16
A Quantum Inductive Logic

16.1. Introduction1

The material theory of induction requires that good inductive inferences be warranted by facts within their domain of application. In earlier chapters, we saw many examples of individual inductive inferences warranted by specific facts. Marie Curie, for example, inferred the crystallographic system of all crystals of radium chloride from inspecting just a few specks of the substance. The inference was warranted by the facts of crystallography established in the preceding century, not by some universal inductive inference schema.

In such cases, there is little sense that the inductive inference forms part of a larger inductive logic whose overall structure could be abstracted in some measure from the specific subject matter. There are cases, however, in which this abstraction is possible. But complete abstraction is impossible; for that would provide a universal logic of induction. Yet we can find cases in which sufficient structure can be abstracted for a rich logic to appear.

The most familiar inductive logic of this type, and the one that is best worked out, is probabilistic logic. The prevalence of probabilistic logic has given the illusion that it is the universal inductive logic and that no other inductive logic is viable. This illusion persists only because of the familiarity of the example and the lack of sufficient efforts to identify alternatives. If we have any domain governed by some well-developed theory, then a compactly expressible inductive logic may be supported. Just which that logic will be, depends on the character of the theory. There will be cases in which the logic supported is not probabilistic.

In the preceding chapters, such cases were illustrated with simple examples: an infinite lottery machine, various forms of indeterministic systems, and nonmeasurable outcomes arising among infinitely many coin tosses. While we can and should demand that an inductive logic applies to these cases, one can be forgiven for finding the examples contrived or abstruse. They were so precisely because it enabled the systems to be simple enough for us to comprehend their physical properties fully.

Might we find an example with an immediate application to present science? This chapter presents such an example. Drawing on the work of Leifer and Spekkens (2013), we shall see that the natural mathematical structures of quantum theory afford a distinctive, non-probabilistic logic, at least for certain quantum systems, such as systems of entangled particles.

This quantum inductive logic differs from a probabilistic inductive logic in its most fundamental quantity. A probabilistic logic uses an additive probability measure to represent degrees of support or, in subjective terms, belief states. In its place, quantum logic uses a structure that arises naturally in quantum theory, a density operator. That this is the appropriate structure derives in turn from a deeper difference. Probabilities arise naturally when all of the distinct states of a system fall under a single probability measure supplied by background facts. While there are probabilities associated with measurement outcomes in quantum theory, each measurement setting is associated with a different probability measure and, crucially, their totality does not form a single probability measure. Rather, the different probability measures are both issued and unified by a single, deeper structure—a density operator. This structure is the fundamental quantity of quantum inductive logic.

It may seem strange at first to replace a probability measure by a density operator when probabilities can also be found in quantum theory, even if in scattered form. For, one might think, a probability measure—when it can be found—is just the right thing to use to represent partial inductive support or uncertain beliefs. This thought is driven more by familiarity and comfort than good reasons. The naturalness of a probability measure is an artifact of hundreds of years of development. It is a rather abstruse notion, as one finds when one engages in the cumbersome task of explicating precisely what it means to say that, for example, some outcome has such-and-such a probability. We shall see below that a density operator is no more abstruse and, since it is the central structure provided by the quantum mechanics, it functions much better as the basis of a quantum inductive logic.

Section 16.2 will sketch some probabilistic inferences on the presence of a rare genetic mutation among siblings. It will serve as a foil for the quantum case introduced in Sections 16.3 and 16.4, inductive inferences over the measured spins of entangled electrons. Section 16.5 to 16.9 develop the mathematical devices needed to treat the spins of entangled electrons. Sections 16.10 and 16.11 will identify one of these devices, a density operator, as the appropriate analog in the quantum case of the probabilities of the foil. Section 16.12 will provide a simple geometric picture of density operators to support this identification. Section 16.13 will briefly review how Matthew Leifer and Robert Spekkens (2013) developed the approach sketched into a fuller calculus with some analogies to the probability calculus. Sections 16.14 and 16.15 explore analogies and disanalogies between the probabilistic and quantum inductive logics. Section 16.16 offers conclusions.

16.2. Probabilistic Inductive Inference

16.2.1. Rare Genetic Mutations

As a foil for the quantum case, let us consider cases in which a probabilistic logic is warranted by prevailing facts. One case arises when we have outcomes generated by physical chances. The simplest case is a gambling casino. By careful design, a roulette wheel (with a 0 and 00) has a physical chance of 18/38 of a red outcome; and a physical chance of 18/38 of a black outcome. This fact, and others like it, warrant using the corresponding probabilities as the measure of inductive support for red and black; and it warrants employing the probability calculus as the logic of induction applicable to casino games.

Population frequencies can also provide a factual warrant for the use of probabilities in an inductive logic. Demographic data consistently shows that low educational levels correlate with unemployment. People in the US without a high school diploma, for example, are the group with the most unemployment. We make the added assumption that some individual has been chosen randomly, where randomly just means that each individual in the population has an equal probability of being chosen. It follows that the probability that the individual selected has a certain property matches the frequency of the property in the population. We can then use these probabilities as the measures of inductive support for the propositions that the individual has various educational levels and various employment statuses. That the individual has no high school diploma increases the inductive support for the proposition that the individual is unemployed; for the probability of unemployment given no high school diploma is greater than the unconditioned probability of unemployment.

Inductive inferences concerning genetic mutations in some population combine the essential features of the last two cases. To make matters concrete, consider a human population in which a mutation of some particular gene arises, but only very rarely. To make the example more interesting, assume that the mutation can arise in n mutually exclusive variations, so we have possible alleles

where N is the overwhelmingly most common case of no mutation—hence, the symbol N for “No.” We have a population of alleles in which the n mutations will arise with varying frequency. Physical chance process will govern the propagation of the alleles through the generations and those physical chances will determine the equilibrium distribution frequency of the various alleles. If standard, idealized conditions are met, these frequencies will conform with the Hardy-Weinberg equilibrium.

16.2.2. Inductive Inference Problems

The conditions specified above are the background facts that warrant inductive inferences over the presence of the mutation in the population. Since the physical chances are probabilistic, these inferences will be within a probabilistic inductive logic.

