Transition-Probability Preserving Maps

For rays $[\psi],[\phi]\in\mathbb P(\mathcal H)$, the intrinsic transition probability is

$$\mathcal P([\psi],[\phi]) = \frac{|\langle\psi|\phi\rangle|^2} {\|\psi\|^2\|\phi\|^2}.$$

A map

$$S:\mathbb P(\mathcal H)\longrightarrow\mathbb P(\mathcal H)$$

is a transition-probability preserver when

$$\mathcal P(S[\psi],S[\phi]) = \mathcal P([\psi],[\phi])$$

for every pair of rays. Unitary and antiunitary operators induce such maps. Wigner’s theorem supplies the converse when $S$ is also surjective: the ray map is implemented by one of those two operator types.

The distinctions among full probability preservation, orthogonality-only preservation, and non-surjective isometric embedding are essential. They have different hypotheses and different conclusions.

Required background. Physical States as Rays defines the projective state space; Transition Probabilities supplies the operational meaning of the invariant.

Helpful background. Probability Amplitudes distinguishes the complex overlap from its phase-invariant squared magnitude.

Transition probability on projective Hilbert space

The expression $\mathcal P$ is independent of representatives. Replacing $\psi$ and $\phi$ by nonzero multiples $a\psi$ and $b\phi$ multiplies the numerator and denominator by the same factor $|a|^2|b|^2$.

In terms of rank-one projectors,

$$\mathcal P([\psi],[\phi]) = \operatorname{Tr}(P_\psi P_\phi).$$

It detects both equality and orthogonality:

$$\mathcal P(r,s)=1 \Longleftrightarrow r=s, \qquad \mathcal P(r,s)=0 \Longleftrightarrow r\perp s.$$

It also determines the Fubini–Study distance up to a fixed convention,

$$d_{\mathrm{FS}}(r,s) = \arccos\sqrt{\mathcal P(r,s)}.$$

Thus preserving every transition probability is equivalent to preserving the full projective metric, not merely its zero-distance and right-angle cases.

Full preservers and immediate consequences

Suppose $S$ preserves $\mathcal P$ for every pair. Then:

• $S$ is injective. If $S(r)=S(s)$, then $\mathcal P(r,s)=\mathcal P(Sr,Ss)=1$, so $r=s$.
• Orthogonality is preserved in both directions. The value zero is preserved exactly.
• Every projective distance is preserved. This follows from the displayed distance formula.
• Finite Gram-matrix magnitudes are preserved. For representatives of a finite set of rays, all normalized overlap magnitudes remain unchanged.

Surjectivity does not follow. A distance-preserving map can embed projective space as a proper subset of itself in infinite dimensions. Standard Wigner theorems therefore state bijectivity or surjectivity explicitly.

No continuity hypothesis is needed in the standard theorem for a bijective map preserving all transition probabilities. Continuity becomes relevant when one studies a symmetry group as a parameterized family, chooses coherent lifts, or weakens the preserved data.

Unitary and antiunitary examples

A unitary operator $U$ is complex-linear and satisfies

$$\langle U\psi|U\phi\rangle = \langle\psi|\phi\rangle.$$

It therefore induces

$$S_U[\psi]=[U\psi]$$

and preserves $\mathcal P$.

An antiunitary operator $A$ is conjugate-linear,

$$A(a\psi+b\phi) = a^*A\psi+b^*A\phi,$$

and is an isometry satisfying, in the site’s inner-product convention,

$$\langle A\psi|A\phi\rangle = \langle\psi|\phi\rangle^*.$$

Complex conjugation changes the phase of an overlap but not its magnitude. Thus $S_A[\psi]=[A\psi]$ also preserves transition probabilities.

Antiunitary does not mean “a special unitary matrix.” Conjugate-linearity is intrinsic and survives every change of complex basis. In a chosen basis an antiunitary can be written $UK$, where $U$ is unitary and $K$ is componentwise complex conjugation.

A non-surjective preserving map

Let $\mathcal H=\ell^2(\mathbb N_0)$ and define the unilateral shift

$$V(c_0,c_1,c_2,\ldots) = (0,c_0,c_1,\ldots).$$

The shift is a linear isometry:

$$\langle V\psi|V\phi\rangle = \langle\psi|\phi\rangle.$$

It therefore induces a transition-probability-preserving ray map. But its range is the proper closed subspace of sequences with zeroth component zero, so $V$ is not surjective and is not unitary.

This example shows why one must separate:

1. a bijective full preserver, covered by the standard Wigner theorem;
2. a non-surjective full-preserving isometric embedding;
3. a map preserving only orthogonality.

A theorem about one class cannot be quoted as though it proved the others.

Orthogonality-only preservation

An orthogonality preserver retains only the zero set of $\mathcal P$:

$$r\perp s \quad\Longleftrightarrow\quad S(r)\perp S(s).$$

Uhlhorn’s theorem states, in a standard form, that for a complex Hilbert space of dimension at least three, a bijection of rays preserving orthogonality in both directions is implemented by a unitary or antiunitary operator. This is a stronger conclusion from weaker preserved data, purchased with a dimension hypothesis and exact bijectivity assumptions.

Dimension two is exceptional. Qubit rays form the Bloch sphere, and orthogonality pairs antipodal points. Many odd bijections of the sphere preserve antipodal pairs without preserving all angles or dot products. They are not Wigner symmetries.

The dimension-at-least-three condition belongs to the orthogonality-only theorem, not to the standard full-transition-probability version. Conflating Wigner and Uhlhorn is a common source of a spurious dimension restriction.

