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Limited Holism and Real-Vector-Space Quantum Theory

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Abstract

Quantum theory has the property of “local tomography”: the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by “bilocal tomography”: the state of any composite system is determined by the statistics of measurements on pairs of components. Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that real-vector-space quantum theory, while not locally tomographic, is bilocally tomographic.

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Notes

  1. Here we are making a further assumption: that measurements on system A which are operationally equivalent when A is taken alone remain operationally equivalent if A is taken in conjunction with another system, B. In other words, we are assuming that K A is a non-contextual property of system A. In the foliable joint probabilities framework [16] such an assumption is unnecessary because system B can be taken as a preparation for A and so any effect of a measurement of B on the value of K A would already be counted.

  2. Note that the inequality (11) does not by itself imply that all the K A K B local contributions are independent. If local tomography is not valid, then K AB could be large because of global degrees of freedom, even if many of the K A K B locally accessible parameters are redundant. But (11) is implied by local independence.

  3. See footnote 2. Equation (28) does not by itself imply that the above contributions are independent, but the independence of these contributions does imply (28).

  4. Already we know that L A L B must always be an integer, but this fact does not rule out, for example, the possibility that every L is an integer multiple of \(\sqrt{2}\).

  5. This equation follows from an inductive argument. Let \({\mathcal{K}}(N_{1}\ldots N_{m})\) be the number of real operators \(\widehat {Q}_{k_{1} \ldots k_{m}}\) for the system A 1A m , and let \({\mathcal{L}}(N_{1}\ldots N_{m})= (N_{1}\ldots N_{m})^{2} -{\mathcal{K}}(N_{1}\ldots N_{m})\) be the number of imaginary operators. If we assume that \({\mathcal{K}}(N)=N(N+1)/2\) for some particular value of m, it follows that this equation must also be true for m+1, since the functions K(N)=N(N+1)/2 and L(N)=N(N−1)/2 satisfy (31). The equation \({\mathcal{K}}(N) = N(N+1)/2\) certainly applies to the single system A 1. So by induction, the equation holds for any number of components.

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Acknowledgements

Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.

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Correspondence to William K. Wootters.

Appendices

Appendix 1: The function L(N)

Our aim here is to prove that we can associate with each system a number L, which is a function of N satisfying K AB =K A K B +L A L B and L AB =K A L B +L A K B . We assume (26), (27), and (29). That is, we assume that K A =K(N A ), that K AB K A K B , and that

$$K_{ABC} = K_AK_{BC} + K_BK_{AC} + K_CK_{AB} - 2K_AK_BK_C. $$
(51)

To begin, consider a collection of four systems A, B, C, and D for which the numbers of perfectly distinguishable states are N A , N B , N C , and N D . The whole collection must satisfy (51) when any two of the subsystems are considered as a single entity. In particular, for the grouping {AD,B,C}, we have

$$K_{ADBC} = K_{AD}K_{BC} + K_BK_{ADC} + K_CK_{ADB} - 2K_{AD}K_BK_C.$$

And for the grouping {A,BD,C}, we have

$$K_{ABDC} = K_AK_{BDC} + K_{BD}K_{AC} + K_CK_{ABD} - 2K_AK_{BD}K_C.$$

The right-hand sides of these two equations must be equal, since the left-hand sides are; so we obtain

We now use (51) to replace the factors K ADC and K BDC with expressions in which each K factor involves only two of the subsystems A,B,C,D. Making these replacements and collecting like terms, we get

A little further manipulation gives us the following equation.

in which both sides factor, leading to the simple equation

$$ h(N_A, N_D)h(N_B,N_C) = h(N_B,N_D)h(N_A,N_C).$$
(52)

Here h is defined by h(N A ,N B )=K AB K A K B . That is, h(N A ,N B ) counts the number of parameters accessible by a joint measurement on AB beyond what one can obtain by separate measurements on A and B.

