Bit Symmetry Entails the Symmetry of the Quantum Transition Probability

Gerd Niestegge

doi:10.2478/qic-2025-0034

Full Article

1.

Introduction

Generalized probabilistic theories (GPTs [1–5]) are commonly used as generic models to reconstruct quantum theory from a few basic principles and to gain a better understanding of the probabilistic and information theoretic foundations of quantum physics and quantum computing. A more specific model is the author’s transition probability framework [6–9]. A distinguishing feature is the direct or indirect postulate that the transition probability must exist for the quantum logical atoms or, equivalently, that there is one unique state for each atom in which the atom carries the probability 1. This equivalent postulate is sometimes called sharpness [5].

A single geometric property of a compact convex set that gives rise to such a model was discovered in Ref. [9]. It is presupposed here as a given fact and we refer to Ref. [9] for background information and motivating considerations.

We use this model and the geometric property to study several well-known symmetry postulates of the GPTs and particularly Müller and Ududec’s bit symmetry, which they motivate by quantum computational needs [4]. We show that bit symmetry implies the symmetry of the transitions probabilities between the atoms. Using a result by Barnum and Hilgert [10], we can then conclude that a stronger form of symmetry rules out all models but the classical cases and the simple Euclidean Jordan algebras.

The transition probability framework and some results from previous papers (particularly [9]) that will be needed here are briefly recapitulated in the next section. The symmetry postulates under consideration are defined and discussed in Section 3. Section 4 is dedicated to the first and most weak one. In Section 5 bit symmetry is studied and our main result (Theorem 1) is presented. The strong form of symmetry is considered in Section 6.

2.

A Brief Synopsis of the Transition Probability Framework

Let Ω be any compact convex set in some finite-dimensional real vector space and let A_Ω denote the order unit space that consists of the affine real-valued functions on Ω; its order unit is the constant function 𝕀 ≡ 1. The state space of A_Ω consists of the positive linear functionals μ on A_Ω with μ(𝕀) = 1 and is isomorphic to the convex set Ω with the mapping ω → δ_ω, ω ∈ Ω, where δ_ω(a) := a(ω) for a ∈ A_Ω. We consider the following compact convex set in A_Ω $[0, I] : = {a \in A_{Ω} ∣ 0 \leq a \leq I} = {a \in A_{Ω} ∣ 0 \leq a, ‖ a ‖ \leq 1}$ [0,]: = \{ a \in {A_\Omega }\mid 0 \le a \le \} = \{ a \in {A_\Omega }\mid 0 \le a,\left\| a \right\| \le 1\}

and the set of its extreme points ext([0, 𝕀]). For each ω ∈ Ω we define the following function e_ω on Ω: $e_{ω} (ζ) : = i n f {a (ζ) : a \in A_{Ω}, 0 \leq a and a (ω) = 1}$ {e_\omega }(\zeta ): = inf\{ a(\zeta ):a \in {A_\Omega },0 \le aanda(\omega ) = 1\}

for ζ ∈ Ω. Since 𝕀(ω) = 1, we have e_ω(ζ) ≤ 1 for all ζ ∈ Ω. Generally, e_ω is not affine and does not belong to A_Ω. The following novel property of a compact convex set Ω was introduced in Ref. [9]:

(∗∗) For each extreme point ω ∈ Ω, the function e_ω is affine (this means e_ω ∈ A_Ω) and $e_{ω} (ζ) \neq 1$ {e_\omega }(\zeta ) \ne 1 for all ζ ∈ Ω with ζ ≠ ω.

This means that there is a smallest non-negative affine function with the value 1 at the extreme point and at no other point. If the condition (∗∗) holds, then A_Ω has the following properties [9]:

