The explicit quantification of O’s information
now permits us to render the distinction between three informational classes of S’s states – which
we already loosely referred to as ‘states of max- imal knowledge’ or ‘states of non-maximal infor- mation’ in previous sections – precise.
Firstly, we determine the maximally attain- able (independent and dependent) information content within a state ~yO→S of a system of N gbits. This can be easily counted: once O knows
the answers toN mutually independent questions
(these do not need to be individuals), he will also know the answers to all their bipartite, tripar- tite,... and N-partite correlation questions – all
5.27and 5.31. But these are then N 1 ! + N 2 ! +· · · N N ! = N X i=1 N i ! = 2N−1
answered questions fromQMN, while all remain-
ing questions inQMN will be maximally comple-
mentary to at least one of the known ones. O’s to-
tal information, as quantified by IO→S (47), thus contains plenty of dependentbitsof information
– a result of the fact that the questions in QMN
are pairwise but not necessarily mutually inde- pendent.
Using this observation, we shall characterize
S’s states according to their information content,
i.e. squared length of the Bloch vector. By rules3
and 4, this distinction applies to all states which are connected via some time evolution to the states above, including those for which the infor- mation may be distributed partially over many elements of a fixed QMN. Specifically, we shall
refer to a state~yO→S as a pure state: if it is a state of maximal information content, i.e. maximal length
IO→S(~yO→S) = DN
X
i=1
(2yi−1)2 = (2N−1)bits,
mixed state: if it is a state of non-extremal information content, i.e. non-extremal length 0bit< IO→S(~yO→S) =
DN
X
i=1
(2yi−1)2 <(2N −1)bits,
totally mixed state: if it is the state of no information~yO→S = 12~1with zero length IO→S ~ yO→S = 1 2~1 = 0bit.
We note that these characterizations of states in terms of their length are indeed true in quantum theory. In particular, N qubit pure states actu-
ally have a Bloch vector squared length equal to 2N −1. Our reconstruction gives this peculiar fact a clear informational interpretation.
One may wonder whether the above definition of a pure state is directly equivalent to being ex- tremal in ΣN and thereby to the usual defini- tion of pure states in GPTs or quantum theory. This is not obvious from what we have estab- lished thus far. However, since clearly we as- sume IO→S = (2N −1) bits to be the maxi- mally attainable information, we can already con- clude that pure states so defined must lie on the boundary of ΣN because a non-constant convex function cannot have its maximum in the inte- rior of its convex domain [89]. Showing that the pure states, as defined above, of the theory sur- viving the imposition of all rules – quantum the-
ory – are, indeed, the extremal states, requires more work. For N = 1 we shall demonstrate this shortly, while the discussion forN >1is deferred to [1].
7 The
N
= 1
case and the Bloch ball
Before closing this toolkit for now, we quickly give a flavor of the capabilities of the newly developed concepts and tools by applying them to show that in the simplest case of a single gbit (N = 1) rules
1–4 indeed only have two solutions within Lgbit,
namely the qubit and the rebit state space includ- ing their respective time evolution groups. This proves the claim of section 4.1for N = 1.
To this end, recall theorem 5.6 which asserts that the dimension of the N = 1state space Σ1
is either D1 = 2 or D1 = 3 which thus far we
suggestively referred to as the ‘rebit’ and ‘qubit case’, respectively.
7.1 A single qubit and the Bloch ball
We begin with theD1 = 3 case. Σ1 will be parametrized by a three-dimensional vector
~yO→S= y1 y2 y2 ,
where y1, y2, y3 are the ‘yes’-probabilities of a mutually maximally complementary question set
Q1, Q2, Q3 constituting an informationally complete Q1 (11).
The informational distinction of states introduced in section 6.10reads in this case pure states: ~yO→S such that
IO→S= (2y1−1)2+ (2y2−1)2+ (2y3−1)2= 1bit,
mixed states: ~yO→S such that
0bit< IO→S = (2y1−1)2+ (2y2−1)2+ (2y3−1)2 <1bit,
totally mixed state: ~yO→S = 12~1 such that
IO→S= (2y1−1)2+ (2y2−1)2+ (2y3−1)2= 0bit.
