Betting on the Outcomes of Measurements:A Bayesian Theory of Quantum Probability

Abstract

The Bayesian approach takes probability to be a measure of ignorance, reflecting our state of knowledge and not merely the state of the world. It follows Ramsey’s contention that “we have the authority both of ordinary language and of many great thinkers for discussing under the heading of probability… the logic of partial belief” (Ramsey 1926. Truth and probability. Cambridge: Cambridge University Press, p. 55). Here we shall assume, furthermore, that probabilities are revealed in rational betting behavior: “The old-established way of measuring a person’s belief … by proposing a bet, and see what are the lowest odds which he will accept, is fundamentally sound.” My aim is to provide an account of the peculiarities of quantum probability in this framework. The approach is intimately related to and inspired by the foundational work on quantum information of Fuchs (2001, Quantum mechanics as quantum information (and only a little more). Quant-ph 0205039), Schack et al. (2001, Physical Review A64 014305: 1–4) and Caves et al. (2002, Physical Review A 65(2305): 1–6).

Itamar Pitowsky†(1950–2010)

Appendix

1.1 The Formalism of Quantum Probability

With each quantum system we associate a complex Hilbert space ${\mathbb{H}}$. The dimension of ${\mathbb{H}}$ represents the number of degrees of freedom of the system. In this paper we consider systems with a finite number of degrees of freedom, hence dim ${\mathbb H} = n < \infty$, and we can identify ${\mathbb{H}}$ with ${\mathbb{C}}^n$.

Following Dirac we denote a column vector in ${\mathbb{C}}^n$ by “ket” $| \alpha \rangle$ and its transpose (row vector) by “bra” $\langle \alpha |$. (Recall that in ${\mathbb{C}}^n$ taking the transpose involves complex conjugation of the coordinates.). The inner product of $| \alpha \rangle$ and $| \beta \rangle$ is then simply $\langle {\beta } \mathrel{ | {\vphantom {\beta \alpha }} \kern-\nulldelimiterspace} {\alpha } \rangle$. Similarly, $| \alpha \rangle \langle \beta |$ is the linear operator defined for each ket vector $| \gamma \rangle$ by $| \alpha \rangle \langle \beta |( {| \gamma \rangle } ) = \langle {\beta } \mathrel{ | {\vphantom {\beta \gamma }} \kern-\nulldelimiterspace} {\gamma } \rangle | \alpha \rangle$. In particular, if $| \alpha \rangle$ is a unit vector $\langle {\alpha } \mathrel{ | {\vphantom {\alpha \alpha }} \kern-\nulldelimiterspace} {\alpha } \rangle = 1$, then $| \alpha \rangle \langle \alpha |$ is the projection operator on the one dimensional subspace of ${\mathbb{C}}^n$ spanned by $| \alpha \rangle$.

A pure state is a projection operator on a one dimensional subspace of ${\mathbb{C}}^n$. A mixture is any non-trivial convex combination of pure states $\Sigma _j \lambda _j | {\alpha _j } \rangle \langle {\alpha _j } |$, where $|{\alpha _j } \rangle$’s are unit vectors, $\lambda _j \ge 0$, and $\Sigma _j \lambda _j = 1$. A state is either a pure state or a mixture. It is not difficult to see that every state W is a Hermitian operator on ${\mathbb{C}}^n$ with non-negative eigenvalues and trace 1.

An observable is simply any Hermitian operator. Let A be Hermitian and let $a_1 ,a_2 , \ldots a_m ,m \le n$, be all the (real) distinct eigenvalues of A. To each eigenvalue a _i corresponds an eigenspace H _i of all eigenvectors associated with the eigenvalue a _i . Then the subspaces H _i are orthogonal in pairs and their direct sum is the entire space: $H_1 \oplus H_2 \oplus \cdots \oplus H_m = {\mathbb{H}}$. Let E _i denote the projection operator on H _i ; then we can represent:

$$A = \sum\limits_{j = 1}^m {a_j E_j }$$

The first bridge between the abstract formalism and experience is given by:

Born’s Rule :

Any measurement of the observable A yields one (and only one) of the outcomes $a_1 ,a_2 ,.\,.\,.\,,a_m$. If the state of the measured system is W then the probability of the outcome a _i is tr(WE _i ).