Consider some randomly selected child. The fact of random selection means that the probability that the child carries mutation mi matches the overall frequency ri of mutation-i carrying individuals in the population. These facts together warrant our use of probabilities as the measure of inductive support, where those probabilities are matched with population frequencies.

Consider two sibling children in some family. The measures of inductive support that each carries the mutation mi is given by the two probability measures:2

These two probabilities must also be related by the rule of total probability:

In general, the two conditional probabilities in this last formula are quite complicated expressions of the various gene frequencies. However, for the case of extremely rare mutations—that is ri << 1—they are approximated very well by

The first conditional probability arises from the circumstance that, if child1 carries mi, then it is overwhelmingly likely that just one of the children’s parents carries mutation mi. It is possible that a parent may carry two copies, or that both parents may carry copies, but these cases are far less likely and can be neglected. If just one of the children’s parents carries mutation mi, then there is a probability of 1/2 that child2 inherits it. The second conditional is recovered from a short application of Bayes’ theorem.3

We can use the conditional probabilities (3) to support an inference from the probability that child1 carries mi to the probability that child2 carries mi. Substituting (3) into (2) we find:

which agrees with (1).

This last inference is a particular case of how the rule of total probability becomes a rule of inductive inference in the probabilistic logic. Consider an outcome space that can be partitioned into mutually exclusive outcomes in two ways; that is as {S0, S1, …, Sn} and as {R0, R1, …, Rn}. We start with the probability distribution P(Rk) for k = 0, …, n as representing the inductive support for the outcomes Rk. The conditional probabilities P(Si | Rk) for i, k = 0, …, n, allow us to infer from the support accrued to the outcomes Bk to the support accrued to the outcomes Ai, by means of the rule of total probability:

16.3. From Mutations to Electrons

Quantum mechanics describes a physical realm that differs from more familiar systems in which probabilistic logics are appropriate. The facts that comprise quantum theory can warrant a rather different inductive logic for certain quantum systems. One of these systems, a pair of entangled particles, is analogous to the pairs of children of the mutation case above in that the pair is comprised of two related systems. However, if we try to carry out inductive inferences analogous to those concerning mutations carried by children, we will find that we need to use a non-probabilistic inductive logic and that this logic can be read off directly from the quantum mechanical formalism.

This is not the place to attempt a self-contained development of the standard formalism of quantum theory.4 However, my concern is that the development is accessible to those who do not work in quantum theory. I will do my best to motivate and explain the least amount needed to convey the main ideas to you, if you have less familiarity with the formalism. So do keep reading—this is written for you.

In the following, we will consider one of the simplest properties of one of the simplest, best-known particles. That is, we will consider electrons and their spins. Electrons carry angular momentum. Classically, angular momentum is a measure of the quantity of rotational motion of a body, like a spinning top. It is the rotational analog of ordinary linear momentum—“mass time velocity”—and, for the spinning top, is “moment of inertia times the angular velocity.” It is a vector quantity and is fully specified when we have fixed its real number magnitude and its direction in space. The magnitude is determined by the speed of rotation and the mass distribution in the top, as expressed by its moment of inertia. The direction is fixed by the axis of rotation.5 Angular momentum acquires its importance in both classical and quantum systems since it is a conserved quantity. The total angular momentum remains constant in all closed interactions.

It is almost the same with the spin of the electron. The angular momentum of the electron has a magnitude. Unlike a classical top that can spin faster and slower and thus can carry more or less angular momentum, all electrons carry the same magnitude of spin angular momentum. It is 1/2 in units of h/2π (where h is Planck’s constant). Since it is the same for all electrons, this magnitude is unimportant for what follows. Like the top, electron spin also has a direction and this direction can take all orientations in space. This direction is the quantity that will interest us. The direction is what is measured in many foundational thought experiments in quantum mechanics.

The major disanalogy between spinning tops and electrons with spin is that there is nothing rotating or spinning inside the electron. An electron carries angular momentum in the same way that it carries electric charge, as a fundamental, irreducible property. There is no deeper story about some hidden, spinning machinery that explains how the angular momentum comes about. It is just there.

16.4. Two Inductive Inference Problems for Electrons

The background facts fix the inductive logic appropriate to some domain. We can find situations involving electrons in which a familiar probabilistic induction is the appropriate one; and we can find situations in which it is not.

16.4.1. Uncertainty over Randomly Selected Positions

Here is an example of the first type that can be handled with ordinary probabilistic inferences. Assume that we have six widely spaced boxes, numbered one to six, and that an electron has been placed in just one of them. We do not know which contains the electron. We will choose one box at random and consider the outcome that this is the box containing the electron. How much inductive support is accrued to this outcome? To ensure that we have background facts that authorize probabilistic reasoning, assume that we choose the box whose number matches the outcome of a roll of an ordinary die. The die’s behavior, we assume, is governed by the probability calculus in the usual way.

A probabilistic inductive logic is warranted by the set up. The evidence just presented affords inductive support at the level of a probability of 1/6 that the box selected contains the electron. All possible outcomes fall under a single probability measure. While this is obvious, this simple feature is worth noting, since it will fail in the quantum problems to be considered below. However, so far, no quantum peculiarity has entered.6 The analysis would be the same if, instead of electrons, we had prepared six boxes with only one containing a marble.

16.4.2. Uncertainty over Measurements on Electrons in Entangled States

Now consider a second problem. It is possible to entangle two electrons so that their states are highly correlated. In the simplest case of two electrons in a “singlet state” (explained below), the two electrons have spins that always point in opposite directions. If one is measured to have a spin that points north, the other will always be measured to point south; and so on for every other possible pairing of opposite directions. This singlet state can persist even when the two electrons are separated by great spatial distances. They are entangled.

If we have access to one of the electrons in this entangled state, we can perform measurements of the direction of its spin. The measurement process is foundationally quite troublesome in quantum theory, as we shall see below. However, for present purposes, all that matters is that the measurement will yield some definite direction. We do not know in advance which that will be. Quantum theory only gives us probabilities for the different possible directions. Once we know the spin direction of one of the electrons in a singlet state, then we know the spin direction of the other electron, no matter how distant that electron is from us.