What consequence does (52) have for the form of K(N)? To answer this question, we consider two cases: either (i) there exists an integer x for which h(x,x)>0, or (ii) h(x,x)=0 for all integers x. (Our assumption of local independence guarantees that h(x,x) cannot be negative.) Consider case (i). In that case, in (52) we set both N B and N C equal to x—here x is a specific integer for which h(x,x)>0 (say the smallest such x)—and write

$$h(N_A,N_D) = \left( \frac{h(N_D,x)}{\sqrt{h(x,x)}} \right) \left( \frac {h(N_A,x)}{\sqrt{h(x,x)}} \right).$$

We can therefore define L A to be \(h(N_{A},x)/\sqrt{h(x,x)}\) and conclude that K AD has the form in (31):

$$K_{AD} = K_AK_D + L_AL_D.$$
(53)

That is, the number of parameters accessible only by a joint measurement on AD is a product, L A L D , of factors characteristic of the individual systems A and D.

Now consider case (ii): h(x,x)=0 for every integer x. Then (52), with N A =N C and N B =N D , gives us h(N A ,N D )2=0. It follows that h(N A ,N D )=0 for all values of N A and N D . This result is still consistent with (31): we simply set L A =0 for all N A . That is, there are no hidden parameters. (This case includes ordinary quantum mechanics.)

Note that according to the above definitions L is a function of N (i.e. L A =L(N A )). At this stage we cannot say that L is necessarily an integer. We know that \(L(N) = \sqrt {h(N,N)}\); so each L(N) must be the square root of an integer.

We now look for an equation analogous to (31) that determines the value of L for a joint system. That is, if we know the values of K and L for each of two systems A and B, what do we know about L AB ?

This question can be answered directly from (51). Let us rewrite that equation, substituting for every factor of the form K XY the expression given in (31). After a little simplification, the equation becomes

$$K_{ABC} = K_AK_BK_C + K_AL_BL_C + K_BL_AL_C + K_CL_AL_B.$$

On the other hand, we can also group the system into AB and C and write

$$K_{ABC} = K_{AB}K_C + L_{AB}L_C=[K_AK_B + L_AL_B]K_C + L_{AB}L_C.$$

Comparing the two equations, we see that

$$L_C[L_{AB}- K_AL_B - L_AK_B] = 0.$$

Thus either L C is zero for every value of N C (as in quantum mechanics), or

$$ L_{AB} = K_AL_B + L_AK_B.$$
(54)

In fact (54) is true even if L is always zero.

Appendix 2: Functional Forms

We first want to establish the form of K(N) for the case of local tomography. We assume that K(N A N B )=K(N A )K(N B ), which follows from (12) and (15), and that K is a monotonically strictly increasing function of N, which follows from (16).

The first assumption, K(N A N B )=K(N A )K(N B ), leaves completely free the value of K for each prime value of N. For each prime p, let k p =K(p). It then follows from the multiplicative assumption that if we write a general value of N in terms of its prime factors—that is, \(N = 2^{n_{2}}3^{n_{3}}\ldots m_{N}^{n_{m_{N}}}\), where m N is the largest prime factor of N—the form of K must be

$$K(N) = k_2^{n_2}k_3^{n_3}\ldots k_{m_N}^{n_{m_N}}. $$
(55)

We now use the monotonicity assumption to show that each k p must have the form k p =p r, for a fixed value of r independent of p. We will do the proof by contradiction.

Suppose that for two primes p and q, we have \(k_{p} = p^{r_{p}}\) and \(k_{q} = q^{r_{q}}\), with r p r q . Let N A =p a and N B =q b. From (55) we have that

$$\frac{\ln K_B}{\ln K_A} = \frac{b r_q \ln q}{a r_p \ln p},$$
(56)

while the corresponding ratio for the N’s is

$$\frac{\ln N_B}{\ln N_A} = \frac{b \ln q}{a \ln p}.$$
(57)

Let us now choose the integers a and b so that the ratio a/b lies strictly between the two real numbers lnq/lnp and (r q /r p )(lnq/lnp), which are distinct by assumption. Then the ratios lnK B /lnK A and lnN B /lnN A will lie on opposite sides of the number 1. That is, a larger value of N will correspond to a smaller value of K, contradicting our monotonicity assumption. It follows that there is a single value of r such that k p =p r for every prime p. Equation (55) then tells us that

$$K(N) = N^r.$$
(58)

The number r must be a non-negative integer to avoid fractional values of K. But strict monotonicity rules out r=0; so r must be a positive integer.