Now, with every physical system (a particle, a pair of particles, an atom, a molecule etc.) physicists associate a Hilbert space and a state on that space. The source of physical systems can be either natural (for example, a radioactive decay) or artificial (an electron gun). The choice of state reflects the physicist’s knowledge of the nature of the source. With every observable of the system (energy, momentum, angular momentum, spin) quantum theory associates an Hermitian operator. Hence, the calculation of the probability of every outcome of every measurement is made possible. Suppose the physicist chooses to test Born’s rule using the operator A and the state W. She prepares many systems in the same state W, and measures A on each. The prediction is then tested using standard statistical methods. (In most cases there is no problem to produce a sample of a very large size). We shall consider several examples below.

When an agent bets on the possible outcomes of a measurement of A the actual eigenvalues $a_1 ,a_2 , \ldots ,a_m$ are merely used as labels. Any other observable $A' = \Sigma_{j = 1}^{m} {b_j E_j }$, with $b_j \ne b_k$ for $j \ne k$, has exactly the same eigenspaces as those of A, and will make the same gambling device as A. This is like putting the numbers 7–12 on the faces of a dice instead of 1–6. In the main text we are interested in the outcomes, not their labels, and we therefore use in the term “observable” to denote the Boolean algebra generated in ${\mathbb{H}}$ by the eigenspaces H _i , as explained in Sections 11.2.1 and 11.2.4. In this appendix we shall keep the traditional meaning. Here observable is a Hermitian operator.

So far there is nothing non-classical about this mathematical description. One can, in fact, model any experiment with a finite number of possible outcomes by choosing an appropriate Hermitian operator and state on a suitable Hilbert space of a finite dimension. But when we consider more than one measurement on the same system, we transcend classical reality.

Heisenberg’s Rule:

Two observables A, B can be measured simultaneously on the same system if and only if $[A,B] = AB - BA = 0$.

Assume that A, B, and C are three Hermitian operators such that $[A,B] = [B,C] = 0$, but $[A,C] \ne 0$. By Heisenberg’s rule we cannot measure A and C together. However, the eigenspaces of B are elements of the Boolean algebra generated in ${\mathbb{H}}$ by the eigenspaces of AB and also in the Boolean algebra generated by the eigenspaces of BC. In other words, although non-commuting observables cannot be measured together, they can have logical relations. The logical relations between non-commuting observables are the source of the uncertainty relations (Section 11.3.1). In fact, the logical relations determine the probability rule (Gleason’s theorem Section 11.3.4). This means that, in a sense, Born’s rule can be derived from Heisenberg’s rule.

Examples:

Examples:

• 1. Spin- ${\frac{1}{2}}$ particles: let x, y, and z be three orthogonal directions in physical space and consider the $2 \times 2$ Hermitian matrices

  $$\sigma _x = \left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\\end{array}} \right), \quad\sigma _y = \left( {\begin{array}{*{20}c} 0 & { - i} \\ i & 0 \\\end{array}} \right), \quad\sigma _z = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & { - 1} \\\end{array}} \right)$$

  ((11.4))

  which satisfy $\sigma _x^2 = \sigma _y^2 = \sigma _z^2 = I$ (where I is the unit matrix). Also

  $$\sigma _x \sigma _y = - \sigma _y \sigma _x = i\sigma _z , \quad\sigma _y \sigma _z = - \sigma _z \sigma _y = i\sigma _x , \quad\sigma _z \sigma _x = - \sigma _x \sigma _z = - i\sigma _y $$

  ((11.5))

