It’s generally held that information does not have mass. However, physicist Melvin Vopson, reflecting on the work of Rolf Landauer (1927–1999), offers a somewhat alarming view: Not only does information have mass but that — at the rate we humans output it now — its energy will outweigh Earth. Yesterday, Eric Holloway offered a response to that claim:

Let’s accept that creation of information can indeed increase the amount of energy and mass in a system. But, according to the conservation of energy, the energy in a closed system remains constant. So, if Vopson is correct we now have a mystery because his theory is in tension with the conservation of energy. The only solution is that the system is not closed. So where is the opening in the system? If the system is physically closed, then the influx of information must come from outside the physical realm.

Eric Holloway, “Does information weigh something after all? What if it does?” at Mind Matters News (April 11, 2022)

So, anyone who thought that a closed universe (nothing but matter) is more likely if information has mass is bound to be disappointed…

Meanwhile, experimental physicist Rob Sheldon, who also follows the question, writes us to say,

Vopson is only half right. What Landauer said was that “one bit of information” took finite energy to erase, which by Einstein’s E=mc², weighs something. But a bit is not the right way to measure information. Claude Shannon took English words and erased one letter at a time to find out if the sentence was still intelligible. In a classic paper, he argues that each letter in English has about 4.5 bits of information.

I had an engineering friend who worked at Ford Motor Company in the 60s. They wanted to put more and more controls on the steering wheel. But each control had to have a sliding contact between the wheel and the steering column. There’s only so much space there, and maybe room for 4 sliding contacts. They wanted more controls than 4 and they were stumped.

My friend pointed out that instead of analog controls, they could have digital controls and 4 sliding contacts would give them 2⁴ (16) possible on/off controls. They thought he was a genius.

In exactly the same way, let us suppose Rolf Landauer says there is enough energy to erase 100 bits. How much information is that? No, it isn’t 2¹⁰⁰=1.2e30, it is actually 100!=1.0e200. That’s because each bit is distinguishable. Apple–orange–banana is not the same as apple–banana–orange. That is to say, a doghouse is not a housedog. Now look at the progression:

Note: The exclamation marks (!) below mean that the number is a factorial, that is, it is multiplied by every number below down to 1.

2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, which is a “geometrical series,” because a multiplicative 2 is the difference. But 1!=1, 2!=2, 3!=6, 4!=24 is an exponential series, because the multiplicative factor is getting bigger all the time.

So how does this affect the claims re information having mass? Sheldon explains,

For numbers less than 4, geometrical series win, but for 4 and greater the exponential series win out.

So as soon as we are talking about real world information, the more information we have, the less the bits weigh, until at very large amounts of information they weigh almost nothing.

The only way Vopson could be vindicated is if we store all that information serially on magnetic tape. But if the bits are not indistinguishable serial QM bits, then in the infinite limit they weigh nothing at all.

Here are Eric Holloway’s reflections on whether information has mass:

Does information weigh something after all? What if it does? At the rate we create information today, one physicist computes that in 350 years, the energy will outweigh the atoms of Earth. Vopson’s idea that creating information also creates mass and energy is fascinating — and it promises even bigger mysteries than the ones we address now.