangling gates are unitary operators that are able to generate entanglement. A bit more specifically, a unitary operator U ∈ Mn ⊗ Mn (where Mn is the space of n × n complex matrices) is called an entangling gate if there exists a separable pure state v = a ⊗ b ∈ CnCn such that Uv is entangled. Conversely, we will say that a unitary operator U preserves separability if Uv is separable whenever v is separable.

In order to answer the question of what unitaries preserve separability, it is instructive to consider some simple examples (this is often a useful way to formulate conjectures regarding preserver problems). For example, it is clear that if U = A ⊗ B for some unitary operators A, B ∈ Mn, then U preserves separability (because U(a ⊗ b) = Aa ⊗ Bb is separable). Another example of a unitary operator that preserves separability is the swap (or flip) operator S defined on separable states by S(a ⊗ b) = b ⊗ a (the action of S on the rest of CnCn is determined by extending linearly). It turns out that these are essentially the only operators that preserve separability [1,2,3]:

Theorem 1. Let U ∈ Mn ⊗ Mn be a unitary operator. Then U preserves separability (i.e., U is not an entangling gate) if and only if there exist unitary operators A, B ∈ Mn such that either U = A ⊗ B or U = S(A ⊗ B).

As we already saw, the “if” direction of the above result is trivial – the meat and potatoes of the theorem comes from the “only if” direction (as is typically the case with results about linear preservers). Theorem 1 was first proved in [1] essentially by case analysis and checking the action of a separability-preserving unitary on a basis of CnCn, and was subsequently re-proved using similar techniques (but with different motivations and connections) in [2]. The result was proved in [3] by using the vector-operator isomorphism and the fact that a linear map Φ : Mn → Mn preserves the set of rank-1 operators if and only if there exist A, B ∈ Mn such that either Φ(X) ≡ AXB or Φ(X) ≡ AXtB [4].

Theorem 1 also follows as a simple corollary of several related results that have recently been proved in [5,6]. A version of Theorem 1 for multipartite systems (i.e., systems that are the tensor product of more than two copies of Cn) can be found in [3] and [7].

Universal Entangling Gates

A universal entangling gate is, as its name suggests, a stronger form of an entangling gate – it is a unitary operator U such that U(a ⊗ b) is entangled for all a, b ∈ Cn (contrast this with entangling gates, which require only that U(a ⊗ b) is entangled for some a, b ∈ Cn). The structure of universal entangling gates is much less well-understood than that of entangling gates, though we can still at least say when they exist.

It is not difficult to convince yourself that universal entangling gates can’t exist in small dimensions. Let’s begin by supposing n = 2. The set of pure states in C2C2 can be regarded as a 7-dimensional real manifold (7 = 2 × (n × n) – 1, where we subtract one because pure states all have unit length), while the set of separable pure states in C2C2 can be regarded as a 5-dimensional real manifold (5 = (2 × n – 1) + (2 × n – 1) – 1, where the final one is subtracted because the overall phase of the first system relative to the second system is irrelevant). Thus, if U ∈ M2 ⊗ M2 were a universal entangler, it would have to send a 5-dimensional manifold into the 7 – 5 = 2 remaining dimensions of the space, which seems unlikely. Similarly, if n = 3 and U ∈ M3 ⊗ M3 were a universal entangler, it would have to send a 9-dimensional manifold into the 17 – 9 = 8 remaining dimensions of the space, which also seems unlikely.

Indeed, this type of argument was made rigorous via methods of algebraic geometry in [8], where the following result was proved:

Theorem 2. There exists a universal entangling gate in Mn ⊗ Mn if and only if n ≥ 4.

Despite knowing when universal entangling gates exist, we still don’t have a characterization of such operators, nor do we even have many explicit examples (does anyone have an explicit example for 3 ⊗ 4 or 4 ⊗ 4 systems?). Similar techniques to those used in the proof of Theorem 2 should also shed light on when universal entangling gates exist in multipartite systems Mn1 ⊗ Mn2 ⊗ … ⊗ Mnk, but to my knowledge this calculation has not been explicitly carried out.