  The eigenvectors of $\sigma _z$ are ${| { + z} \rangle = {\scriptsize\left( {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array}} \right)}}$ and $| { - z} \rangle = {\scriptsize\left( {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array}} \right)}$ corresponding to the eigenvalues +1 and −1 respectively. In other words $\sigma _z = | { + z} \rangle \langle { + z} | - | { - z} \rangle \langle { - z} |$. The (normalized) eigenvectors of $\sigma _x$ are then $| { + x} \rangle = {\tfrac{1}{{\sqrt 2 }}}( {| { + z} \rangle + | { - z} \rangle } )$ and $| { - x} \rangle = {\frac{1}{{\sqrt 2 }}}( {| { + z} \rangle - | { - z} \rangle } )$ corresponding to the eigenvalues +1 and −1 respectively; and the eigenvectors of σ _y corresponding to the eigenvalues +1 ,−1 are, respectively, $| { + y} \rangle = {\frac{1}{{\sqrt 2 }}}( {| { + z} \rangle + i| { - z} \rangle } )$ and $| { - y} \rangle = {\frac{1}{{\sqrt 2 }}}( { - | { + z} \rangle + i| { - z} \rangle } )$. To measure the observable σ _z we subject the particle (which should be a spin-${\frac{1}{2}}$ particle, for example, an electron or a proton) to a magnetic field oriented in the z direction. The particle is then deflected above (eigenvalue +1, or spin-up in the z-direction) or below (eigenvalue −1 spin-down in the z-direction) its previous plane of motion, where it can be detected. To measure σ _x we do exactly the same thing, only with a magnetic field oriented along the x axis, and similarly for σ _y . Since none of the observables $\sigma _x,\;\sigma _y$, and σ _z commute with the other, only one of them can be measured at one time on the same particle.

  Consider the unit vector $| \alpha \rangle = {\textrm{\it{a}}}| { + z} \rangle + b| { - z} \rangle ,| {\textrm{\it{a}}} |^2 + | b |^2 = 1$. If the particle is in the pure state $W = | \alpha \rangle \langle \alpha |$, then the measurement of σ _z gives a spin-up (+1) result with probability $| {\textrm{\it{a}}} |^2$ and spin down result with probability $| b |^2$. A measurement of σ _x yields a +1 result with probability ${\frac{1}{2}}| {a + b} |^2$ and a −1 result with probability ${\frac{1}{2}}| {a - b} |^2$. A σ _y measurement yields +1 with probability ${\frac{1}{2}}| {a - {\textrm{i}}b} |^2$ and −1 with probability ${\frac{1}{2}}| {a + {\textrm{i}}b} |^2$.

• 2. Spin-1 particles: Define

  $$ S_x = \frac{1}{{\sqrt 2 }}\left( {\arraycolsep4pt\begin{array}{*{20}c} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\\end{array}} \right),\,S_y = \frac{1}{{\sqrt 2 }}\left( {\arraycolsep4pt\begin{array}{*{20}c} 0 & i & 0 \\ { - {{i}}} & 0 & {{i}} \\ 0 & { - {i}} & 0 \\\end{array}} \right),\, S_z\, = \frac{1}{{\sqrt 2 }}\left( {\arraycolsep4pt\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & { - 1} \\\end{array}} \right) $$

  ((11.6))

  They satisfy $[S_x ,S_y ] = iS_z ,[S_y ,S_z ] = iS_x ,[S_z ,S_x ] = iS_y$. Each operator has three eigenvalues −1, 0, +1. To measure S _z we subject a spin-1 (massive) particle to a magnetic field along the z-direction. This time there are three possible outcomes: the particle may be deflected below (−1), or above (+1) its plane of motion or simply remain on it (0). For a given intensity of the magnetic field, the amount of deflection in this case is twice as big as the spin-${\frac{1}{2}}$ case. Similar considerations apply for S _x and S _y . Since S _x , S _y , S _z do not commute they cannot be measured together. However, there is an interesting feature to this system, the squares of the operators commute: $[ {S_x^2 ,S_y^2 } ] = [ {S_y^2 ,S_z^2 } ] = [ {S_z^2 ,S_x^2 } ] = 0$. Also, $S_x^2 + S_y^2 + S_z^2 = 2I$, meaning that in a simultaneous measurement of $S_x^2 ,S_y^2 ,S_z^2$ one and only one of these observables will have the value 0, and the other two the value 1. To measure the simultaneous values of $\scriptsize{S_x^2 ,S_y^2 ,S_z^2}$ we measure the observable $H = S_x^2 - S_y^2$ using an electrostatic field. The three possible outcomes are 1, 0, −1, corresponding respectively to the cases where the values of $S_y^2$ is 0, of $S_z^2$ is 0, of $S_x^2$ is 0.