The inductive inference problem starts with the evidence that we have two electrons in some state, such as a singlet state. How much support does this evidence give to the various spin direction measurement outcomes that may arise for each of the electrons? How much support does this evidence give to possible connections between the spin direction measurements of the two electrons? These questions are the analogs of those asked above about the children and rare mutations. Given the background facts of the distribution of the random mutation, what is the probability that the first child carries the mutation? Given that one is a carrier, what is the probability for the other?

There are probabilities in the quantum inductive problem. However they prove not to be the fundamental quantities. The uncertainty is not the sort of probabilistic uncertainty that arises with random selection. For in random selection, there is a single probability measure that covers all possible outcomes. In the quantum case, there is no single probability measure covering all outcomes.

To proceed, we need to develop the elements of the quantum theory of electron spin.

16.5. Vector Spaces

An electron spin can point in any direction in space. It turns out that we can recover all possibilities if we start with two states, a spin that points up and a spin that points in the opposite direction, down. All other possibilities are recovered by adding together or subtracting—“superposing”—these states. Left and right pointing spin states are recovered by adding and, respectively, subtracting the up and down spin states.

This is not the way more familiar displacement vectors in space add and subtract. If we add a displacement of one foot north to a displacement of one foot south, they cancel each other out. They do not give us a displacement to the east or the west, as would spin vectors. In this respect, spin vectors are not quite like ordinary displacement vectors. However spin vectors do share the essential property with displacement vectors that we can always add two vectors to produce another with an intermediate direction. What counts as an intermediate direction, however, will be different in the two cases.

To keep track of these different directions, we will label them in the familiar way with Cartesian coordinate axes, x, y, and z and identify the “up” direction as the positive z direction. That we can add and subtract the different spin states to produce new ones, relying on the fact that they form a vector space.

Dirac’s “ket” notation is a convenient and compact way to write the vectors. The vectors of unit length corresponding to the to the +z (up) and −z (down) directions are written as kets |z> and |−z>. The x and −x pointing vectors of unit length, |x> and |−x>, are recovered by superposition as7

The summations can be pictured in the familiar vector diagram of Figure 16.1.

Figure 16.1. Superposition of vectors.

The figure also makes apparent the need for the factor of 1/√2. For simply adding vectors |z> and |−z> of unit length produces a vector of length √2. It must be rescaled by this factor to recover a unit vector.

So far we have spin states pointing in the x and z directions. We can also introduce spin states in the y direction by means of superpositions that employ i =√−1.

In general any superposition of these vector states produces a new vector state. There is a symmetry among them all; none is more fundamental. We can start by labeling any direction as the z direction and use the above formulae to produce the complete spin space.

Figure 16.1 allows the vector addition to look like the familiar addition of vector displacements in space. But it is in other ways a poor representation of the spin space. It allows us to draw the vectors |−z> and (−1)|−z> = −|−z> as two separate vectors, with the second pointing in a direction opposite to the first. This gives the appearance of a difference where there is no physical difference. The distinguishing phase factor (−1) in quantum theory has no physical import so that |−z> and (−1)|−z> represent the same state. A simpler picture eradicates the duplication. It is the Bloch sphere shown in Figure 16.2.

Figure 16.2. The Bloch sphere.

The figure looks so familiar that it is easy to misread. What is orthogonal—“perpendicular”—to what differs from Euclidean expectations. In this space, |z> and |−z> are orthogonal; as are |x> and |−x>; and |y> and |−y>. Yet |x> and |z> are not orthogonal, even though Euclidean expectations suggest otherwise. The sphere also looks like it is a three dimensional vector space that must be built from three independent basis vectors. However, it is a two dimensional space, with |z> and |−z> as its basis vectors. Their linear superpositions can span the whole sphere since complex numbers can be used in forming linear superpositions; and this shift from real to complex numbers gives the added degree of freedom needed.

16.6. Measurement

16.6.1. An Oddity in Quantum Theory

In non-quantum systems, measuring the state of a system is merely a technical challenge, not a foundational problem. If we have a spinning top, in principle, we can determine the direction of its axis of spin without having to destroy the top. Things are different in quantum theory.

We can learn something of the direction of the spin axis of an electron by passing it through an inhomogeneous magnetic field in a Stern-Gerlach apparatus. The magnetic dipole moment of the electron aligns with its spin and that moment determines how the electron is deflected by the magnetic field. The direction of the deflection tells us the direction of the spin. We need not delay with further details of this measuring operation except to mention one point:

To perform the measurement, we must choose in advance some direction in space along which to align the magnetic field of the Stern-Gerlach apparatus. Our measurement will be performed along that direction. The curious and foundationally troublesome property of measurement in the quantum context is that the measurement will always return a definite result along the direction chosen, no matter what the spin state of the electron.

If we measure the z-spin of an electron that has z-spin up, that is, its state is |z>, we will measure z-spin up with certainty. If we measure the z-spin of an electron with z-spin down, that is, its state is |−z> we will measure z-spin down with certainty. So far, there is nothing unexpected. But if we measure the z-spin of an electron in state |x> with x spin up, something odd happens. Since a state of x-spin up is different from either z-spin up or z-spin down, you might expect the measurement to fail in some way. It might, perhaps, give a muddled answer of both z-spin up and z-spin down and the same time; or perhaps no result at all. This does not happen. We still get a definite z-spin measurement outcome. It will be either z-spin up or z-spin down, without any confounding. Which of the two will happen? The formalism gives us a probability of 0.5 for each.

16.6.2. The Born Rule

In general, a z-spin measurement always returns either a z-spin up or z-spin down outcome. The probability of each will vary according to the state measured. Standard quantum theory provides a simple rule—the “Born rule”—for computing these probabilities. Assume that we are measuring the z-spin of an electron with some general state |ϕ>. We can decompose the state vector |ϕ> into two components in the |z> and |−z> directions.

Figure 16.3. Components of |ϕ>.

The two components are Pz|ϕ> and P−z|ϕ>, where the projection operator Pz picks out the component of |ϕ> in the |z> direction; and the projection operator P−z picks out the component of |ϕ> in the |−z> direction. The vector |ϕ> is the sum of these two components:

The Born rule tells us that the probability of measuring each outcome is given by the (length)2 of each of these two component vectors, where we recall that by supposition |ϕ> has unit length.