(a)
The set of extreme points ext([0, 𝕀]) is an atomic orthomodular lattice with the orthocomplementation $p \to p^{'} : = I - p$ p \to {p^\prime }: = - p for p ∈ ext([0, 𝕀]). This set thus becomes a quantum logic; its atoms are the minimal non-zero elements.
(b)
For each atom e in ext([0, 𝕀]) there is one unique state ℙ_e with ℙ_e(e) = 1. This state is pure (an extreme point of the state space).
(c)
For each pure state μ there is an atom e with μ = ℙ_e.
(d)
Two atoms e₁ and e₂ are orthogonal if one of the following three equivalent conditions holds: (1) e₁ + e₂ ≤ 𝕀, (2) ℙ_e₁(e₂) = 0, (3) ℙ_e₂(e₁) = 0. Note that the orthogonality of atoms implies their linear independence.
(e)
A_Ω is spectral; this means that each a ∈ A_Ω can be represented as $a = \sum_{k = 1}^{n} s_{k} e_{k}$ {e_1} + {e_2} \le with s_k ∈ ℝ and pairwise orthogonal atoms $e_{k} \in e x t ([0, I]), k = 1, 2, \dots, n, n \in ℕ$ a = \sum\nolimits_{k = 1}^n {{s_k}{e_k}} . Here we have 0 ≤ a iff 0 ≤ s_k for each k = 1, …, n.

Property (b), which is sometimes called sharpness [5], means that the transition probability [7] exists for each atom. It is invariant under the automorphisms U of the order-unit space $A_{Ω} : ℙ_{e} (a) = ℙ_{U e} (U a)$ {e_k} \in ext([0,]),k = 1,2, \ldots ,n,n \in for any atom e and any element a in A_Ω [7]. The transition probability is called symmetric if $ℙ_{e_{2}} (e_{1}) = ℙ_{e_{1}} (e_{2})$ {A_\Omega }:{_e}(a) = {_{Ue}}(Ua) holds for each pair of atoms e₁ and e₂.

Following the notation of Ref. [1], the maximum number m of pairwise orthogonal atoms is called the information capacity; m is finite, since A_Ω has a finite dimension. Then 𝕀 = e₁ + … + e_m with some pairwise orthogonal atoms e₁,…,e_m and the sum of any m pairwise orthogonal atoms equals 𝕀. Furthermore we have n ≤ m for the number n in the spectral decomposition (e).

First examples of compact convex sets with (∗∗) are the strictly convex and smooth compact convex sets [8]; they all have information capacity m = 2. Here the transition probability is symmetric only for the Euclidean unit balls.

Further examples are the state spaces of the finite-dimensional Euclidean (formally real) Jordan algebras [9,11,12]. Their transition probabilities are always symmetric and they include the finite-dimensional classical state spaces (simplexes) as well as the state spaces of finite-dimensional quantum theory.

A more exotic example is the triangular pillow [11]. Its information capacity is m = 3, its transition probabilities are not symmetric [8] and it satisfies (∗∗) [9].

The mathematical property (∗∗) appears to emerge without any motivation here. However, it does follow from four physically more plausible conditions [9]; these are the above conditions (b), (c) and (e) together with a further condition that is not discussed here and that more closely ties the order relation to the state space [9].

3.

The Symmetry Postulates

We consider the automorphism groups Aut(Ω) and Aut(A_Ω) of Ω and A_Ω, repectively; Aut(Ω) consists of the bijective affine transformations of Ω and Aut(A_Ω) consists of the bijective positive linear transformations U of A_Ω with a positive inverse and U(𝕀) = 𝕀. Each one is a compact group [13,14].

For each T ∈ Aut(Ω) we get $T^{*} \in A u t (A_{Ω})$ {_{{e_2}}}({e_1}) = {_{{e_1}}}({e_2}) by defining $(T^{*} a) (ω) : = a (T ω)$ {T^*} \in Aut({A_\Omega }) for a ∈ A_Ω and ω ∈ Ω. Vice versa, for each U ∈ Aut(A_Ω) we get $U^{*} \in A u t (Ω)$ ({T^*}a)(\omega ): = a(T\omega ) by defining U* ω := θ, where θ is the element in Ω with δ_θ = δ_ωU.

The first symmetry postulate becomes that Aut(Ω) acts transitively on the extreme points of Ω [14,15]: For any two extreme points ω₁ and ω₂ there shall be T ∈ Aut(Ω) with Tω₁ = Tω₂. Because of (b) and (c) in section 2 this is equivalent to the property that Aut(A_Ω) acts transitively on the atoms in ext([0, 𝕀]). This property is often called transitivity, but here it shall be named weak symmetry, which better matches our terminology of symmetry conditions.

We say that the exchange symmetry holds if there is T ∈ Aut(Ω) with Tω₁ = Tω₂ and Tω₂ = Tω₁ for any two extreme points ω₁ and ω₂ in Ω. This is equivalent to the property that there is U ∈ Aut(A_Ω) with Up = q and Uq = p for any two atoms p and q in ext([0, 𝕀]).