  Now let x′, y′ be two orthogonal directions so that x, y, z and x′, y′, z form two orthogonal triples of directions with the z-direction in common. The operators $H = S_x^2 - S_y^2$ and $H' = S_{\tiny x'}^2 - S_{\tiny y'}^2$ do not commute, but the (one-dimensional) eigenspace corresponding to the eigenvalues 0 of H and 0 for H′ are identical. This situation is depicted in Fig. 11.1. The logical relations depicted in Fig. 11.2 can also be realized by the same spin-1 system. We simply choose the orthogonal triples of directions in the end of Section 11.3.1.

1.2 Composite Systems, Kochen and Specker’s Theorem,and the EPR Paradox

Given a system whose Hilbert space is ${\mathbb{H}}_1$ and another system with a Hilbert space ${\mathbb{H}}_2$, the space associated with the combined system is the tensor product ${\mathbb{H}}_1 \otimes {\mathbb{H}}_2$. If $| \alpha \rangle \in {\mathbb{H}}_1$ and $| \beta \rangle \in {\mathbb{H}}_2$, we shall denote by $| \alpha \rangle | \beta \rangle \in{\mathbb{H}}_1 \otimes {\mathbb{H}}_2$ their tensor product. Let $| {\alpha _1 } \rangle ,| {\alpha _2 } \rangle, \,.\,.\,.\,,| {\alpha _n } \rangle$ and $| {\beta _1 } \rangle ,| {\beta _2 } \rangle, \,.\,.\,.\,,| {\beta _m } \rangle$ be orthonormal bases in ${\mathbb{H}}_1$ and ${\mathbb{H}}_2$; then every vector in ${\mathbb{H}}_1 \otimes {\mathbb{H}}_2$ has the form $| \phi \rangle = \sum {c_{jk} | {\alpha _j } \rangle } | {\beta _k } \rangle$. Applying the polar decomposition theorem to the matrix of coefficients c _jk we can find bases $| {\alpha'_1 } \rangle ,| {\alpha'_2 } \rangle , \ldots ,| {\alpha'_n } \rangle$ and $| {\beta'_1 } \rangle ,| {\beta'_2 } \rangle , \ldots ,| {\beta'_m } \rangle$ in which $| \phi \rangle$ has the form $| \phi \rangle = \Sigma d_j | {\alpha'_j } \rangle | {\beta'_j } \rangle$, where the d _j are real and the sum extends to min(m, n). Any Hermitian operator on ${\mathbb{H}}_1 \otimes {\mathbb{H}}_2$ is an observable; those which have the special form $A \otimes B$, where A and B are Hermitian operators on ${\mathbb{H}}_1$ and ${\mathbb{H}}_2$ respectively, are called “local observables”. The reason is that they are measured by separately performing A on the first system and B on the second. Notice that if $[A,A'] = 0$ and $[B,B'] = 0$ then $[A \otimes B,A' \otimes B'] = 0$. The extension of these observations to three or more systems are straightforward.