  1. M. Marcus and B. N. Moyls, Transformations on tensor product spaces. Pacific Journal of Mathematics 9, 1215–1221 (1959).
  2. F. Hulpke, U. V. Poulsen, A. Sanpera, A. Sen De, U. Sen, and M. Lewenstein, Unitarity as preservation of entropy and entanglement in quantum systems. Foundations of Physics 36, 477–499 (2006). E-print: arXiv:quant-ph/0407118
  3. N. Johnston, Characterizing Operations Preserving Separability Measures via Linear Preserver Problems. To appear in Linear and Multilinear Algebra (2011). E-print: arXiv:1008.3633 [quant-ph]
  4. L. Beasley, Linear operators on matrices: the invariance of rank k matrices. Linear Algebra and its Applications 107, 161–167 (1988).
  5. E. Alfsen and F. Shultz, Unique decompositions, faces, and automorphisms of separable states. Journal of Mathematical Physics 51, 052201 (2010). E-print: arXiv:0906.1761 [math.OA]
  6. S. Friedland, C.-K. Li, Y.-T. Poon, and N.-S. Sze, The automorphism group of separable states in quantum information theory. Journal of Mathematical Physics 52, 042203 (2011). E-print: arXiv:1012.4221 [quant-ph]
  7. R. Westwick, Transformations on tensor spaces. Pacific Journal of Mathematics 23, 613–620 (1967).
  8. J. Chen, R. Duan, Z. Ji, M. Ying, J. Yu, Existence of Universal Entangler. Journal of Mathematical Physics 49, 012103 (2008). E-print: arXiv:0704.1473 [quant-ph]

The Q-Toothpick Cellular Automaton

March 26th, 2011

The Q-toothpick cellular automaton (defined earlier this month by Omar E. Pol) is described by the following simple rules:

  1. On an infinite square grid, draw a quarter circle from one corner of a square to the opposite corner of that square:
  2. Call an endpoint of a quarter circle (or a “Q-toothpick”) exposed if it does not touch the endpoint of any other quarter circle.
  3. From each exposed endpoint, draw two more quarter circles, each of the same size as the first quarter circle you drew. Furthermore, the two quarter circles that you draw are the ones that can be drawn “smoothly” (without creating a 90° or 180° corner). Thus the next two generations of the automaton are (already-placed quarter circles are green, newly-added quarter circles are red):

The name “Q-toothpick” comes from its analogy to the more well-studied toothpick automaton (see Sloane’s A139250 and this paper), in which toothpicks (rather than quarter circles) are repeatedly placed on a grid where exposed ends of other toothpicks lie. In this post, we will examine how this automaton evolves over time, and in particular we will investigate the types of shapes that it produces.

Counting Q-Toothpicks

While the Q-toothpick automaton appears quite random and unpredictable for the first few generations, evolving past generation 6 or so reveals several patterns. The following image depicts the evolution of the automaton for its first 19 generations.

The first 19 generations of the Q-toothpick cellular automaton (red segments are pieces that are newly added in the current generation)

Perhaps the most notable pattern is that the grid is more or less filled up in an expanding square starting from the initial Q-toothpick. In fact, by inspecting generations 4, 6, 10, 18, we see that at generation 2n + 2 (n = 1, 2, 3, …) the automaton has roughly filled in a square of side length 2n+1 + 1, and then evolution continues from there on out of the corners of that square. Also, the number of cells added (A187211) at these generations can now easily be computed:

A187211(2n + 2) = 16 + 8(2n-1 – 1) for n ≥ 3.

Furthermore, the growth in the following generations repeats itself. In particular, we have:

A187211(2n + 3) = 22 for n ≥ 1,
A187211(2n + 4) = 40 for n ≥ 2,
A187211(2n + 5) = 54 for n ≥ 2.

Similarly, for n ≥ 3, the four values of A187211(2n + 6) through A187211(2n + 9) are similarly constant (their values are 56, 70, 120, and 134). In general, for n ≥ k the 2k-1 values of A187211(2n + 2k-1 + 2) through A187211(2n + 2k + 1) are constant in n, though I am not aware of a general formula for what these constants are. If we ignore the first four generations and arrange the number of Q-toothpicks added in each generation in rows of length 2n, we obtain a table that begins as follows:

22, 20
22, 40, 54, 40
22, 40, 54, 56, 70, 120, 134, 72
22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136

C scripts are provided at the end of this post for computing the values of A187210 and A187211 (and hence the values in the above table).