For the general case of a |ψ> measurement on a state |ϕ>, we have

For the case of |ψ> = |z> and |ϕ>=|x>, we have from (6) that

so that

as shown in Figure 16.4. The probability of each outcome is just (length)2 = = 0.5.

16.6.3. The Basis of the Difference between Probabilistic and Quantum Inductive Logics

That the Born rule gives us the correct probabilities for measurement outcomes is well established by experiment. How it does so and what happens during the measurement process, however, remains a troublesome issue in the foundations of quantum theory.

In the standard, textbook account, the electron state vector of the electron undergoing measurement “collapses” onto one of the two measurement states |z> or |−z>, with the probabilities given by the Born rule. That is, measurement instantly transforms a |x> state into a different one, a |z> state or a |−z> state, according to the outcome. Measurement changes the state. That measurement can do this is odd and puzzling. Yet it is an essential part of the standard account of quantum theory. An expansive literature has sought to find alternative accounts of measurement that avoid this oddity. None has produced a view that has been accepted widely enough to be the new standard.

Fortunately, my present purposes require no decision on how the measurement problem should be solved. I need to adopt only the bare account in which the Born rule gives us the correct probabilities for measurement outcomes.

This oddity of quantum theory is decisive as far as inductive logics are concerned. For the probabilities introduced by measurement do not merely reflection an uncertainty over which prior, existing state is at hand. Measurement changes the state and then attaches probabilities to the result. As a result, the probabilities of outcomes associated with different measurement scenarios cannot be combined into a single probability measure. Rather, a different quantity synthesizes these measures, and that quantity forms the basis of a quantum inductive logic.

16.7. Density Operators

The goal here is to find the inductive logic warranted by the quantum facts concerning electrons in entangled states. To proceed, we need to identify the structure in the quantum case that is analogous to the probability measure of probabilistic logic. This inductive structure is the density operator. It arises as follows.

For a single particle, in the most definite case, we assuredly have just one quantum state, such as |z>. It is called a “pure state.” What if we are uncertain as to which of two such pure states, |z> and |−z>, is at hand? It would be nice if our uncertainty could be captured merely by taking a suitably weighted sum of the two pure state vectors. This simple option fails. We already saw that adding these two vectors just gives us another pure state vector. If we add them with equal weight, for example, we merely recover |x>, as (6) shows.

While this simple option fails, something very close to it succeeds. An alternative way of representing a pure state is by a projection operator. There is a one-to-one correspondence between them, so picking one amounts to picking the other. We have already seen projection operators in the context of the Born rule of measurement above in equation (8). They pick out the component of a vector parallel to the direction of projection. For each unit vector, such as |z>, there will be just one projection operator that finds all of |z> to be in the direction in which it projects. We have written that unique projection operator as Pz. More compactly, the pure state |z> is associated uniquely with the projection operator Pz that has the property that Pz |z> = |z>.

Since these projection operators are a special case of density operators, let us explore them a little more. Operators in vector spaces are the analogs of functions in ordinary algebra. A function maps numbers to numbers. The square function maps 2 to 4, 3 to 9, and so on. An operator in the vector space maps vectors to vectors. The projection operator is one of the simplest. The behavior of the projection operator Pz associated with the vector |z> is fully specified by two facts. First is that the operator takes |z> back to itself and takes the vector |−z> to zero:

Second is that the projection operator is linear, so that

for all complex numbers A and B. Since an arbitrary vector |ψ> can always be written as this sort of linear sum |ψ> = (A |z> + B |−z>), linearity and (10) fix how the projection operator acts on any vector.

Now we return to the original problem. What if we are unsure as to which of |z> and |−z> is at hand? As long as we represent the states directly by vectors, we cannot just add the two vectors in a suitably weighted summation. We saw that would give us a new vector, which is just a different pure state. If we represent states with projection operators, then we can add them without this happening. If we weight the two states equally, then we produce the new operator for a so-called “mixed state,” in contrast to the pure states with which we started:

The subscript “max” indicates that the state is maximally mixed—that is, as far away as possible—from a pure state. (We will see how this comes about below.) This new operator is no longer a projection operator.8 It is a density operator. We do not need to use the ½ to ½ weighting. We merely need to use two positive real weights that sum to unity. The ½ to ½ weighting, however, is the case that will interest us most. We arrive at the most general density operator for the single electron spin by choosing arbitrary positive, real number weights wz and w−z,

such that the weights sum to unity, wz + w−z = 1.

At this stage, it looks as if the density operators of (11) and (12) are behaving just like probability measures. We appear to be uncertain over which of |z> or |−z> we have with probabilities wz and w−z, respectively. That appearance is reinforced by the term “mixed state.” Something like this is correct. But it is not quite like this. The unqualified term “mixed state” is misleading and it is in the qualifications needed that the novelty of the quantum logic will be found.

16.8. Tensor Product Spaces

A density operator is the appropriate structure for an inductive logic when we are inferring inductively over the properties of electrons in entangled states. These states arise as follows. Consider two electrons. Each has its own spin vector space. The first is formed by taking all linear superpositions of the states |z>1 and |−z>1 of the first electron. The second is formed by taking all linear superpositions of the states |z>2 and |−z>2 of the second particle. (The subscripts 1 and 2 just number the particles.) The two electrons together form a combined physical system with its own vector space. One state in it will be a product state such as |z>1|z>2. That is, the first electron state is z-spin up and the second is z-spin up also. All four of these possibilities are

We form a new vector space, the combined space of all possible states of the two particles, by taking all linear superpositions of these four states. The space is formed in the same way as we formed the one electron vector space by taking all linear superpositions of |z> and |−z>. This new space is the tensor product of the vector spaces associated with the individual particles.

This new vector space contains many new states. We will investigate one, the singlet state of total spin angular momentum of zero. It is 9

It is a superposition of two states: |z>1|−z>2 in which the first particle spin points “up” and the second “down”; and |−z>1|z>2 in which the first particle spin points “down” and the second “up.”