Consider now three spin-${\frac{1}{2}}$ particles. They are associated with the space ${\mathbb{C}}^2 \otimes {\mathbb{C}}^2 \otimes {\mathbb{C}}^2 \simeq {\mathbb{C}}^8$. Denote by $\sigma _k^{( j )}$ the operator $\sigma _k^{}$, k = x, y, z acting on particle $j,\,j = 1,2,3$. In other words: $\sigma _x^{( 1 )} = \sigma _x \otimes I \otimes I$, or $\sigma _y^{( 2 )} = I \otimes \sigma _y \otimes I$, and so on. In particular $\sigma _x^{( 1 )} \sigma _y^{( 2 )} \sigma _y^{( 3 )} = \sigma _x \otimes \sigma _y \otimes \sigma _y$ etc. Consider the following table of observables

$$\begin{array}{lllll} {\sigma _x^{\left(1\right)} ,} & \quad{\sigma _x^{\left(2\right)} ,} & \quad{\sigma _x^{\left(3\right)} ,} & \quad{\sigma _x^{\left(1\right)} \sigma _x^{\left(2\right)} \sigma _x^{\left(3\right)} } \\ {\sigma _x^{\left(1\right)} ,} & \quad{\sigma _y^{\left(2\right)} ,} & \quad{\sigma _y^{\left(3\right)} ,} & \quad{\sigma _x^{\left(1\right)} \sigma _y^{\left(2\right)} \sigma _y^{\left(3\right)} } \\ {\sigma _y^{\left(1\right)} ,} & \quad{\sigma _x^{\left(2\right)} ,} & \quad{\sigma _y^{\left(3\right)} ,} & \quad{\sigma _y^{\left(1\right)} \sigma _x^{\left(2\right)} \sigma _y^{\left(3\right)} } \\ {\sigma _y^{\left(1\right)} ,} & \quad{\sigma _y^{\left(2\right)} ,} & \quad{\sigma _x^{\left(3\right)} ,} & \quad{\sigma _y^{\left(1\right)} \sigma _y^{\left(2\right)} \sigma _x^{\left(3\right)} } \\ \sigma _x^{\left(1\right)} \sigma _y^{\left(2\right)} \sigma _y^{\left(3\right)}, & \quad\sigma _y^{\left(1\right)} \sigma _x^{\left(2\right)} \sigma _y^{\left(3\right)}, & \quad\sigma _y^{\left(1\right)} \sigma _y^{\left(2\right)} \sigma _x^{\left(3\right)}, & \quad-\sigma _x^{\left(1\right)} \sigma _x^{\left(2\right)} \sigma _x^{\left(3\right)}\\ \end{array}$$

((11.7))

The observables in each row in (11.7) commute in pairs, and the product of the first three in each row equals the fourth. This is obvious for the first four rows; as for the fifth row the equation

$$( {\sigma _x^{\left( 1 \right)} \sigma _y^{\left( 2 \right)} \sigma _y^{\left( 3 \right)} } )\ ( {\sigma _y^{\left( 1 \right)} \sigma _x^{\left( 2 \right)} \sigma _y^{\left( 3 \right)} } )\ ( {\sigma _y^{\left( 1 \right)} \sigma _y^{\left( 2 \right)} \sigma _x^{\left( 3 \right)} } ) = - ( {\sigma _x^{\left( 1 \right)} \sigma _x^{\left( 2 \right)} \sigma _x^{\left( 3 \right)} } )$$

((11.8))

as well as the fact that the operators commute in pairs, follows from (11.5).

Following Mermin (1990) we shall use this system to prove two major results. The first is originally due to Kochen and Specker (1967) who used a single spin-1 particle. The second result is due to Bell (1964) who used a pair of spin-${\frac{1}{2}}$ particle. In both cases Mermin’s proof is much simpler than the original.

Can we assign each quantum mechanical observable a value at all times, and regardless of whether it is actually being measured? In classical mechanics we associate with every observable (position, energy, momentum, angular momentum, etc.) a value at all times. We can consistently maintain that the system possesses the value, and a measurement merely reveals the possessed value. Can we do likewise in quantum mechanics?

Suppose (contrary to Bohr , see Section 11.4.1) that we can. To every observable A of the system we ascribe a value $\nu ( A )$ which may depend on time. Two conditions seem natural.

