Wigner’s friend

Quantum mechanics describes a system that nothing has measured as a sum of its possible results, and a measurement returns one of them. In 1961 Eugene Wigner asked what happens when a friend does the measuring and Wigner himself has not looked. In his example the friend watches for a flash of light, and Wigner asks afterwards:

For the friend the measurement is over: there is a result. For Wigner, who has not looked, the textbook’s rules, applied to the friend’s look as one more step, describe the whole laboratory, friend and notebook included, as a sum of both results. A sum is not a list of possibilities. Its two parts can be made to interfere, the way the two paths through a double slit make bands on a screen, and on those rules someone outside who controls the laboratory completely could, in principle, do that to the friend’s two results. So which is it: one result, or a sum of two?

1. 1961

   Eugene Wigner

   A friend who looks while Wigner does not, and the question of which description is right.

2. 1985

   David Deutsch

   An observer who writes down only that he knows the result, not which one, and whose two results would then be made to interfere. On the record that entry does not tell the two results apart, so it does not stop them interfering.

3. 2018

   Časlav Brukner

   With two friends and two Wigners: if every result, the friends’ and the Wigners’, has a value in every run whichever measurement each Wigner makes, if choices are free and if nothing travels between the sides, then a score made of four correlations cannot pass 2. The textbook’s rules, applied to the friends too, pass it.

4. 2018

   Daniela Frauchiger and Renato Renner

   Two friends and two Wigners, each using the textbook’s rules to reason about the others’ results, reach conclusions that contradict each other. On the record their Wigners can undo a friend’s laboratory only if the friend posted nothing.

Who has stood in for the friend

In each test, whatever held the friend’s result was something Wigner could reach and undo or read jointly: on the record, nothing about it was posted before the final read. None of these tests closed the loopholes that Bell tests without friends closed in 2015: the two sides were not far enough apart to rule out a signal between them, and in the photon tests only runs in which every photon was detected were counted, which takes the counted runs as a fair sample, and settings were not chosen at random run by run.

A friend here is whatever makes the measurement and holds its result: in the three experiments above, a photon, a photon’s path or a group of up to 17 qubits. Here is the experiment at the top of the page in the words of the record, the ledger of every event, written one posting at a time; the textbook’s word follows in brackets. In the model on this page the photon is a held register: nothing has been posted about what it holds [an unmeasured quantum system]. Its two values, one for each path, add into the sum of section 01.

The friend’s contact with the photon [the friend’s interaction with it] leaves a mark: a value held in a second register, the friend’s channel, that tells the photon’s two values apart. A mark is not a posting. The friend’s channel is a held register too, so Wigner, who can act on every held register inside the sealed laboratory, can run the contact backward: a contact of the opposite strength undoes it (the theorem “A contact leaves the state or applies the crossing”, Theorem 25; section 03), because nothing about the mark has been posted. The mark is gone, the photon’s two values are one history again, and his read at a setting φ, the turn he gives the lower path, rises and falls as he turns φ: fringes [a read is the textbook’s probability, once the total is set to one]. That is what happens on the right.

Writing the result down is different. It is a posting: one entry in the record, and no act Wigner can make erases it. The entry tells the photon’s two values apart for good, so his read comes out the same at every setting: no fringes. That is what happens on the left. Section 05 says what this rests on.

The paper proves what a contact does to a held state (its theorem “A contact leaves the state or applies the crossing”, Theorem 25). Write X for the crossing, the operator that one posting applies across the boundary of a register, and θ for the strength of the contact; a whole crossing is θ = 90°. The contact is

C(θ) = cos θ + i sin θ X,

and the theorem says exactly what it does to a state lying in a part of either of two kinds:

• Where the crossing moves the state off where it stood, the contact divides it in two parts: one left exactly as it was, with weight cos²θ, and one with a single crossing applied, with weight sin²θ; the two weights add to one.
• Where the crossing keeps the state where it stood, nothing is moved: the contact turns the two parts of definite sign in opposite senses and changes the weight of neither. A state whose sign is already definite is left alone by a contact of any strength.