16.9. Reduced Density Operators

Consider two entangled electrons, such as the singlet state (13). The two electrons can remain entangled in the singlet state, even when they are widely separated spatially. If we have access to just one of these electrons, we can make a measurement of the spin direction of that one electron. The entanglement means that whatever measurement outcomes we obtain on our nearby electron will be correlated with the measurement outcomes that someone else finds on the other remote electron. We read that correlation directly from the two terms in the singlet formula (13). The first term |z>1|−z>2 tells us that whenever the first electron produces z-spin up on measurement, the second electron produces z-spin down (and conversely). The second term |−z>1|z>2 tell us that whenever the first electron produces z-spin down on measurement, the second produces z-spin up (and conversely). In short, our measurement on the nearby electron will always give a spin of the opposite direction from the result of a measurement on the remote electron.

When we make our measurements on the nearby electron, we will know nothing of these remote outcomes. Let us set them aside and ask what outcomes we should expect for measurements on the one electron to which we have access. Quantum theory provides the following recipe for determining the probabilities of the various outcomes.

The first step is to eliminate explicit appearance of the second, remote electron from the description of the two-electron system to arrive at a reduced description of the first, nearby electron only. We begin by replacing the vector representation of the entangled state by its corresponding projection operator, P12. For example, the projection operator associated with the pure singlet state |s> can be written as a sum that includes projection operators associated with the individual particles that comprise it:

where the “, 1” and “, 2” notation labels the nearby and remote electrons (respectively) to which the individual projection operators belong. The “further cross terms” contain operators that are not projection operators. While important in some applications, these further terms drop out of the calculations below.10

We now suppress the details of the second remote particle “2” by means of a “trace” operation “Tr.” This linear operator replaces the degrees of freedom in its scope by their expectation values. The trace operator Tr2 of the remote electron vector space suppresses the properties of the remote electron. If P12 is the projection operator associated with the entangled pair of electrons, we arrive at an operator that represents the properties of the first electron only by means of

The operator r1 need no longer be a projection operator but will in general be a density operator. Since they are produced in the reducing of the two-electron vector space to a one-electron space, they are called reduced density operators. For the case of the singlet state when P12 = Ps, we have 11

The operator ρs1 is not a projection operator.

That the reduced density operator for the nearby electron is not a projection operator captures the fact that the electron is in no definite spin state. If the entangled pair is in a singlet state, then the reduced density operator of the nearby electron (16) is the maximally mixed state (11). One might expect that the two factors of ½ are just the probabilities of measuring z-spin up and measuring z-spin down. They are.

This follows from the Born rule (9) for measurement outcomes for density operators. In its general form, the rule says that the probability of measuring a spin state |ψ> when we have an electron described by a density operator ρ is12

The projection operator Pψ is just the projection operator associated with the vector |ψ>. Applying this formula to the maximally mixed state ρmax we find:13

16.10. Density Operators Do Not Represent Probabilistic Ignorance of a Unique, True State

16.10.1. Many Probability Measures

The density operator ρmax for the maximally mixed state looks initially as if it just represents a familiar probabilistic uncertainty over whether the true state is |z> or |−z>. The two coefficents of for the states |z> and |−z> in the expression (11) reappear as the probabilities of measuring these states according to the Born rule (18).

What makes this mixed state different from mere probabilistic uncertainty is an important fact about the density operators of mixed states: they can be written in many ways, each indicating a different sort of uncertainty with a distinct probability measure associated with it. That makes the term “mixed state” potentially quite misleading. The state is not a simple mixture that can be decomposed uniquely into its components. It is not like a mixture of sand and iron filings that can be unmixed uniquely with a magnet.

Since this is the key point for all that follows, let us be clear on how this comes about. The density operator is simply a map that takes vectors to vectors. Two density operators are the same if they map the same vectors to the same vectors. In this respect, they are no different from ordinary functions. Take f(x) = x2. It is a function that maps numbers to their squares. While their expressions look different when written down, the functions g(x) = (x + 1)(x − 1) + 1 and h(x) = (x + 2)(x − 2) + 4 perform exactly the same mappings. So they are the same function.

It turns out that the mapping of the maximally mixed state ρmax of (11) can be represented equally well by many equivalent expressions

Here, Px is the projection operator associated with |x>, Py with |y>, etc., and Pψ is the projection operator associated with some arbitrarily chosen unit vector |ψ> in the Bloch sphere, pointing in any direction. I is the identity map that takes each vector back to itself.

Each of the expressions for ρmax in (19) represent the same map on the vector space, which is written most simply as the last expression on the list, I. That is, ρmax is the map that merely takes each vector in the space back to a half-sized version of itself. To see that they are equivalent, we need only recall from (7) that an arbitrary vector |ϕ> is the sum of its two components, when decomposed in the + ψ and – ψ directions:

It follows that (Pψ + P−ψ ) is just the identity operator I—that is the operator that merely maps a vector back to itself. Thus (Pψ + P−ψ ) = Pψ + P−ψ = I. This is true no matter which unit vector |ψ> is used to define it. Thus, the maximally mixed state density operator ρmax is defined equally well by any of the formulae in (19).

These equivalent representations of the maximally mixed state ρmax provide further probabilities for measurement outcomes analogous to (18)

16.10.2. No Single Probability Measure Unifies Them

The combined measurement outcomes of (18) and (20) are incompatible with the ordinary notion of probabilistic uncertainty as mere ignorance of some definite but unknown state. This sort of ignorance can be captured by a single probability measure, whereas there can be no single probability measure covering all the results of (20). For each of the states returned by measurement are incompatible with all the others. An x-spin up state is different from either a y-spin up and a z-spin up state. An effort to treat these probabilities as generated by ignorance over some true but unknown state fails and does so rapidly.

Take the probabilities of (18). If we interpret them as this sort of ignorance, then we have with probability one that the true state of the system is |z> or |−z>. For the two states are mutually exclusive so that

It now follows that the probabilities of all the other states must be zero, which contradicts the probabilities reported in (20).