• 1. $v( A )$ is always among the values which are actually observed when we measure A, in other words $v( A )$ is an eigenvalue of A.

• 2. If A, B, C, … all commute, and if they satisfy a (matrix) functional equation. $f(A,B,C, \ldots ) = 0$, then they also satisfy the (numerical) equation $f( {v( A ),v( B ),v( C ),...} ) = 0$.

Both conditions follow from the requirement that the possessed values $v( A )$, $\nu ( B )$,… are the ones that are actually found upon measurement. In particular, in the second condition we assume that all the operators satisfying the functional relation commute in pairs. This means that they can be measured simultaneously, and the measured values indeed satisfy the corresponding numerical equation.

The Kochen and Specker’s theorem asserts that conditions 1 and 2 are inconsistent. In fact, one cannot assign values satisfying these conditions to the ten observables in (11.7). To see why, suppose by negation that we have assigned such values. By condition 1 we have $\nu ( {\sigma _k^{( j )} } ) \in \{ { - 1,1} \}$ for $k = x,y,z$ and $j = 1,2,3$. By condition 2 we have $\nu ( {\sigma _y^{( 1 )} \sigma _x^{( 2 )} \sigma _y^{( 3 )} } ) = \nu ( {\sigma _y^{( 1 )} } )\nu ( {\sigma _x^{( 2 )} } )\nu ( {\sigma _y^{( 3 )} } )$, and similar equations for the other triples. But this is impossible: take the product of the values of the first three operators in the fifth row: It is $+ \nu ( {\sigma _x^{( 1 )} } )\nu ( {\sigma _x^{( 2 )} } )\nu ( {\sigma _y^{( 3 )} } )$ since each of the $\nu ( {\sigma _y^{( j )} } )$’s occurs twice and $\nu ( {\sigma _y^{( j )} } ) \in \{ { - 1,1} \}$. This, however, contradicts the functional relation (11.8).

To translate this result to the language of the main text we consider a gamble ${\mathcal M}$ with five possible measurements, one for each row in (11.7). We write down the Boolean algebras of the possible outcomes of each measurement. There are many logical relations among the five Boolean algebras in ${\mathcal M}$, as each one of the ten operators appears in two different measurements of ${\mathcal M}$. The result is that the gamble ${\mathcal M}$ cannot itself be imbedded in a single Boolean algebra. This fact is actually equivalent to the Kochen and Specker’s theorem, as explained in their 1967 paper. In the main text I use a different simple example to derive the same conclusion.

It seems therefore that we cannot universally assign values to observables independently of their measurements. However, this does not prevent us from doing that in special cases when certain reasonable principles apply. A principle of that kind was proposed by Einstein Podolsky and Rosen (EPR) in their classical 1935 paper:

Principle R (reality) : If, without in any way disturbing a system, we can predict with certainty (that is, probability 1) that a measurement of A will give the result a, then we can say that $v( A ) = a$ independently of the measurement.

This principle stems from common sense: If I can predict with certainty that every time I open my office door the desk will be there, it means that the desk is there, regardless of whether I (or anyone else) sees it. Next, EPR explain what they mean by “disturbing the system”. To be more precise, they specify a necessary condition under which a disturbance can occur.

Principle L (locality) : A (singular) event that occurs at point x in space at time t can influence another event at point x′ at time t′ only if $\| {{\textit{x}} - {\textit{x}{'}}} \| \le c| {t - t'} |$, where c is the velocity of light.