Shapes Traced Out by Q-Toothpicks

In the graphic above that depicts the initial 19 generations of the Q-toothpick automaton, several shapes are traced out, including circles, diamonds, hearts, and several nameless blobs:

By far the most common of these shapes are circles, diamonds and hearts. The fourth shape appears only on the diagonal and it’s not difficult to see that it forever will make up the entirety of the diagonal (with the exception of the circle in the center). The fifth and sixth objects are the first two members of an infinite family of objects that appear as the automaton evolves. The fifth object first appears in generation 9, and sixth object (which is basically two copies of the fifth object) first appears in generation 17. The following object, which is basically made up of two copies of the sixth object (i.e., four copies of the fifth object) first appears in generation 33:

In general, a new object of this type (made of 2n copies of the fifth object above) first appears in generation 2n+3 + 1. In fact, these objects are the only ones that are traced out by this automaton. [Edit: this final claim is not true! See ebcube’s great post that shows a double-heart shape in generation 31.]

Update [March 28, 2011]: I have added a script that counts the number of circles, diamonds, and hearts in the nth generation of the Q-toothpick automaton, and another script that computes Sloane’s A187212.


  • A187210.c – computes the total number of Q-toothpicks present in the nth generation
  • A187211.c – computes the number of Q-toothpicks added in the nth generation
  • A187212.c – computes the number of Q-toothpicks if we restrict them to the positive quadrant
  • count_shapes.c – computes the number of circles, diamonds, and hearts in the nth generation

The Maximum Score in the Game “Entanglement” is 9080

January 21st, 2011

Entanglement is a browser-based game that has gained a fair bit of popularity lately due to its recent inclusion in Google’s Chrome Web Store and Chrome 9. The way the game works is probably best understood by actually playing it, but here is my brief attempt:

  • You are given a hexagonal tile with six paths printed on it, with two path ends touching each side of the hexagon. One such tile is as follows:

  • You may rotate, but not move the hexagon that has been provided to you.
  • Once you have selected an orientation of the hexagon, a path is traced along that hexagon, and you are provided a new hexagon that you may rotate at the end of your current path.
  • The goal of the game is to create the longest path possible without running into either the centre hexagon or the outer edge of the game board.

To make things a bit more interesting, the game was updated in November 2010 to include a new scoring system that gives you 1 + 2 + 3 + … + n (the nth triangular number) points on a turn if you extend the length of your path by n on that turn. This encourages clever moves that significantly extend the length of the path all at once. The question that I am going to answer today is what the maximum score in Entanglement is under this scoring system (inspired by this reddit thread).

On a Standard-Size Game Board

The standard Entanglement game board is made up of a hexagonal ring of 6 hexagons, surrounded by a hexagonal ring of 12 hexagons, surrounded by a hexagonal ring of 18 hexagons, for a total of 36 hexagons. In order to maximize our score, we want to maximize how much we increase the length of our path on our final move. Thus, we want to just extend our path by a length of one on each of our first 35 moves, and then score big on the 36th move.

Well, each hexagon that we lay has six paths on it, for a total of 6*36 = 216 paths on the board. 35 of those paths will be used up by our first 35 moves. It is not possible to use all of the remaining 181 paths, however, because many of them lead into the edge of the game board or the central hexagon, and connecting to such a path immediately ends the game. Because there are 12 path ends that touch the central hexagon and 84 path ends that touch the outer border, there must be at least (12+84)/2 – 1 = 47 unused paths on the game board (we divided by 2 because each unused path takes up two path ends and we subtracted 1 because one of the paths will be used by us).

Thus we can add a length of at most 181 – 47 = 134 to our path on the 36th and final move of the game, giving a total score of at most 35 (from the first 35 moves of the game) + 1 + 2 + 3 + … + 134 = 35 + 9045 = 9080. Not only is this an upper bound of the possible scores, but it is actually attainable, as demonstrated by the following optimal game board:

Paths in red are unused, the green line depicts the portion of the path laid by the first 35 moves of the game, and the blue line depicts the portion of the path (of length 134) gained on the 36th move. One fun property of the above game board is that it is actually completely “unentangled” – no paths cross over any other paths.