Recall from section 6.9that the set of possible time evolutions T1 must be contained in SO(3)
and that the time evolution rule 4 requires the set of states into whichanystate~yO→Scan evolve to be maximal, while being compatible with the rules. Thus, in particular, the image of the cer- tainly legal pure state (1,0,0) (rules1 and2 im- ply its existence in Σ1) under T1 must be maxi-
mal. Applying all ofSO(3)to this state generates
all states with |2~y −~1|2 = 1 bit and these cer-
tainly abide by the rules. Consequently, the set of all possible time evolutions, compatible with rules 1–4, is the rotation group
T1 ≃SO(3)≃PSU(2).
This is the component of the isometry group of the Bloch ball which is connected to the identity and it is required in full in order to maximize the number of states into which (1,0,0) can evolve. However, since time evolution is state indepen- dent (c.f. assumption 8), this is the time evolu- tion group for all states. PSU(2) is precisely the adjoint action of SU(2) on density matrices ρ2×2
over C2, ρ2×2 7→U ρ2×2U†,U ∈SU(2) and thus coincides with the set of all possible unitary time evolutions of a single qubit in standard quantum
theory.
The set of allowed states populates the entire unit ball in the three-dimensional Bloch ball, i.e. Σ1 ≃B3. This follows from the fact that apply-
ing the full group T1 = SO(3) to (1,0,0) gener-
ates the entire Bloch sphere as a closed set of ex- tremal states and the fact thatΣ1 must be closed
convex according to assumption 2. Hence, we recover the well-known three-dimensional Bloch ball state space of a single qubit of standard quan- tum theory with the set of all pure states defining the boundary sphere S2, the totally mixed state
(as the state of no information) constituting the center and the set of mixed states filling the inte- rior in between, as illustrated in figure 6a. This is precisely the geometry of the set of all normal- ized density matrices onC2. Notice that the pure state space S2 ≃ CP1 indeed coincides with the set of all unit vectors inC2 (modulo phase).
Given the complete symmetry of the Bloch ball as the state space forN = 1, there should not ex- ist a distinguished informationally complete ques- tion setQM1, corresponding to a distinguished or-
thonormal Bloch vector basis, by means of which
O can interrogate S. While it is an additional
PSfrag replacements
totally mixed state mixed states
pure states
~r
~r= 2~yO→S−~1
(a) 3D qubit Bloch ball
PSfrag replacements
totally mixed state mixed states
pure states
~r
~r= 2~yO→S−~1
(b) 2D rebit Bloch disc
Figure 6: The three-dimensional Bloch ball (a) and the two-dimensional Bloch disc (b) are the correct state spacesΣ1 of a single qubit in standard quantum theory and a single rebit in real quantum theory, respectively. The vector~rparametrizing the states is the Bloch vector
2~yO→S−~1.
to every Q∈ Q1 there exists a unique pure state
in Σ1 which represents the truth value Q =‘yes’
and, conversely, that every pure state of this sys- tem corresponds to the definite answer to one question in Q1. But then Q1 ≃ S2 which also
coincides with the set of all possible projective measurements onto the +1(or, equivalently, the −1) eigenspaces of the Pauli operators ~n·~σ over
C2 which is parametrized by ~n ∈ R3, |~n|2 = 1, and where ~σ = (σx, σy, σz) are the usual Pauli matrices. This set of permissible questions QN for allN will be discussed more thoroughly in [1]
together with a derivation of the Born rule for projective measurements.
The ‘ballness’ and three-dimensionality of the
state space of a single qubit can also be derived from various operational axioms within GPTs [15,16,18,21,23,59,93] and constitutes a cru- cial step in most GPT based reconstructions of quantum theory [14–16,20,21]. The principle of
continuous reversibility, according to which every
pure state of the convex set can be mapped into any other by means of a continuous and reversible transformation, usually assumes a crucial role in such derivations. Here we offer a novel perspec- tive on the origin of the Bloch ball by deriving it from elementary rules for the informational rela- tion between O and S; in particular, we recover
continuous reversibility as a by-product.