We call a pair of extreme points ω₁ and ω₂ in Ω orthogonal if the atoms belonging to the corresponding pure states are orthogonal. In the generalized probabilistic theories, the term ”perfectly distinguishable” [1,4,10] is often used in this case. The set Ω is called bit-symmetric [4] if there is a transformation T ∈ Aut(Ω) with Tω₁ = Tθ₁ and Tω₂ = Tθ₂, whenever ω₁ and ω₂ form any orthogonal pair of extreme points and θ₁ and θ₂ form any further orthogonal pair of extreme points in Ω. This becomes equivalent to the condition that any two orthogonal pairs of atoms in ext([0, 𝕀]) can be mapped to each other by a transformation in Aut(A_Ω). Bit symmetry is considered a quantum computational requirement, since it is thought that a quantum computer must be capable to reversibly transfer any logical bit to any other logical bit.

Strong symmetry [10] means that any family of pairwise orthogonal extreme points of Ω (equivalently, atoms in ext([0, 𝕀])) can be mapped to any other such family with the same number of elements by a transformation in Aut(Ω) (or Aut(A_Ω)). Such families are called frames in the GPTs [1,5,10].

Obviously, the strong symmetry implies the bit symmetry which again implies weak symmetry. Moreover, the exchange symmetry implies the weak symmetry, but its relation to the bit symmetry and the strong symmetry is not clear.

An immediate important consequence of the exchange symmetry is that the transition probability becomes symmetric. For any two atoms e₁ and e₂ we have an automorphism U with Ue₁ = e₂ and Ue₂ = e₁ and then $ℙ_{e_{1}} (e_{2}) = ℙ_{U e_{1}} (U e_{2}) = ℙ_{e_{2}} (e_{1})$ {U^*} \in Aut(\Omega ). As shown in Ref. [9] the symmetric transition probability then results in an inner product 〈 | 〉 on A_Ω with a self-dual cone and $ℙ_{e_{1}} (e_{2}) = 〈 e_{1} | e_{2} 〉$ {_{{e_1}}}({e_2}) = {_{U{e_1}}}(U{e_2}) = {_{{e_2}}}({e_1}) for any atoms e₁ and e₂. The main result of this paper will be that bit symmetry has the same consequences, but this is somewhat more difficult to show.

The finite-dimensional Euclidean Jordan algebras do not generally satisfy the above symmetry postulates. Bit symmetry and strong symmetry hold only in the simple (or non-decomposable or irreducible) algebras and in the abelian algebras. Weak symmetry and exchange symmetry hold also in the decomposable algebras, if they are direct sums of isomorphic factors, but do not hold in direct sums of non-isomorphic factors.

Note that, with information capacity m = 2, weak symmetry, bit symmetry and strong symmetry become immediately equivalent. If Up = Uq holds for two atoms p and q and U ∈ Aut(A_Ω), then U(𝕀 − p) = U(𝕀 − q), but any orthogonal pair of atoms consists of an atom e and the atom $e^{'} = I - e$ {_{{e_1}}}({e_2}) = {e_1}|{e_2} in this case. Moreover frames with more than two elements do not exist.

Several further symmetry postulates for the compact convex sets and the GPTs can be found in the mathematical and physical literature, but do not play any role here.

4.

Weak Symmetry

Lemma 1

Let Ω be any finite-dimensional weakly symmetric compact convex set with the property (∗∗) and information capacity m.

(i)
There exists an Aut(A_Ω)-invariant state μ_inv on A_Ω and μ_inv (e) = 1/m for every atom e in ext([0, 𝕀]).
(ii)
There exists an Aut(A_Ω)-invariant inner product 〈 | 〉_o on A_Ω with 〈e|e〉_o = 1 for every atom e.
(iii)
If $I = q_{1} + \dots + q_{n}$ {e^\prime } = - e holds with pairwise orthogonal atoms q₁,…,q_n, then n = m.

Proof

(i) and (iii): Using the normalized Haar measure on Aut(A_Ω) and arbitrarily selecting any state μ₀, we define $μ_{i n v} (a) : = \int_{U \in A u t (A_{Ω})} μ_{0} (U a) d U$ = {q_1} + \ldots + {q_n}

for a ∈ A_Ω. This becomes an Aut(A_Ω)-invariant state. Since Aut(A_Ω) acts transitively on the atoms, we get μ_inv (p) = μ_inv (q) for any atoms p and q.