This principle is a cornerstone of Einstein’s theory of relativity , a highly corroborated theory. It says that no disturbance, or influence, or any form of information can travel space at a speed greater than c. Bell (1964) proved that the conjunction of principle R and principle L is inconsistent with quantum mechanics. Here is a simple version of the proof:

In the Hilbert space of three spin-${\frac{1}{2}}$ particles ${\mathbb{C}}^2 \,\otimes\, {\mathbb{C}}^2 \,\otimes\, {\mathbb{C}}^2$ consider the unit vector $| \phi \rangle = \sqrt {1/2} ( {| { + z} \rangle _1 | { + z} \rangle _2 | { + z} \rangle _3 - | { - z} \rangle _1 | { - z} \rangle _2 | { - z} \rangle _3 } )$. It is a simultaneous eigenvector of $\sigma _x^{( 1 )} \sigma _y^{( 2 )} \sigma _y^{( 3 )} ,\sigma _y^{( 1 )} \sigma _x^{( 2 )} \sigma _y^{( 3 )},\sigma _y^{( 1 )} \sigma _y^{( 2 )} \sigma _x^{( 3 )}$, all corresponding to the eigenvalue +1. Consequently, by (11.8), it is also an eigenvector of $\sigma _x^{( 1 )} \sigma _x^{( 2 )} \sigma _x^{( 3 )}$ with eigenvalue −1. Suppose a source emits a triple of particles prepared in the state $| \phi \rangle \langle \phi |$. The particles emerge from the source and travel away from one another, forming trajectories 120° apart in a single plane. After the particles have travelled sufficient distance, say a few light years, each arrives at a measurement device with an observer. Call the observers Alice, Bob and Carol. We assume that the observers also move away from one another at a lower speed, being chased by the particles. Each observer performs a measurement, and all measurements are simultaneous in a frame of reference which is at rest relative to the source. This means that it will take a long time for any disturbance that might have been caused by Alice’s measurement to reach Bob’s or Carol’s location and vice versa. Assume that the observers know that the state is $| \phi \rangle \langle \phi |$. Assume also that they choose which measurement to perform, σ _x or σ _y , only at the last moment, and that, as a matter of fact, they all chose to measure σ _x . Now Alice correctly argues: “my result is $\nu ( {\sigma _x^{( 1 )} } )$, if Bob and Carol each measure σ _y then with probability one they will have $\nu ( {\sigma _y^{( 2 )} } )\nu ( {\sigma _y^{( 3 )} } ) = \nu ( {\sigma _x^{( 1 )} } )$”. Using the conjunction of R and L we conclude that the observable $\sigma _y^{( 2 )} \sigma _y^{( 3 )}$ has a value and it is $\nu ( {\sigma _x^{( 1 )} } )$. By a completely symmetrical reasoning we conclude that $\sigma _y^{( 1 )} \sigma _y^{( 3 )}$ has the value $\nu ( {\sigma _x^{( 2 )} } )$, and $\sigma _y^{( 1 )} \sigma _y^{( 2 )}$ has the value $\nu ( {\sigma _x^{( 3 )} } )$. The subtle point here is to see that there is a whole space-time region in which all three conclusions are warranted together (given R and L).⁹ But this is a contradiction, since

$$ \begin{aligned} 1 &= \nu {} ( {\sigma _y^{\left( 1 \right)} } ) \, \nu {}\, ( {\sigma _y^{\left( 2 \right)} } )\, {} \nu {}\, ( {\sigma _y^{\left( 1 \right)} } ) \,{} \nu {}\, ( {\sigma _y^{\left( 3 \right)} } )\, {} \nu {}\, ( {\sigma _y^{\left( 2 \right)} } ) \, {} \nu {}\, ( {\sigma _y^{\left( 3 \right)} } )\\ &= \nu {} ( {\sigma _x^{\left( 3 \right)} } )\, {} \nu {}\, ( {\sigma _x^{\left( 2 \right)} } ) \, {} \nu {}\, ( {\sigma _x^{\left( 1 \right)} } ) = - 1 \\ \end{aligned}$$

Some physicists prefer to avoid this dilemma by assuming that L is false. Bohm has taken this approach, and in his theory there are faster than light disturbances which, however, cannot be used for communication. In the main text I argue, on a more general basis, that R is the principle that should go.

More on these subjects may be found in Redhead (1987), and Bub (1997).