For the register itself, the record’s working reading is the second case. One state can be of both kinds against different parts. In the model on this page the crossing takes the shape the record’s working statement gives it, a sign on the register (here the photon’s path) times a flip on the friend’s channel, which starts in a state the flip takes to one it does not overlap; on that channel the division into cos²θ and sin²θ holds under either reading: with weight cos²θ the channel stands where it stood and the photon is exactly as it was; with weight sin²θ the channel is flipped and the photon carries the sign. At θ = 45° the two marks the contact leaves for the photon’s two values do not overlap at all, so the channel tells them apart completely. Standing or flipped is not the friend’s result: the result is which of the two marks the channel holds. That is the contact the figure at the top runs.

Weak contacts add up to strong ones, C(θ)C(θ′) = C(θ + θ′), and a state crossed twice has not been crossed. What the theorem does not give, in either case, is the event: that one part is posted and the other is not. That is owed (O1).

The number at the top is the score of a correlation test of the kind Clauser, Horne, Shimony and Holt wrote down in 1969: two parties each choose one of two acts, and four averages of their paired outcomes are added, three with a plus sign and one with a minus. In the Wigner’s-friend form there are two sealed laboratories, and each party is a Wigner who either asks the friend for the result or reads the whole laboratory in a way that makes the friend’s two results interfere.

The paper proves three ceilings for this score (its theorem “Three ceilings”, Theorem 27), each reached: the first two for two parties acting on two blocks of one record cube, the third for any table of outcome probabilities; that two laboratories are two such blocks is owed (O3):

• 2 for acts that only read a value already there;
• 2√2 = 2.828… for acts that write, made on one state, commuting across the two parties and read with squared weights (one record cube reaches it exactly: “One cube reaches the middle ceiling”, Theorem 28);
• 4 for any table of outcome probabilities in which neither party’s choice shows at the other.

The value measured with photon friends, at the top of the page, sits above 2 and below 2√2. If choices are free and nothing travels between the sides, passing 2 excludes outcomes that were all values already there to be read: that is what the first ceiling of “Three ceilings” (Theorem 27) gives. On the record’s account, what is left is that some outcomes are written by the act, as in every test of Bell’s inequality that passes 2.

Bong and colleagues made the test sharper. Their bounds assume that every observed result is a single, absolute fact, that free choices are uncorrelated with events that came before them, and that a choice cannot affect a result too far away for a signal to reach it; and they add that a Wigner can carry out any operation the textbook allows on a friend and its surroundings. The record keeps the single result for a posted result and denies that last assumption for a friend who has posted (section 05). The authors leave room for a limit of this kind themselves (“some other limitation on coherent quantum control”). The page gives no number for a friend who has posted (not claimed).

Is any of this different from ordinary quantum mechanics?

No number is. What differs is what each takes as given: the textbook’s rules, applied to everything, let a Wigner with complete control undo any friend, notebook included, while the definition of the record lets no one undo a posting (section 05). Wigner’s friend is the one place among the twelve paradoxes where the record says something the unitary step alone does not.

Isn’t this known?

The mathematics is: the first case of “A contact leaves the state or applies the crossing” (Theorem 25) has the form of the textbook’s measurement rule, the paper’s theorem “Acts can be read together exactly when they commute” (Theorem 24) is Lüders’ theorem of 1951, the ceiling 2 is Bell’s theorem in the form Clauser, Horne, Shimony and Holt gave it, and 2√2 is Tsirelson’s. The shape of the answer has company too: objective-collapse theories also make a registered result final (the paper’s own working candidate for when a massive body posts has the form of Penrose’s rule), while on decoherence alone, many-worlds and relational readings a Wigner with complete control could still undo it; on the record that finality is the definition of the record (section 05).

What is not proved?

The three owed statements this paradox waits on, set out in section 05: attachment (O1), the position read (O2) and composition (O3). Not proved either: the working statement that gives the crossing its form, and the premises and hypotheses of the theorems used. The answer’s starting point is the definition of the record, not a theorem. Until O1 says what counts as writing in a laboratory, the page cannot say which friend in a real laboratory has written, only that a friend who has cannot be made to interfere.