We now have sufficient structure in our hand to determine the functional relation between αi and yi. Given that the generalized Bloch vector
2~yO→S−~1transforms nicely under time evolution (53), it is useful to parametrize αi by 2yi−1, i.e. αi=α(2yi−1). Rule3 entails thatO’s total in- formation about (an otherwise non-interacting)S
is a ‘conserved charge’ of time evolution
IO→S(~yO→S(∆t)) =IO→S(~yO→S(0)) which translates into the condition
IO→S T(∆t)2~yO→S(0)−~1 = DN X i=1 α DN X j=1 Tij(∆t) (2yj(0)−1) = DN X i=1 α(2yi(0)−1) =IO→S(2~yO→S(0)−~1). (59)
If T(∆t) was a permutation matrix, (59) would hold for any function α(2yi −1). For exam-
ple, N classical bits are governed by the evolu-
form a discrete group, while in our present case {T(∆t),∆t ∈ R} constitutes a continuous one- parameter group. This is wherecontinuityof time
evolution, as asserted by rule4, becomes crucial. Under a reasonable assumption on the informa- tion measure, we shall now show that continuity of time evolution, together with conditions (i)–
(vi) of subsection 6.2, enforces the quadratic re- lationαi = (2yi−1)2. To this end, we once more invoke the coin flip scenario.45
Given our parametrization in terms of the Bloch vector 2~yO→S−~1, O’s information about the outcomes of his questions in the coin flip sce- nario can be written as follows
IO→S12 2 (λ ~yO→S1+ (1−λ)~yO→S2)−~1 =IO→S12 λ(2~yO→S1 −~1) + (1−λ)(2~yO→S2−~1) = DN X i=1 αλ(2yi1−1) + (1−λ)(2y2i −1).
It is instructive to consider the case in which O
is entirely oblivious about S2 such that the lat-
ter is in the state of no information~yO→S2 =
1 2~1
relative to him, but that O has some informa-
tion about S1. In this case, 2~yO→S12 −~1 = λ(2~yO→S1−~1)and (assuming the outcome of the
coin flip is not certain) strict convexity of IO→S (see subsection 6.2) implies
IO→S12(λ(2~yO→S1−~1))< λ IO→S1(2~yO→S1−~1)
or, equivalently,
IO→S12(λ(2~yO→S1−~1)) (60)
=f · IO→S1(2~yO→S1 −~1),
where f < 1 is a factor parametrizing O’s in-
formation loss relative to the case in which he does not toss a coin and, instead, directly asks S1.46 The reason O experiences such a relative
information loss about the outcome of his inter- rogation is, of course, entirely due to the random- ness of the coin flip. But the coin flip is indepen- dent of the systemsS1,2 and, in particular, of the
states in which these are relative to O; the fac-
tor λ by which the probabilities ~yO→S1 become
45We suspect that this result may be derivable from
purely group theoretic argumentswithout an operational setup by employing the mathematical fact that to every continuous matrix group acting linearly on some space there corresponds a conserved inner product which is quadratic in the components of the vectors.
46In fact, one can equivalently interpret the situation as
follows: Oonly considers a systemS1 which would be in the state~yO→S1if it was present with certainty. However,
λ in the state λ(2~yO→S1−~1) represents the probability
that S1 ‘is there’ at all. Indeed, in this case, (4) can be written asλ(~yO→S1+~nO→S1) =λ·~1 such thatλ(~yO→S1−
~nO→S1) =λ(2~yO→S1−~1).
rescaled is state independent. For that reason, the relative information loss should likewise de- pend only on the coin flip, quantified by λ, and not on the state ~yO→S1. For instance, if we also
considered the case that the coin flip was certain, i.e. λ= 0,1, then clearly for λ= 1 we must have
f = 1 ∀~yO→S1 ∈ΣN and for λ= 0 it must hold f = 0 ∀~yO→S1 ∈ ΣN. We shall make this into
a requirement on the information measure for all values ofλ:
Requirement 1. The relative information loss factor f in (61) is a state independent (contin- uous) function of the coin flip probability λ with f(λ)<1 for λ∈(0,1).
The binary Shannon entropy H(y) = −ylogy−(1−y) log(1−y)in the role of αwould
fail this requirement. Namely, the measureαthus
factorizes,α(λ(2yi−1)) =f(λ)α(2yi−1), while H(y) would not. Setting λ=λ1·λ2 yields47
f(λ1·λ2)α(2yi−1) =α(λ1·λ2(2yi−1))
=f(λ1)α(λ2(2yi−1))
=f(λ1)f(λ2)α(2yi−1) and therefore f(λ1)·f(λ2) = f(λ1 ·λ2), which
implies f(λ) = λp, for some power p ∈ R. But then,αmust be a homogeneous functionα(2yi− 47Such a factorization of coin flip probabilities could be
achieved, e.g., ifOdecided to use one coin, with ‘heads’ probabilityλ1, to firstly decide which of two possible con- vex mixtures to prepare where both possible mixtures are generated with a second coin with ‘heads’ probabilityλ2. If three of the four states within the two mixtures are chosen as the state of no information, one would obtain precisely such an equation.
1) = k(2yi−1)p with some constant k∈ R. In consequence, the informationIO→S(2~yO→S−~1)is (up tok) thep-norm of the Bloch vector2~yO→S− ~1.
We can rule out that p ∈ (−∞,0] because in this case, as one can easily check, it is impossi- ble to satisfy all the consistency conditions (i)– (vi) of subsection 6.2. Hence, p > 0. At this stage we can make use of (59) and a result by Aaronson [90] which implies that the only vector
p-norm withp >0which is preserved by acontin- uousmatrix group is the2-norm. Since any given time evolution of the Bloch vector 2~yO→S−~1 is governed by a continuous, one-parameter matrix group, we conclude thatα(2yi−1) =k(2yi−1)2. Imposing condition (iv) yields k = 1 and there- fore ultimately IO→S(~yO→S) = DN X i=1 (2yi−1)2. (61)
It is straightforward to convince oneself that all of (i)–(vi) of subsection 6.2 are satisfied by this quadratic information measure. O’s total amount
of information aboutS is thus the squared length
of the generalized Bloch vector, thereby assuming a geometric flavour. It is important to emphasize that, had we not imposed continuity of time evo- lution in rule 4, we would not have been able to arrive at (61); if time evolution was not continu-
ous, many solutions to αin terms of yi would be possible.
The quadratic information measure (61) has been proposed earlier by Brukner and Zeilinger in [32,34,91,92] from a different perspective, em- phasizing that this is the most natural measure taking into account an observer’s uncertainty – due to statistical fluctuations – about the out- come of the next trial of measurements on a system in a multiple shot experiment. Further- more, taking the formalism of quantum theory as given, Brukner and Zeilinger [73] later singled out the quadratic measure from the set of Tsal- lis entropies by imposing an ‘information invari- ance principle’, according to which a continuous transformation among any two complete sets of mutually complementary measurements in quan- tum theory should leave an observer’s informa- tion about the system invariant. While [73] is certainly compatible with the present framework, here we come from farther away to the same re- sult: we do not pre-suppose quantum theory and
derive the quadratic measure more generally by starting from the landscape of information infer- ence theories and imposing rule 3 of information preservation and rule4of maximality of time evo- lution thereon. In this regard, the present deriva- tion may similarly be taken as a strong justifica- tion for the original Brukner-Zeilinger proposal.
6.9 The set of all time evolutions is a subgroup