There are m pairwise orthogonal atoms e₁,…,e_m with 𝕀 = e₁ +…+ e_m. Then $1 = μ_{i n v} (I) = μ_{i n v} (e_{1}) + \dots + μ_{i n v} (e_{m}) = m μ (p)$ {\mu _{inv}}(a): = \int_{U \in Aut\left( {{A_\Omega }} \right)} {{\mu _0}} (Ua)dU and thus μ(p) = 1/m for every atom p. So far we have (i). Now suppose 𝕀 = q₁ +…+ q_n with any further pairwise orthogonal atoms q₁,…,q_n. Then $1 = μ_{i n v} (I) = μ_{i n v} (q_{1}) + \dots + μ_{i n v} (q_{n}) = n / m$ 1 = {\mu _{inv}}() = {\mu _{inv}}({e_1}) + \ldots + {\mu _{inv}}({e_m}) = m\mu (p). Therefore n = m and we have (iii).

(ii): Let ( | ) be any inner product on A_Ω. Using again the normalized Haar measure on Aut(A_Ω), we construct a further Aut(A_Ω)-invariant inner product via ${〈 a ∣ b 〉}_{o} : = \int_{U \in A u t (A_{Ω})} (U a ∣ U b) d U$ 1 = {\mu _{inv}}() = {\mu _{inv}}({q_1}) + \ldots + {\mu _{inv}}({q_n}) = n/m

for any a,b ∈ A_Ω. Since Aut(A_Ω) acts transitively on the atoms, we have 〈p|p〉_o = 〈q|q〉_o for any two atoms p and q and we can normalize 〈 | 〉_o in such a way that 〈e|e〉_o = 1 for every atom e.

Lemma 1 and its proof constitute a small piece of a proof in Müller and Ududec’s paper [4]. The same thing has been shown by Wilce in another way [16].

5.

Bit Symmetry

So far there is no connection between the orthogonality in the quantum logic ext([0, 𝕀]) and the orthogonality in the Euclidean space A_Ω with the inner product 〈 | 〉_o from Lemma 1 (ii). Here we will construct a further inner product with such a connection, preassuming bit symmetry and using the state μ_inv and the inner product 〈 | 〉_o from Lemma 1.

Theorem 1

Let Ω be any bit-symmetric finite-dimensional compact convex set with the property (∗∗) and information capacity m.

Then A_Ω possesses an inner product 〈 | 〉 such that the positive cone becomes self-dual and $ℙ_{p} (q) = 〈 p | q 〉$ {\langle a\mid b\rangle _o}: = \int_{U \in Aut\left( {{A_\Omega }} \right)} {(Ua\mid Ub)} dU holds for any atoms p and q. Therefore the transition probability is symmetric: $ℙ_{p} (q) = ℙ_{q} (p)$ {_p}(q) = \langle p|q\rangle .

Furthermore, $〈 I | x 〉 = 〈 x | I 〉 = m μ_{i n v} (x)$ {_p}(q) = {_q}(p) for x ∈ A_Ω. The atoms are the extreme points of the set ${a \in A_{Ω} | 0 \leq a, μ_{i n v} (a) = 1 / m}$ \langle |x\rangle = \langle x|\rangle = m{\mu _{inv}}(x) and this set is affinely isomorphic to the state space of A_Ω as well as to Ω itself.

Proof

We use the state μ_inv and the Aut(A_Ω)-invariant inner product 〈 | 〉_o from Lemma 1. The information capacity is again denoted by m. From the bit symmetry we get a real number ϵ such that ϵ = 〈p|q〉_o for any atoms p,q with p + q ≤ 𝕀. Together with Lemma 1 (ii) the Cauchy-Schwarz inequality implicates |ϵ| ≤ 1. Moreover, the case |ϵ| = 1 is impossible, since atoms p,q with p + q ≤ 𝕀 are linearly independent, and we thus have |ϵ| < 1. We now define $〈 a | b 〉 : = \frac{1}{1 - ϵ} [〈 a | b 〉_{o} - m^{2} ϵ μ_{i n v} (a) μ_{i n v} (b)]$ \{ a \in {A_\Omega }|0 \le a,{\mu _{inv}}(a) = 1/m\}

for a,b ∈ A_Ω. Then 〈p|p〉 = 1 for each atom p and 〈p|q〉 = 0 for atoms p and q with p + q ≤ 𝕀.