Might we solve the problem with a simple expedient? Take a large outcome space whose primitive events are of the following form:

We can form a single probability measure over this larger outcome space, such that the probabilities of (20) can be recovered as conditional probabilities. For example,

The difficulty with this proposal is that our space now includes probabilities over our freely chosen actions, such as:14

The probabilities of (20) are provided directly by quantum theory itself. These new probabilities over our actions bring nothing but trouble. What grounds these new probabilities? To secure a grounding in physical chances, we might employ some physical randomizer to instruct us in which measurement to make. Then our inductive logic has been restricted to this special case. Or if we wish to leave the setting as open as possible, then this very openness means that there are no specific facts that warrant the introduction of the probabilities. In the worst case, they are arbitrarily chosen subjective probabilities and we corrupt the objectivity of our inductive logic by mingling them with the objective probabilities of (20). Setting aside this extreme case, we have still compromised the quantum inductive logic by interweaving inductive support from two distinct arenas: the inductive support for various quantum measurement outcomes as guided by quantum theory; and the inductive support for certain of our choices as guided by the vagaries of the human circumstances surrounding our choices.

These are serious difficulties and best avoided. Inductive support for quantum outcomes ought to be independent of human affairs. There is no need for us to face these difficulties. For nothing compels us to combine the probability measures of (20) into a single huge measure. We can arrive at an inductive logic that does not need them, as long as we are willing to give up the idea that an inductive logic must be probabilistic.

16.10.3. Density Operators as the Fundamental Inductive Structures

The maximally mixed state ρmax already represents some sort of uncertainty over the electron state. It is not the same as the probabilistic uncertainty familiar from cases of ignorance arising through random sampling, for such uncertainty cannot issue in the measurement probabilities (20). The direct way to understand the sort of uncertainty represented by ρmax is that it is the inductive structure that manifests as the infinite list of the measurement probabilities (20). It is a compact representation of them all.

For many, the predisposition to favor probabilities is strong. They might be inclined to say that this means that the logic is still probabilistic—here, finally we have probabilities. However, these probabilities are not the central quantities. They are intermediates that mediate between the density operator and the measurement outcomes. To capture the inductive situation fully, we need the entire infinite set. It is insufficient merely to report a subset associated with fewer than all directions of measurement. One cannot use the rules of the probability calculus to infer from the measurement probabilities for x-spin measurements, for example, to those for y-spin measurements.

The density operator is the natural and compact representation of the capacity of the electron to deliver different measurement results. When we form the new inductive logic adapted to this quantum case, the density operator is the central quantity that replaces the probability measure of the more familiar probabilistic inductive logics. It is the quantity that figures centrally in the physics of entangled electrons, in the same way as physical chances figure centrally in the physics of roulette wheels. It is the quantity around which we should build an inductive logic for entangled electrons, just as we build an inductive logic for roulette wheel outcomes around physical chances.15

16.11. Is the Density Operator Really an Inductive Structure?

Is it really admissible to treat density operators as inductive structures that can serve in an inductive logic? They seem to be a poor choice, for it is hard to say precisely what sort of uncertainty they represent. They do not represent the familiar sort of uncertainty captured by probabilities. Why should we erect an inductive logic for quantum theory around density operators when, perhaps with some effort, we might find a way to replace them with probability measures?

The short answer is that we should use these density operators since they are the appropriate structures delivered by the applicable physics. The uncertainty they represent is more opaque to us than that represented by a probability measure merely because the latter are familiar and their problems largely tamed. We should not mistake the resulting transparency of probability measures for their necessity in inductive logics. Indeed, the sorts of analyses that make probability measures interpretationally transparent can be applied equally successfully to density operators.

To see this, note that probability measures initially require considerable interpretive work before their meaning becomes clear or clear enough. If we are unprepared, we encounter severe difficulties when we try to give an explicit definition of probability talk. The challenge is to complete the formula:

The difficulty is that “probability” always seems to creep into the text requested. We cannot complete the formula by saying that the frequency of success in repeated, independent trials approaches 0.65 in the limit of arbitrarily many trials. We must say that this limit is approached with probability one.

While these are serious difficulties, they do not mean that probability talk is meaningless. Recalling the lengthier discussion in Section 13.8.1, we can constrain the meaning of probability talk quite effectively with a simple device.16 First, we recall that a probability measure conforms to the standard axioms of probability theory. Second, we give interpretations of near certainty to the probabilistic extremes: a probability near one is a near certainty of occurring; a probability near zero is a near certainty of not occurring. We can use these components to provide interpretations for cases of intermediate probability. The trick is to embed the probability talk into a larger discourse in which the already interpreted cases of near unit or near zero probability arise.

For example, take the proposition that an outcome has probability 0.65. The weak law of large numbers is a most useful theorem of the probability calculus. It tell us that we can always carry out sufficiently many repeated, independent trials to bring the frequency of success as close as we like to 0.65 with a probability as close as we like to one. Most people find that this gives them enough to grasp the difference between the two propositions:

An outcome has probability 0.65.

An outcome has probability 0.05.

Loosely speaking, the first outcome happens thirteen times as often in repeated, independent trials.

If this sort of interpretive apparatus is sufficient to dispel the clouds around probability talk, then the clouds surrounding the density operator as an inductive structure can also be dispelled. For a quite analogous interpretive apparatus can be employed for them.

First, density operators obey a quite definite axiom set and thereby accrue meaning implicitly, just as do probabilities.17 Second, we can identify extreme cases. The most definite is the density operator corresponding to a pure state such as |z>, a projection operator such as Pz. This projection operator expresses certainty that we do have the state |z>; and certainty that we do not have the state |−z>. It is analogous to an outcome of probability one if |z> is the true state at hand. The maximally mixed density operator of ρmax is the least definite. It favors all spin directions equally, for under it every possible spin direction has the same probability upon measurement. It is the analog of a uniform probability measure that assigns the same probability to all simple outcomes.

These most and least definite density operators are the extreme cases. For all the intermediate cases, we will be able to give a list analogous to (75) of the probabilities of all possible measurement results. That list gives us the same sort of interpretive purchase on the associated density operator as does saying something like “probability 0.65 means that the outcome happens roughly 65% of the time.” Analogously, we can say that having some particular density operator entails that we have such and such probabilities of outcomes on this or that measurement, where the list includes all possible measurement and outcomes. That is, we know probabilistically what it is to have some density operator as an inductive structure in terms or all possible measurement experiences in the world. If we are confident in our understanding of probabilities as inductive structures, then we should be confident in our understanding of these density operators as inductive structures.