On a Larger or Smaller Game Board

Other than being a good size for playability purposes, there is no reason why we couldn’t play Entanglement on a game board of larger or smaller radius (by radius I mean the number of rings of hexagons around the central hexagon – the standard game board has a radius of 3). We will compute the maximum score simply by mimicking our previous analysis for the standard game board. If the board has radius n, then there are 6 + 12 + 18 + … + 6n = 3n(n+1) hexagons, each of which contains 6 paths. Thus there are 18n(n+1) lengths of path, 3n(n+1)-1 of which are used in the first 3n(n+1)-1 moves of the game, and we want to add as many as possible of the remaining 15n(n+1)+1 lengths of path in the final move of the game. There are 12 path ends that touch the central hexagon and 12 + 24n path ends that touch the outer edge of the game board. Thus there are at least (12 + 12 + 24n)/2 – 1 = 11 + 12n unused paths on the game board.

Tallying the numbers up, we see that on the final move, we can add at most 15n(n+1)+1 – (11 + 12n) = 15n2 + 3n – 10 lengths of path. If T(n) = n(n+1)/2 is the nth triangular number, then we see that it’s not possible to obtain more than 3n(n+1)-1 + T(15n2 + 3n – 10) = (225/2)n4 + 45n3 – 135n2 – (51/2)n + 44 points. In fact, this score is obtainable via the exact same construction as the optimal board in the n = 3 case – just extend the (counter)clockwise rotation of the path in the obvious way. Thus, the maximum score for a game of Entanglement on a board of radius n for n = 1, 2, 3, … is given by the sequence 41, 1613, 9080, 29462, 72479, … (A180667 in the OEIS).

Further Variants of the “Look-and-Say” Sequence

January 13th, 2011

In two previous posts, I explored Conway’s famous “look-and-say” sequence 1, 11, 21, 1211, 111221, 312211, …, obtained by repeatedly describing the sequence’s previous term, as well as a simple binary variant of the sequence. In this post I will use similar techniques to explore some further variations of the sequence – a version where each term in the sequence is read in ternary, and a related sequence where no digit larger than 2 may be used when describing its terms.

As with the regular look-and-say sequence, the way we will attack these sequences is by constructing a “periodic table” of elementary non-interacting subsequences that all terms in the sequence are made up of. Then standard recurrence relation techniques will allow us to determine the rate of growth of the length of the terms in the sequences as well as the limiting distribution of the different digits in the sequence.

The Ternary Look-and-Say Sequence

Since we have already looked at the regular (i.e., decimal) look-and-say sequence, which is equivalent to the base-4 version of the sequence since it never contains a digit of 4 or larger, and we have also looked at the binary version of the sequence, it makes sense to ask what happens in the intermediate case of the ternary (base-3) version of the sequence: 1, 11, 21, 1211, 111221, 1012211, … (see A001388).

As always, we begin by listing the noninteracting subsequences that make this version of the sequence tick. Not surprisingly, it is more complicated than the corresponding table (of 10 subsequences) in the binary case, but not as complicated as the corresponding table (of 92 subsequences) in the decimal case.

# Subsequence Evolves Into
1 1 (3)
2 10 (5)
3 11 (19)
4 110 (21)
5 1110 (2)(4)
6 111210 (2)(8)
7 111221 (2)(16)
8 1121110 (22)(4)
9 112211 (23)
10 112221 (21)(20)
11 11222110 (21)(24)
12 1122211210 (21)(25)
13 1211 (7)
14 121110 (6)(4)
15 1221 (9)
16 12211 (10)
17 122110 (11)
18 1221121110 (12)(4)
19 21 (13)
20 211 (15)
21 2110 (17)
22 211210 (18)
23 212221 (14)(20)
24 22110 (26)
25 221121110 (27)(4)
26 222110 (2)(24)
27 22211210 (2)(25)

The (27×27) transition matrix for this evolution rule is included in the text file at the end of this post. Its characteristic polynomial is

The maximal eigenvalue of the transition matrix is thus the largest root of x3 – x – 1, which is approximately 1.324718. It follows that the number of digits in the terms of this sequence grows on average by about 32.5% from one term to the next.