Each a ∈ A_Ω has a spectral decomposition $a = s_{1} e_{1} + \dots + s_{n} e_{n}$ \langle a|b\rangle : = {1 \over {1 - }}[{\langle a|b\rangle _o} - {m^2}{\mu _{inv}}(a){\mu _{inv}}(b)] with pairwise orthogonal atoms e_k and s_k ∈ ℝ. Thus $〈 a | a 〉 = s_{1}^{2} + \dots + s_{n}^{2} \geq 0$ a = {s_1}{e_1} + \ldots + {s_n}{e_n} and 〈a|a〉 = 0 iff a = 0. If 0 ≤ 〈a|b〉 for all b ∈ A_Ω with 0 ≤ b, select b = e_k and get 0 ≤ s_k for each k; thus 0 ≤ a.

Now let p_o be any atom. By the Riesz representation theorem there is an element a_o ∈ A_Ω with $ℙ_{p_{o}} (x) = 〈 a_{o} | x 〉$ \langle a|a\rangle = s_1^2 + \ldots + s_n^2 \ge 0 for all x ∈ A_Ω. Let $a_{o} = s_{1} e_{1} + \dots + s_{n} e_{n}$ {_{{p_o}}}(x) = {a_o}|x be its spectral decomposition with pairwise orthogonal atoms e₁,…,e_n and real numbers s₁,…,s_n. From $ℙ_{p_{o}} (e_{j}) = 〈 a_{o} ∣ e_{j} 〉 = \sum_{k} s_{k} 〈 e_{k} ∣ e_{j} 〉 = s_{j}$ {a_o} = {s_1}{e_1} + \ldots + {s_n}{e_n} we get 0 ≤ s_j ≤ 1 for j = 1,…,n. Moreover $1 = ℙ_{p_{o}} (I) = 〈 a_{o} | e_{1} + \dots + e_{n} 〉 = s_{1} + \dots + s_{n}$ {_{{p_o}}}\left( {{e_j}} \right) = \left\langle {{a_o}\mid {e_j}} \right\rangle = \sum\nolimits_k {{s_k}} \left\langle {{e_k}\mid {e_j}} \right\rangle = {s_j} and $1 = ℙ_{p_{0}} (p_{0}) = 〈 a_{0} ∣ p_{0} 〉 = \sum_{k} s_{k} 〈 e_{k} ∣ p_{0} 〉$ 1 = {_{{p_o}}}() = {a_o}|{e_1} + \ldots + {e_n} = {s_1} + \ldots + {s_n}. The Cauchy-Schwarz inequality implicates |〈e_k|p_o〉| ≤ 1 for each k.

For those k with s_k ≠ 0 we then get 〈e_k|p_o〉 = 1. Thus e_k and p_o become linearly dependent. From $‖ e_{k} ‖ = 1 = ‖ p_{o} ‖$ 1 = {_{{p_0}}}\left( {{p_0}} \right) = \left\langle {{a_0}\mid {p_0}} \right\rangle = \sum\nolimits_k {{s_k}} \left\langle {{e_k}\mid {p_0}} \right\rangle we get e_k = ±p_o. Since 0 ≤ e_k and 0 ≤ p_o we have e_k = p_o. However e_k ≠ e_j for k ≠ j and e_k can coincide with p_o for only one single k_o. Therefore s_k = 0 for k ≠ k_o, s_{k_o} = 1 and a_o = e_{k_o} = p_o.

We now have that ℙ_p(x) = 〈p|x〉 holds for each atom p and for all x ∈ A_Ω. For any two atoms we get $ℙ_{p} (q) = 〈 p | q 〉 = 〈 q | p 〉 = ℙ_{q} (p)$ \left\| {{e_k}} \right\| = 1 = \left\| {{p_o}} \right\|. It remains to show that 0 ≤ 〈a|b〉 holds for 0 ≤ a and 0 ≤ b(a,b ∈ A_Ω). Due to the spectral decomposition it is sufficient to prove this inequality for atoms p an q and here we have already $〈 p | q 〉 = ℙ_{p} (q) \geq 0$ {_p}(q) = \langle p|q\rangle = \langle q|p\rangle = {_q}(p).