16.12. A Geometric Picture of an Electron Spin Density Operator

Part of our comfort with probability measures is that there are simple physical or geometric models for them. For example, distributing probabilities over different outcomes is akin to dividing a unit mass into parts and locating different parts on the different outcomes. The weight of evidence appears directly in the analogy as a mass. Additivity of the probability measure is captured by the fact that we can only increase the mass on one outcome by reducing the mass on others by exactly the same amount.18 If we have a probability density over some continuous space of outcomes, we can picture the space as an area and the probability density at each point as the altitude of some mountainous surface spread over it.

A fertile picture of all possible probability distributions over n + 1 mutually exclusive outcomes is provided by an n-simplex. For three mutually exclusive outcomes, A, B, and C, the n-simplex is a triangle, as shown in Figure 16.5. The three vertices represent A, B, and C and each point in the triangle represents a distinct probability measure. The probabilities of each of A, B, or C increase with the proximity of the point to the corresponding vertices. The figure shows contours of constant probability for P(A), P(B), and P(C).

The interior point shown in the figure represents a probability measure for which P(A) = 0.5 and P(B) = P(C) = 0.25. For this measure, P(A) is greater than P(B) or P(C) since the representative point is closer to the A vertex than to the B or C vertex.

In general, there are no correspondingly simple geometric pictures for density operators. The exception is the special case of the spin space of an electron. All possible density operators can be represented elegantly in a three-dimensional sphere, as shown in Figure 16.6.19 Each density operator is represented by a single point in or on the sphere.

The pure states, corresponding to projection operators, occupy the surface of the sphere. (This surface by itself is the Bloch sphere we saw in Figure 16.2 above.) These surface points correspond to the most definite cases of a single pure state. The points inside the sphere represent density operators that are not also projection operators. They represent mixed states. The ones closest to the surface are least mixed and closest in their properties to pure states. The deeper one proceeds inside the sphere the more mixed the states become. The central point is the maximally mixed state ρmax.

The sphere representation also affords a simple picture of which pure states are mixed to yield each density operator. The maximally mixed density operator ρmax lies at the center of the sphere. Any diameter through the center connects two opposite points on the surface of the sphere, as shown in Figure 16.7. The points connected are two pure states that form ρmax.

We read directly from the figure that ρmax can be formed by equal mixtures of pure states |x> and |−x>; or |y> and |−y>; and so on, as summarized in (19). The multiplicity of possible decompositions of the mixture is represented by the multiplicity of possible diameters through the center.

There is a corresponding representation for the remaining density operators. Consider another density operator ρ that is not the maximally mixed ρmax. Any chord through it will intersect the surface of the sphere at two points, as shown in Figure 16.8.

The two pure states at either end of the chord, |a> and |b>, are the two that are mixed to form ρ. Since there are infinitely many chords through an arbitrary point inside a sphere, a given density operator ρ can be constituted from infinitely many distinct pairs of pure states.

For each case, the geometric construction provides the weighting. The point representing ρ on the chord divides it into two lengths, λa and λb, where the lengths have been scaled so that λa + λb = 1. (Note that λa is the length of the chord segment between ρ and |b>; and similarly for λb.) These are the two weights used to form ρ. That is, if Pa and Pb are the projection operators associated with pure states |a> and |b>, then20

The general density operator of (21) can no longer by recovered by tracing away the degrees of freedom of a remote particle in a singlet state (13). We need to replace the entangled singlet state by another. Many choices are possible. A simple one is:

When we trace away the degrees of freedom of the second particle, this operator reduces to the density operator (21).

The maximally mixed ρmax divides each unit diameter into two equal parts of length 1/2 and these weighting factors correspond to the probability of measurement outcomes coinciding with the pure states at either end of the diameter. Something similar holds for the general case of (21), in which the density operator lies on the chord connecting pure states Pa and Pb. We have21

where

That is, the probability of an a-outcome on an a-measurement is given by the weighting factor la, with the addition of a correction factor in P(a | b). This correction factor arises only when the two states mixed, |a> and |b>, are not orthogonal, that is, not mutually exclusive. It does not appear in the case of the maximally mixed ρmax, since ρmax arises from mixing orthogonal states such as |z> and |−z>.

Combining all these considerations, we recover a quite serviceable representation of the sort of uncertainty represented by density operators in this simple case. A density operator Pa on the sphere’s surface is a projection operator associated with a pure state |a>. It is the most definite case. For an a-measurement, it will assuredly give us an a-outcome. A density operator close to Pa will give an a-outcome on an a-measurement with high probability. For it will most commonly be associated with a value of λa close to one.22 As the location of the density operator approaches the mid-point, the probability of an a-outcome on a-measurement will approach 0.5, which is the probability associated with the maximally mixed density operator at the center of the sphere. This maximally mixed density operator treats all pure states alike: the probability of an a-outcome on a-measurement is 0.5, no matter what |a> is. That it must do this is immediately clear from the fact that the sphere has a rotational symmetry about the center of the sphere. From that central point, no pure state is closer than any other. It must treat all alike.

16.13. Leifer and Spekkens’ System of Quantum Inference

So far, we have seen only a part of the inductive logic appropriate to entangled electrons. We have identified the reduced density operator in each single electron’s vector space as the structure corresponding to the probability measure in a probabilistic logic. We need to do only a little more to specify the full logic. That is, we need a full specification of which density operators arise in which circumstances. As it happens, no further theorizing is needed to arrive at this specification. It is given to us by the standard formalism of quantum theory. When the theory lays out the physics of how the reduced density operators of entangled electrons relate, it is also giving us the inductive logic.

One may wonder, however, if what results really is an inductive logic. If one is used to and is expecting a probabilistic logic, it will be unfamiliar, just as density operators are not quite like probability measures. But that is no reason to dismiss it. Lack of familiarity is not the same as failure.

Leifer and Spekkens (2013) have shown, however, that the inductive logic based on density operators is not so unfamiliar after all. Once we adopt the density operator as the basic inductive structure, they have shown how we can rewrite basic results in quantum theory so that they are structurally analogous to formulae in a probabilistic logic. Their system is elaborate and distinguishes connections between variables according to whether they are causally or acausally related. To give a quite preliminary sense of the system, I will describe how it treats the case of acausally related systems, such as the two particles in a singlet state.