The Look-and-Say Sequence with Digits 1 and 2

Closely related to the ternary version of the sequence is the sequence obtained by reading the previous term in the sequence, but with the restriction that you can never use a number larger than 2 (see A110393). This sequence begins 1, 11, 21, 1211, 111221, 21112211, …, and the sixth term is obtained by reading the fifth term as “two ones, one one, two twos, one one”. Because only two different digits appear in this sequence, it is perhaps not surprising that its table of noninteracting subsequences is quite simple:

# Subsequence Evolves Into
1 1 (2)
2 11 (5)
3 111 (7)
4 1211 (3)(6)(1)
5 21 (4)
6 22 (6)
7 2111 (1)(6)(3)

The transition matrix associated with this evolution rule is

As before, the average rate of growth of the number of digits in the terms of this sequence is determined by the magnitude of the largest eigenvalue of this matrix. A simple calculation reveals that this eigenvalue is √φ = 1.272…, where φ = (1 + √5)/2 is the golden ratio. Furthermore, we can answer the question of how many 1s there are in the terms of this sequence compared to 2s by looking at the eigenvector corresponding to the maximal eigenvalue:

What this means is, for example, that the second elementary subsequence (11) occurs φ times as frequently as the fourth elementary subsequence (1211). By weighting the subsequences by the entries in this vector appropriately, we can calculate the limiting ratio of the number of ones to the number of twos as

Download: Transition matrices [plaintext file]

Statistical Analysis of Password Strength via Gawker’s Leaked Database

December 15th, 2010

This past weekend, Gawker Media was hacked and its user account database was leaked online. The database contained about 1.3 million rows of information containing usernames, e-mail addresses, and passwords (encrypted via DES). This security breach is unfortunate for people whose information is contained within that database, but the silver lining is that it provides a rare opportunity for statistics nerds like me to analyze some otherwise completely unobtainable data.

Because the passwords were encrypted using such an out-of-date scheme (tsk, tsk, Gawker), about 200,000 of the passwords contained in the database have been decrypted. Of course, the passwords that were cracked were relatively weak. For example, all 2641 accounts that used some trivial modification of “password” or “querty” as their password were of course decrypted. In this post I will look at trends in which users’ passwords were cracked to gain insight into which users do and do not create strong passwords.

It should of course be made clear that, because this data comes from a single database, the results that follow may not be representative of the population as a whole, but rather may be skewed by the fact that people with Gawker accounts are generally more “techy” than the average internet user.

Preliminaries: Cleaning Up the Database

The database of course had to be significantly cleaned before it could be of too much use statistically, so some of the numbers here may differ slightly from the raw numbers you see from news outlets or if you download the raw database yourself. The numbers here are the result of removing any incomplete rows from the database (i.e., rows missing a password, e-mail address, or both) and removing any accounts that were clearly created by SPAMbots (I’m only interested in the password strength of real users).

Also, I will only look at accounts that contain an e-mail address with a domain that was registered in the database at least 50 times. This restriction is in place partly because it is extremely difficult to compute any sort of meaningful statistics on something with a sample size that is much smaller than 50, and it is partly due to the fact that Gawker doesn’t require verified e-mail addresses (so 46993 of the 52593 domain names listed in the database were used by exactly one person, many of which are clearly fake and/or for SPAM).

After making the aforementioned “fixes” to the database, there are 412670 accounts, 157794 (38.2%) of which had their password decrypted.

Password Strength by Domain Name

The following table displays the 10 most frequently-occurring domain names used for e-mail addresses in the database along with how many users of the domain had their password cracked.

Domain Total Accounts Decrypted Passwords Decryption % 158031 50530 32.0% 94147 40964 43.5% 66752 27332 40.9% 17534 8151 46.5% 7222 2801 38.8% 5544 2250 40.6% 4951 1750 35.3% 3896 1667 42.8% 3204 1476 46.1% 2211 860 38.9%

The following table shows the z-values associated with the statistical test that the two given domains have the same proportion of users with strong passwords. Differences that are statistically significant at the α = 0.01 level are in bold. Click on a z-value to see a normal distribution showing the associated p-value. Notice in particular that users have stronger passwords than users of any of the other top-10 domain names, while and users have the weakest passwords.