We now turn to the case of bilocal tomography. Our starting point is very similar to what we started with in the case of local tomography: multiplicativity and monotonicity. But now these assumptions apply separately to the two quantities K(N)+L(N) and K(N)−L(N). (See (35), (36), (38), and (39).) In the case of the difference, before we can use the above argument, we need to show that K(N)−L(N)≥0. To see this, note first that by applying (39) to the case N A =N B =1, we conclude that K(1)−L(1) is either 0 or 1. That K(N)−L(N) is never negative then follows from the monotonicity equation, (36).

We can now apply the above argument, which shows us that

$$K(N) + L(N) = N^r \quad \hbox{and}\quad K(N) - L(N) = N^s,$$
(59)

from which, as we have seen, it follows that

$$ K= \frac{1}{2} (N^r +N ^s) \quad \hbox{and} \quad L= \frac {1}{2}(N^r-N^s).$$
(60)

In order that K always be an integer, we must take r and s to be nonnegative integers, and as before, strict monotonicity requires that each be positive. Moreover we must have rs since we have defined L to be nonnegative. (In principle, we could have defined L(N) to be always negative or zero. That is, in Appendix 1 we could have written \(L(N)=-h(N,x)/\sqrt{h(x,x)}\) instead of \(L(N)=h(N,x)/\sqrt{h(x,x)}\). Then in (60) we would have sr. Equations (31) and (32) make clear that this change would have no effect on K(N). But from an interpretational point of view this would have been an odd choice.)

Appendix 3: An Approach to n-Local Independence

Do the local and bilocal “independence” equations, (12) and (29), generalize in a natural way to n-local tomography? Here we present one approach to this question, which will lead us to a specific equation that one might take as a candidate for the expression of 3-local independence in a 3-locally tomographic theory.

Let us use the expression “locally ideal” to describe any theory that satisfies (12). (A locally tomographic theory satisfying the local independence condition is thus locally ideal—the number of conceivably independent parameters accessible by local measurements is exactly equal to the number of parameters needed to specify the global state.) Similarly, let us use “bilocally ideal” for any theory satisfying (29). We now ask whether one can define a reasonable notion of “n-local ideality.” Presumably this condition should be expressed by an equation of the form

$$ K_{A_1A_2\ldots A_{n+1}} = \sum_P \alpha_P \hbox{ (product of the $K$'sof the subsystems given by $P$)},$$
(61)

where the sum is over all partitions P of the system into subsystems, and α P is a real number associated with the partition P. In our earlier examples, each partition corresponded to a particular set of measurements performed concurrently on the subsystems specified by the partition. But here we focus more on the mathematical form of the condition rather than on its interpretation in terms of measurement.

We begin with an alternative set of assumptions leading to (12) and (29).

Permutation invariance: We assume that the right-hand side of (61) is symmetric under all permutations of the basic components.

Triviality: We assume that there exists a “trivial system,” that is, a system such that (i) K=1 and (ii) when the system is included as part of a larger system, it does not affect the value of K.

Novelty: We assume that the condition for n-local ideality, together with the triviality assumption, does not imply the condition for m-local ideality for any m<n.

Let us show how these assumptions give rise to the condition of local ideality, that is, (12). For n=1, (61) has the form

$$K_{AB} = \alpha K_AK_B. $$
(62)

We now let component B be the trivial system, so that K AB =K A and K B =1. Then (62) becomes

$$K_A = \alpha K_A,$$
(63)

so that α must be unity and we recover (12). (Note that for this simple case we did not need to assume either permutation invariance or novelty.)

To derive the condition of bilocal ideality (see (29)), we start with

$$K_{ABC} = \alpha(K_{AB}K_C + K_{AC}K_B + K_{BC}K_A) +\beta K_AK_BK_C, $$
(64)

which is the most general form allowed by permutation invariance. We now let C be the trivial system. Then (64) becomes

$$(1-\alpha)K_{AB} = (2\alpha+ \beta)K_AK_B. $$
(65)

There are now two possibilities: either α=1, or the equation reduces to the form K AB =cK A K B for some constant c. In the latter case c must be 1 (since B could be the trivial system), and our 2-local ideality condition would reduce to the 1-local ideality condition, contradicting the novelty assumption. We conclude that α=1, from which it follows from (65) that β=−2, so that we indeed recover (29).