The following Table 16.1, based on Leifer and Spekkens (2013, p. 7), summarizes the correspondences:

Table 16.1. Correspondences between probabilistic and quantum logics

These correspondences are fairly straightforward. In the mutation example, the classical variables R, S, … are the genetic make-ups of each child. When R, S, etc. take specific values, then the genetic makeup of the child is specified as a particular mutation, m1, m2, … and their totality forms the outcome space. In the case of entangled electrons, systems A, B, … correspond to electron1, electron2, …, and the vector spaces HA, HB, … are the vector spaces of electron states described above.

The remaining formulae have been written in a way that emphasizes the parallels between the two cases. The classical summation operation “ΣS…” sums away the variable S. Correspondingly, the trace operator TrB (...) averages away the degrees of freedom associated with B. The star operation  is a particular multiplication operation designed to keep the parallel in the formulae as close as possible. The goal is to find the quantum analog of P(S & R) = P(S | R) × P(R), where the “×” is just ordinary arithmetic multiplication, since the probabilities P(.) are real numbers. One might write ρAB = ρB|A ⋅ ρA, as a direct analog of the probabilistic formula. But caution is needed, since there are important disanalogies. The operation joining ρB|A and ρA is not simple multiplication, but the sequential application of operators, since ρB|A and ρA are operators that act on vectors. This produces two problems.

The first is that the two operators act on different vector spaces. ρB|A acts on vectors in HA⊕ HB. ρA acts on vectors in HA. If they are to be combined, they must act on the same vector space. The simple remedy is to expand ρA to ρA⊕ IB, where the addition of …⊕ IB makes a new operator that acts as ρA on HA and as the identity (“do nothing”) on HB.

The second is that the order in which we combine the operators will matter, whereas it does not matter when we multiply real numbers. The formula ρB|A ⋅ ρA says first act with ρA and then with ρB|A. The formula ρA ⋅ ρB|A says first act with ρB|A and then with ρA. There is no assurance that the two will yield the same result; and in general they will not. Which is the correct order? It turns out that neither is correct if the resulting product is to be a new density operator. To make sure it will be a density operator, we split the operator ρA into a product of its square root, so that ρA = ρA1/2 ⋅ ρA1/2. Instead of multiplying ρB|A by ρA, we multiply it from either side by ρA1/2. The formula that results from both changes is the definition of the star operator:

An inversion gives an explicit expression for ρB|A

16.14. Analogous Inferences: Mutations and Electrons

In the case of mutations among children, we used the rule of total probability (5) in the series of computation (2), (3), and (4), to infer from the probabilities of various mutations in one child to the corresponding probabilities for a second child. We can use these new quantum formulae to display the corresponding inference for pairs of electrons in the singlet state.

Take two electrons in the singlet state (13)/(14) with projection operator P12. Using (16) and (19), the reduced density operator representing each of the electrons individually is

These are the quantum analogs of the probabilistic equation (1) of the mutation case:

A short calculation shows that24

The analog of the rule of total probability in Table 1 is

Substituting for P1|2 and r2, we use this rule to infer from state ρ2 of the second electron to that of the first ρ1. We find

in agreement with (25). This last computation (27) is the quantum analog of the application of the classical rule of total probability in (2), (3), and (4).

16.15. Disanalogies

These last comparisons underscore the analogies between a probabilistic inductive logic and the quantum inductive logic induced by the laws of quantum theory onto electrons in entangled states. That these analogies are present shows that the quantum logic is of comparable richness to the probabilistic logic. The key point for our purposes, however, is that the analogies are incomplete. The quantum inductive logic is a distinct inductive logic.

That the analogies are incomplete is already established by the investigation of the properties of density operators in Section 16.10. When the formal properties of the quantum inductive logic are explored, further disanalogies emerge. They derive from the fact that probabilities are numbers, whose products are insensitive to the order of multiplication, whereas density operators are sensitive to the order of multiplication. Switch that order and one may get a different result.

One consequence of the lack of commutativity of operators is the following disanalogy discussed in Leifer and Spekkens (2013, p. 33). The probability P(S & R) can be expanded as a simple product

The rule is robust and holds if all the probabilities are themselves further conditionalized on another variable T:

This is extremely useful in probabilistic analysis since it means that we can collect all background information into some huge proposition T and then treat all probabilities conditionalized on T, P(. | T), as if they were unconditional probabilities P(.).

The first of these two formulae has a quantum analog

However, the second does not. That is, we do not in general have

This means that the rule for forming conditional states will differ according to whether or not we begin with a state that is itself already conditional.

16.16. Conclusion

The material theory of induction requires that the inductive logic applicable in some domain be dictated by the facts that prevail in that domain. In many domains, facts do warrant a probabilistic inductive logic. The prevalence of such domains has helped foster the mistaken impression that a probabilistic logic is the universal logic of induction.

The burden of this chapter has been to illustrate how a formally rich, alternative inductive logic can be warranted. The domain is that of entangled quantum mechanical particles. The inductive logic appropriate to them employs density operators where a probabilistic inductive logic employs probability measures. This new logic looks very different, initially, from a probabilistic logic. There is no single real-valued measure of support that tells us which state is more or less well supported. Differences of support are expressed by density operators. In the most definite case of narrowest support, the density operator is a projection operator. It identifies a unique state as the true state. At the opposite extreme of the most distributed support, the maximally mixed density operator accords equal support to all states, just as does a uniform probability measure in the probabilistic case. The intermediate cases are captured by density operators between these extremes. For the case of a single electron, the range of cases and their properties are represented in readily interpretable form by the spheres of Section 16.12.

There are further structural analogies between the quantum inductive logic and a probabilistic logic, as described in Leifer and Spekkens (2013). This assures us that we do have a logic of comparable richness. Eventually, the analogies break down, for the two logics are different.

References

Leifer, Matthew S. and Robert W. Spekkens. 2013. “Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference.” Physical Review A 88, http://dx.doi.org/10.1103/PhysRevA.88.052130

Nielsen, Michael A. and Isaac L. Chuang. 2010. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press.

Penrose, Roger. 2004. The Road to Reality. London: Jonathan Cape.