Yahoo Hotmail AOL Comcast MSN Mac SBC HotmailUK Verizon
GMail 58.28 40.84 38.65 12.10 13.48 5.00 14.27 16.89 6.92
Yahoo -10.26 7.29 -7.81 -4.27 -11.31 -0.89 2.87 -4.33
Hotmail 13.23 -3.55 -0.53 -7.74 2.27 5.75 -1.93
AOL -11.09 -7.70 -13.94 -4.19 -0.44 -6.75
Comcast 2.06 -3.85 4.11 6.98 0.09
MSN -5.52 2.14 5.00 -1.37
Mac 7.14 9.67 2.88
SBC 2.77 -2.97
HotmailUK -5.24

Educational Institutions

Not surprisingly, users who entered an e-mail address from an educational institution typically had stronger passwords than the general population. Of the 2092 users who provided a college or university-based e-mail address, only 697 (33.3%) were decrypted. This proportion is significantly lower than the corresponding proportion for the general population (z = 4.64, p < 0.001).

However, two universities stood out as having particularly weak passwords: of the 56 users who used a University of Texas e-mail address, 27 (48.2%) had their password decrypted, and similarly 101 (45.1%) of 224 New York University passwords were decrypted.

ISP-Provided E-Mail Users

Users who used an e-mail address provided to them by their ISP (such as typically had weaker passwords than the general population, a fact that can perhaps be explained by the fact that tech-unsavvy folks are less likely to go out and get a new e-mail address for themselves at a place like GMail. Of the 31667 users who provided an ISP-based e-mail address, 13053 (41.2%) of them had their password decrypted. This proportion is significantly higher than the corresponding proportion for the general population (z = -11.36, p < 0.001).

E-Mail Addresses with Typos

Also unsurprisingly, users who entered an obvious typo in their e-mail address were much more likely to have a weak password than people who entered their e-mail address correctly (by “obvious typo” I basically mean an e-mail address containing a typo of a common domain name, such as “fred@yahoo,com” or “fred@hotmail”). Of the 530 users with a typo in their e-mail address, 280 (52.8%) had passwords that were decrypted. This proportion is significantly higher than the average (z = -6.87, p < 0.001).

Password Strength by Country

The following table shows the strength of user passwords based on the country associated with their e-mail address. Of course some e-mail addresses provide no information about the user’s country, so domains that serve a largely international market (such as, and are excluded from this analysis.

Country Total Accounts Decrypted Passwords Decryption %
India 3129 1448 46.3%
United Kingdom 6874 3057 44.5%
China 1411 600 42.5%
Canada 2825 1160 41.1%
United States 30891 12507 40.5%
Germany 1378 484 35.1%
Russia 2223 533 24.0%

So Russia and Germany are the big winners when it comes to password strength, while India and the United Kingdom seem to have the weakest passwords. The following table shows the z-values associated with the statistical test that the two given countries have the same proportion of users with strong passwords. Differences that are statistically significant at the α = 0.01 level are in bold. Click on a z-value to see a normal distribution showing the associated p-value.

UK China Canada US Germany Russia
India -1.67 -2.32 -4.03 -6.26 -6.94 -16.62
UK -1.31 -3.06 -6.05 -6.37 -17.16
China -0.88 -1.49 -3.97 -11.72
Canada -0.57 -3.67 -12.73
United States -3.95 -15.37
Germany -7.18

Attached below is an Excel Spreadsheet containing significantly more detailed information than the snippets contained in this post (though of course all passwords, e-mail addresses and personally-identifiable information has been removed).

Download: Gawker Database Statistics [Excel spreadsheet]

The Binary “Look-and-Say” Sequence

November 7th, 2010

The look-and-say sequence (which I talked about here) is the sequence that you get by starting with the number 1 and constructing the next term in the sequence by “reading” the previous term. So 1 becomes “one one”, or 11. That becomes “two ones”, or 21. That becomes “one two, one one”, or 1211, and so on.

In this post, I am going to investigate the related binary version of the sequence, which starts off 1, 11 much like the regular sequence. But then when reading 11, we read it as “two ones”. Since two in binary is 10, the next term in the sequence is 101. When reading that term, we read it as “one one, one zero, one one”, so the next term is 111011. That term is read as “three ones, one zero, two ones”, and since three is 11 in binary and two is 10 in binary, the next term is 11110101, and so on. In this post we will answer two questions in particular about this sequence:

1) On average, how much longer is the (n+1)th term in the sequence than the nth term in the sequence?

2) On average, what is the ratio of the number of ones to the number of zeroes in the sequence?