For x ∈ A_Ω with the spectral decomposition $x = s_{1} e_{1} + \dots + s_{n} e_{n} (n \leq m)$ \langle p|q\rangle = {_p}(q) \ge 0 with pairwise orthogonal atoms e_k and s_k ∈ ℝ we get $〈 x | I 〉 = \sum^{} s_{k} 〈 e_{k} | I 〉 = \sum^{} s_{k} ℙ_{e_{k}} (I) = \sum^{} s_{k} = m μ_{i n v} (x)$ \langle x|\rangle = \mathop \sum \nolimits^ {s_k}\left\langle {{e_k}|} \right\rangle = \mathop \sum \nolimits^ {s_k}{_{{e_k}}}() = \mathop \sum \nolimits^ {s_k} = m{\mu _{inv}}(x). For 0 ≤ x we have 0 ≤ s_k for each k and $‖ x ‖ = m a x {s_{1}, \dots, s_{n}} \leq \sum s_{k} = m μ_{i n v} (x)$ \langle x|\rangle = \sum {{s_k}{e_k}| = } \sum {{s_k}{_{{e_k}}}() = } \sum {{s_k} = m{\mu _{inv}}(x)} . Therefore ${a \in A_{Ω} | 0 \leq a, μ_{i n v} (a) = 1 / m} \subseteq [0, I]$ \left\| x \right\| = max\{ {s_1}, \ldots ,{s_n}\} \le \mathop \sum \nolimits^ {s_k} = m{\mu _{inv}}(x).

Each atom is extreme in [0, 𝕀] and thus in ${a \in A_{Ω} | 0 \leq a, μ_{i n v} (a) = 1 / m}$ \{ a \in {A_\Omega }|0 \le a,{\mu _{inv}}(a) = 1/m\} \subseteq [0,]. Vice versa, if x is an extreme point in ${a \in A_{Ω} | 0 \leq a, μ_{i n v} (a) = 1 / m}$ \{ a \in {A_\Omega }|0 \le a,{\mu _{inv}}(a) = 1/m\} , its spectral decomposition becomes a non-trivial convex combination of pairwise orthogonal atoms unless x itself is an atom.

For a ∈ A_Ω with 0 ≤ a and μ_inv(a) = 1/m the map A_Ω ∋ x → 〈a|x〉 defines a state μ_a and the map a → μ_a yields an affine isomorphism onto the state space. The Riesz representation theorem and the self-duality make sure that each state has this form.

With Theorem 1 we come rather close to the familiar situation of Hilbert space quantum mechanics, where μ_inv (x) = trace(x)/m, 〈x|y〉 = trace(xy) for the self-adjoint operators x and y (the observables) and where the positive operators with normalized trace represent the (mixed) states.

The cases with information capacity m = 2 are the generalized qubit models (or binary models) considered in Ref. [8]. Their state spaces are the smooth and strictly convex compact convex sets. If the automorphism group acts transitively on the pure states (weak symmetry), we get the bit symmetry from the remarks at the end of Section 3, the transition probability then becomes symmetric by Theorem 1 and the state space is affinely isomorphic to the unit ball in an Euclidean space, as already shown in Ref. [8]. We can thus conclude that the ellipsoids are the only smooth and strictly convex compact convex sets where the automorphism group acts transitively on the extreme points. In these cases, A_Ω becomes a so-called spin factor which is a special type of Euclidean (formally real) Jordan algebra [8]. Since the exchange symmetry holds in the spin factors, we now know that, in the case of information capacity m = 2 or the smooth and strictly convex compact convex sets, the exchange symmetry is equivalent to the other three symmetry postulates considered here.

The above proof is partly adopted from Müller and Ududec [4], who derive a weaker version of Theorem 1 for a more general situation. They show that any bit-symmetric compact convex set is self-dual. They do not have the property (∗∗) with its implications like a reasonable atomic quantum logic and spectrality and they do not consider the transition probability between the atoms. Moreover their proof becomes more tricky and must use other methods. A simple example of a bit-symmetric compact convex set that does not have the property (∗∗) is the pentagon [4].

6.

Strong Symmetry

Corollary 1

Let Ω be any strongly symmetric finite-dimensional compact convex set with the property (∗∗). Then Ω is either a simplex or the state space of a simple Euclidean Jordan algebra.