The qubit case

Represent a qubit ray by a unit Bloch vector $\mathbf r$. Then

$$\mathcal P(\mathbf r,\mathbf s) = \frac{1+\mathbf r\cdot\mathbf s}{2}.$$

A full transition-probability preserver therefore preserves every Euclidean dot product on the sphere. A bijective one is induced by an orthogonal transformation of $\mathbb R^3$. Proper rotations correspond to unitary qubit operators modulo phase; improper orthogonal transformations require an antiunitary implementation.

By contrast, an antipode-preserving distortion can preserve orthogonality without preserving intermediate dot products. The Bloch sphere makes the full-versus-orthogonality-only distinction visible.

From one map to a group action

Suppose a group $G$ acts on rays by preservers $S_g$ with

$$S_gS_h=S_{gh}.$$

Wigner’s theorem can lift each $S_g$ individually to a unitary or antiunitary operator $U_g$, but the lift is only unique up to phase. Consequently,

$$U_gU_h = \omega(g,h)U_{gh}, \qquad |\omega(g,h)|=1,$$

in a purely unitary sector. The multiplier $\omega$ obeys a cocycle condition. Additional representation theory is needed to decide whether phases can be chosen so that $\omega=1$.

For a connected continuous symmetry group containing the identity, a continuous choice cannot switch locally between linear and conjugate-linear implementation; the identity component is represented unitarily. Discrete symmetries such as time reversal can occupy an antiunitary component. This group-level conclusion uses continuity beyond the hypothesis of one isolated Wigner transformation.

Hypothesis ledger for Wigner’s theorem

The next theorem page will use the following standard formulation:

Item	Hypothesis or conclusion
scalar field	complex Hilbert space
dimension	at least two for the standard nontrivial statement
map	bijection of the entire ray space
preserved data	every transition probability
continuity	not required for one isolated full preserver
conclusion	unitary or antiunitary implementation
uniqueness	implementing operator unique up to one global phase

Removing surjectivity leads to isometric-embedding variants. Preserving only orthogonality leads to Uhlhorn-type variants, normally with dimension at least three. Acting on mixed states or quantum channels requires different preserver theorems.

Scope relative to transition calculations

The Core Formalism transition-probability page owns preparation, dynamics, final tests, and calculations. This page treats $\mathcal P$ as projective geometry and classifies the map hypothesis used in symmetry theory. It stops before proving Wigner’s converse.

Common pitfalls

Omitting surjectivity from the standard conclusion. The unilateral shift preserves all transition probabilities but is not unitary because its range is proper.

Using Uhlhorn’s dimension condition for Wigner’s full-preserver theorem. They preserve different amounts of data and have different hypotheses.

Calling antiunitary operators unitary matrices. Antiunitaries are conjugate-linear isometries; the conjugation cannot be removed by a complex basis change.

Assuming one lift is automatically a group representation. Ray-dependent phase freedom produces multipliers and projective representations.

Applying a pure-ray theorem to channels. Mixed-state affine maps and completely positive maps belong to different classification problems.

Exercises

1. Phase invariance

Verify directly that $\mathcal P([\psi],[\phi])$ is unchanged under $\psi\mapsto a\psi$ and $\phi\mapsto b\phi$ for nonzero complex $a,b$.

Solution

The numerator becomes $|a|^2|b|^2|\langle\psi|\phi\rangle|^2$, while the denominator becomes $|a|^2\|\psi\|^2|b|^2\|\phi\|^2$. The factors cancel.

2. Antiunitary preservation

Let $A$ be antiunitary. Prove that its induced ray map preserves transition probabilities.

Solution

Antiunitarity gives

$$\langle A\psi|A\phi\rangle = \langle\psi|\phi\rangle^*, \qquad \|A\psi\|=\|\psi\|.$$

Taking the squared magnitude and dividing by the norms therefore leaves $\mathcal P$ unchanged.

3. Audit the unilateral shift

Show that the unilateral shift induces an injective, non-surjective ray map that preserves all transition probabilities.

Solution

The shift preserves inner products and norms, so it preserves $\mathcal P$. An isometry has zero kernel, hence the ray map is injective. No shifted vector has a nonzero zeroth component, so the ray of $(1,0,0,\ldots)$ is absent from the image. The map is not surjective.

4. Qubit probabilities

Derive the Bloch-sphere formula and explain why a full preserver must preserve dot products, whereas an orthogonality-only preserver need preserve only antipodes.

Solution

For $P_{\mathbf r}=(I+\mathbf r\cdot\boldsymbol\sigma)/2$,

$$\operatorname{Tr}(P_{\mathbf r}P_{\mathbf s}) = \frac{1+\mathbf r\cdot\mathbf s}{2}.$$

Preserving every probability therefore preserves every dot product. Qubit orthogonality is the special case $\mathbf s=-\mathbf r$, so an orthogonality-only map need preserve only antipodal pairing.

5. Injectivity from probability preservation

Prove that every full transition-probability preserver is injective, even if injectivity is not stated separately.

Solution

If $S(r)=S(s)$, then

$$1 = \mathcal P(S(r),S(s)) = \mathcal P(r,s).$$

Transition probability one holds exactly for equal rays, so $r=s$.

References

• V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” Journal of Mathematical Physics 5, 862–868, 1964, doi:10.1063/1.1704188.
• N. P. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Springer, 2017.
• M. Pankov, Wigner-Type Theorems for Hilbert Grassmannians, Cambridge University Press, 2020.
• U. Uhlhorn, “Representation of symmetry transformations in quantum mechanics,” Arkiv för Fysik 23, 307–340, 1963.