Non-Interacting Subsequences

Much like the regular look-and-say sequence, we are able to study this sequence by constructing a “basis” of non-interacting subsequences that every term in the binary look-and-say sequence is made up of. Fortunately, constructing such a family of subsequences for the binary version of the look-and-say sequence is much simpler than it is for the decimal version of the sequence – here we only need ten different basic subsequences (whereas we needed 92 different subsequences for the regular look-and-say sequence!). These ten subsequences, and the subsequences they evolve into, are summarized in the following table.

# Subsequence Evolves Into
1 1 (2)
2 11 (3)(1)
3 10 (5)
4 110 (3)(4)
5 1110 (6)
6 11110 (7)(4)
7 100 (9)
8 1100 (3)(8)
9 11100 (10)
10 111100 (7)(8)

So for example, the first term in the sequence, 1, evolves into the subsequence (2), which is 11. That term then evolves into subsequence (3) followed by subsequence (1), or 101. That term then evolves into the subsequence (5) followed by the subsequence (2), or 111011, and so on. The reason that this representation of the sequence is useful is we can use it to describe the evolution of the binary look-and-say sequence entirely within a matrix T. In particular, we let T be the matrix with 1 in its (i,j) entry if the subsequence (i) appears in the evolution rule for subsequence (j), and 0 in its (i,j) entry otherwise:

Now if v is a 10-dimensional vector whose ith entry indicates how many times the subsequence (i) appears in a particular term of the binary look-and-say sequence, it follows that the entries of Tv tell us how many times each subsequence appears in the next term of the binary look-and-say sequence. So it follows from standard theory of linear homogeneous recurrence relations that we can now read off all of the long-term behaviour of the binary look-and-say sequence from the eigenvalues and eigenvectors of T.

Rate of Growth of the Sequence

The asymptotic rate of growth of the number of digits in the terms of the binary look-and-say sequence is simply the magnitude of the largest eigenvalue of the transition matrix T above. Using Maple it is simple to derive this value. If Ln is the number of digits in the nth term of the binary look-and-say sequence, then

This limit is approximately 1.465571, which means that the binary version of this sequence grows much faster than the decimal version of the sequence (recall that the growth rate of the number of digits of the regular look and say sequence is approximately 1.303577). This limit is also the unique real root of the cubic x3 – x2 – 1, which follows from the fact that the characteristic polynomial of T is

Ratio of Number of Ones to Zeroes

If we let Nn denote the number of ones in the nth term of the binary look-and-say sequence, and if we let Zn denote the number of zeroes in the nth term of the sequence, what is

In other words, what is the average ratio of ones to zeroes in this sequence? The following table shows the value of Nn/Zn for n = 3, 4, …, 25, which might give some intuition to the problem:

n Nn/Zn
3 2.000
4 5.000
5 3.000
6 2.000
7 2.000
8 2.000
9 1.786
10 1.762
11 1.742
12 1.717
13 1.691
14 1.690
15 1.680
16 1.676
17 1.672
18 1.671
19 1.669
20 1.668
21 1.667
22 1.667
23 1.666
24 1.666
25 1.666

Based on numerical estimates like those given in the table above, it has been conjectured that the limiting ratio is 5/3 (or some nearby value). We will now show that the limit does indeed exist, but its value is not 5/3 — it just happens to be really close to 5/3.

Much like the maximal eigenvalue of T tells us the overall growth rate of the sequence, the corresponding eigenvector tells us the distribution of the different subsequences that are present in the limit. Once we know the distribution of the individual subsequences, it is not difficult to find out the overall ratio of ones to zeroes by weighing the different subsequences appropriately. So our first step is to find the eigenvector corresponding to the maximal eigenvalue. To this end, it will be convenient to let

α is the same as in the previous section, and β is exactly the growth rate limit that we computed. Then the eigenvector corresponding to the maximal eigenvalue of T is:

What this means is that, in the limit, the fifth subsequence, 1110, is β times as frequently-occurring as the sixth subsequence, 11110 (for example). Now we just weigh each subsequence according to how many zeros and ones they contain, and we find the limiting ratio of ones to zeroes is

In particular, this ratio does not equal 5/3, but rather its decimal expansion begins 1.6657272222676… (which is less than 1/1000 away from 5/3).