Extending this approach to the case n=3, we start with the equation

(66)

where the ellipses indicate similar but distinct terms with the components permuted. (For example, β multiplies three terms, each being the product of a pair of two-component K values.) Setting D equal to the trivial system, we get

$$ (1-\alpha)K_{ABC} = (\alpha+\beta+\gamma)(K_{AB}K_C + \cdots) +(3\gamma+ \delta)K_AK_BK_C.$$
(67)

And setting both C and D equal to the trivial system, we get

$$ [(1-\alpha) - (\alpha+ \beta+ \gamma)]K_{AB}=[2(\alpha+ \beta+\gamma) + (3\gamma+ \delta)]K_AK_B.$$
(68)

Now, in order that (68) not reduce to the condition for 1-local ideality, we must have (1−α)=(α+β+γ) and (3γ+δ)=−2(α+β+γ). But then (67) reduces to the 2-local ideality condition unless α=1. So by the novelty assumption, we must have

(69)

These equations do not uniquely determine the values of the coefficients. Rather, we are left with the equation

(70)

which has a single undetermined parameter.

To try to pin down the remaining parameter, we tentatively consider an additional assumption, which is not obviously consistent with the assumptions we have already made.

Inclusion: Any theory that is n-locally ideal is also (n+1)-locally ideal.

The two assumptions “novelty” and “inclusion” can be naturally merged into a single assumption that could be called “strict inclusion”: n-local ideality implies (n+1)-local ideality, but the implication does not go in the other direction for any value of n.

Note that the inclusion assumption is true for n=1: any theory satisfying (12) automatically satisfies (29). (So standard quantum theory, which is 1-locally ideal, is also 2-locally ideal.) We now insist that the assumption also be true for n=2. Our hope is that this requirement will lead us to a unique value of γ in (70).

Consider, then, any 2-locally ideal theory, that is, any theory satisfying

$$K_{ABC}= K_{AB}K_C + K_{AC}K_B + K_{BC}K_A - 2K_AK_BK_C. $$
(71)

The equation must still be satisfied if we replace C with a pair CD:

$$K_{ABCD} = K_{AB}K_{CD} + K_{ACD}K_B + K_{BCD}K_A - 2K_AK_BK_{CD}.$$
(72)

We now symmetrize this equation over all permutations (all of the permuted versions of the equation must also be true), arriving at

(73)

In the first of the three terms on the right-hand side, we are free to use (71) again. Let us make the replacement

$$K_{ABC} \rightarrow\epsilon K_{ABC} + (1-\epsilon)[(K_{AB}K_C + \cdots ) - 2 K_AK_BK_C]$$
(74)

(and similarly for each other triple of components), where ϵ is a parameter we can freely choose. We then arrive at the equation

(75)

So the theory in question will also be 3-locally ideal if our condition for 3-local ideality is of the form of (75) for some value of ϵ. That is, the inclusion assumption will be true for n=2 in that case.

It seems reasonable, then, to assume the form (75) as long as it does not conflict with our earlier assumptions. One can see that (75) is at least consistent with (70) and that there is exactly one solution: ϵ must have the value 1/2, and γ must have the value −4/3.

Adopting these values, we arrive at the following candidate equation that might express the notion of 3-local ideality:

(76)

That is, in a 3-local tomographic theory, (76) would presumably express the condition of 3-local independence.

But we do not yet know enough about 3-local tomographic theories to have a great deal of confidence in this equation. In particular, it is conceivable that when some of our other standard assumptions are in place (e.g., that K is N-specified and that N is multiplicative), (76) will turn out to imply the condition for 2-local ideality. In that case, it may be that there is no natural class of theories that could be called n-locally ideal for n>2, beyond those classes that are already covered by n=1 and n=2. So the notion of n-local ideality remains a matter for future investigation.

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Hardy, L., Wootters, W.K. Limited Holism and Real-Vector-Space Quantum Theory. Found Phys 42, 454–473 (2012). https://doi.org/10.1007/s10701-011-9616-6

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