Proof

Since the strong symmetry implies the bit symmetry, we can apply Theorem 1 and get the inner product 〈 | 〉. Now let μ be any state. By the Riesz representation theorem there is a ∈ A_Ω with μ(x) = 〈a|x〉 for all x ∈ A_Ω. The self-duality gives 0 ≤ a. Let a = s₁e₁ +…+ s_ne_n be the spectral decomposition of a, where 0 ≤ s_k holds for each k and the e_k are pairwise orthogonal atoms (section 2 (e)). Then $μ (x) = 〈 a | x 〉 = \sum s_{k} 〈 e_{k} | x 〉 = \sum s_{k} ℙ_{e_{k}} (x)$ \mu (x) = \langle a|x\rangle = \mathop \sum \nolimits^ {s_k}{e_k}|x = \mathop \sum \nolimits^ {s_k}{_{{e_k}}}(x)

for all x ∈ A_Ω and $1 = μ (I) = \sum s_{k}$ 1 = \mu () = \mathop \sum \nolimits^ {s_k}. From 0 ≤ s_k we get that μ becomes the convex combination of the orthogonal (perfectly distinguishable) pure states $ℙ_{e_{k}}, k = 1, \dots, n$ {_{{e_k}}},k = 1, \ldots ,n. This is another type of spectrality, which differs from (e) in section 2 and which is used by Barnum and Hilgert [10]. We are now able to apply their theorem (Every strongly symmetric compact convex set with their type of spectrality is a simplex or the state space of a simple Euclidean Jordan algebra) and get the desired result.

The simplexes represent the state spaces of classical probability theory with finite dimension. An n-simplex is the set ${(s_{1}, \dots, s_{n}) \in ℝ^{n} | 0 \leq s_{1}, \dots, s_{n}, s_{1} + \dots + s_{n} = 1}$ \{ ({s_1}, \ldots ,{s_n}) \in {^n}|0 \le {s_1}, \ldots ,{s_n},{s_1} + \ldots + {s_n} = 1\}

or any other set that is affinely isomorphic to this one.

In Ref. [9] it was concluded at the end of section 6 that the property (∗∗), strong symmetry and the symmetry of the transition probability are possible only with the simplexes and the state spaces of the simple Euclidean Jordan algebras. Corollary 1 now reveals that the assumption that the transition probability is symmetric was redundant there, since this follows from the strong symmetry.

Thus we have almost reconstructed finite dimensional quantum theory from the strong symmetry and the property (∗∗). Still included is the exceptional Jordan algebra that consists of the Hermitian 3 × 3-matrices over the octonions and that does not possess a representation as Hilbert space operators.

7.

Conclusions

The usual transition probability in Hilbert space quantum theory is symmetric. Is there a deeper physical, probabilistic or information theoretical reason why nature chose this or are extensions of the theory with non-symmetric transition probability conceivable? Here we have revealed a connection to another kind of symmetry and we now know that we must drop the bit symmetry when we abandon the symmetry of the transition probability.

In quantum computation reversible transformations from any logical bit to any other logical bit are usually regarded as necessary and this is considered an information theoretic reason for the bit symmetry [4]. A deeper look at some quantum informational procedures, however, shows that this rationale is not as clear as it seems. Grover’s search algorithm and teleportation require assumptions that implicate the symmetry of the transition probability, but not the bit symmetry [17]. The no-cloning theorem and the quantum key distribution protocols need neither the bit symmetry nor the general validity of the symmetry of the transition probability [7,18].

From the physical point of view, a continuous reversible time evolution from one atom p to any other atom q (or from one pure state to any other one) might be a crucial requirement. This means that there are automorphisms U_t, t ∈ [0, 1], such that U₀ is the identity, U₁ (p) = q and the function t → U_t is continuous. This requirement, which has not been considered here, implicates only the weak symmetry among the symmetry features we have looked at.

So we must finally conclude that, neither for the bit symmetry (and even less for the strong symmetry) nor for the symmetry of the transition probability, any truly convincing physical or information theoretical reason is seen. The symmetry of the transition probability appears to be needed in some cases of the quantum informational procedures mentioned above, whereas it may surprise that the bit symmetry plays no role there.

Bit Symmetry Entails the Symmetry of the Quantum Transition Probability

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