GAP Measures and Wave Function Collapse

GAP Measures and W a v e F unction Collapse Ro deric h T um ulk a ∗ F ebruary 23, 2026 Abstract GAP measures (also kno wn as Scro oge measures) are a natural class of proba- bilit y distributions on the unit sphere of a Hilb ert space that come up in quantum statistical mechanics; for eac h densit y matrix ρ there is a unique measure GAP ρ . W e describ e and prov e a prop erty of these measures that w as not recognized so far: If a wa ve function Ψ is GAP ρ distributed and a collapse o ccurs, then the col- lapsed wa ve function Ψ ′ is again GAP distributed (relative to the appropriate ρ ′ ). This fact applies to collapses due to a quantum measuremen t carried out by an observ er, as w ell as to sp on taneous collapse theories such as CSL or GR W. More precisely , it is the conditional distribution of Ψ ′ , giv en the measurement outcome (resp ectiv ely , the noise in CSL or the collapse history in GR W), that is GAP ρ ′ . Key w ords: ensem bles of wa ve functions; Scrooge measure; Ghirardi-Rimini- W eb er (GR W) theory; contin uous sp ontaneous lo calization (CSL) theory . 1 In tro duction W e rep ort a connection b etw een quantum statistical mechanics and the foundations of quan tum mec hanics that w as surprising to us. It is a connection b et ween GAP measures, certain probabilit y distributions that play a role in quan tum statistical mec hanics as the thermal equilibrium distribution of w a ve functions [7], and w a v e function collapse, either in the form of a collapse p ostulate in the quan tum formalism or in collapse theories [1, 4], [18, Sec. 3.3] (i.e., theories that pro vide foundations for the quantum formalism b y p ostulating equations for how the collapse o ccurs). F or ev ery densit y matrix ρ on a Hilb ert space H , there is a unique measure GAP ρ on the unit sphere S ( H ) := { ψ ∈ H : ∥ ψ ∥ = 1 } . (1) In a precise sense, it is the most spread-out probability distribution with density matrix ρ [10]. These measures are also known under the name “Scrooge measure” [10, 11, 12, 13], ∗ F ach b ereic h Mathematik, Eb erhard Karls Universit y T¨ ubingen, Auf der Morgenstelle 10, 72076 T ¨ ubingen, Germany . E-mail: ro derich.tum ulk a@uni-tuebingen.de 1 and their prop erties and applications ha ve b een studied extensively [19, 15, 8, 17, 16, 11, 20, 12, 13, 9]. F or their definition, see Section 2. F or ρ prop ortional to a pro jection P (i.e., for ρ = P / tr P ), GAP ρ is in fact the uniform distribution o v er S (range P ). Our result applies to v arious kinds of collapses: • First, to the collapse asso ciated with the ideal quantum measuremen t of a self- adjoin t observ able with discrete sp ectrum, A = X α ∈ spectrum( A ) αP α , (2) where P α is the pro jection to the eigenspace with eigen v alue α ; the collapse then replaces the w av e function Ψ of a system by the collapsed wa ve function Ψ ′ = P α Ψ ∥ P α Ψ ∥ (3) if α was the observed v alue. • Second, to more general collapses of the form Ψ ′ = L z Ψ ∥ L z Ψ ∥ (4) where L z can be any op erator (not necessarily a pro jection) describing ho w to collapse Ψ up on observing the v alue z , sub ject to the condition X z L † z L z = I (5) needed to ensure that the asso ciated formula for the probabilit y distribution of the outcome z , P ( z ) = ∥ L z Ψ ∥ 2 (6) do es define a probabilit y distribution. • Third, to collapse theories [1, 4] such as GR W [6, 2] [18, Sec. 3.3] and CSL [14, 5, 1, 4]. In these cases we condition, not on the observed outcome, but on the times and lo cations of the GR W collapses, resp ectiv ely on the history of the noise in CSL. The remainder of this pap er is organized as follows. In Section 2, w e recall the definition of the GAP measure and form ulate and prov e our result as a mathematical theorem. In Section 3, we describ e some examples of how collapse theories are co vered b y our theorem and provide further discussion. 2 2 Main Result GAP measures can b e defined as follo ws. Let ρ b e a densit y matrix (i.e., a positive op erator with trace 1) and G ρ the Gaussian measure on H with mean 0 and cov ariance op erator ρ [7, 17]. Define GA ρ as the measure on H with densit y function ∥ · ∥ 2 relativ e to G ρ , GA ρ ( dψ ) = ∥ ψ ∥ 2 G ρ ( dψ ) . (7) Using tr ρ = 1, one finds that GA ρ is a probabilit y measure [7]. Let Φ be a random v ector with distribution GA ρ ; then GAP ρ is defined as the distribution of Ψ := Φ / ∥ Φ ∥ . Clearly , GAP ρ is concen trated on the unit sphere S ( H ). In fact, the name “GAP” stands for “Gaussian adjusted pro jected,” where the adjustment is the m ultiplication b y ∥ · ∥ 2 and the pro jection maps to the unit sphere. F or further discussion of GAP measures and ho w they arise in thermal equilibrium, see [7, 8, 16]. W e can no w form ulate our main result as a theorem. W e use the “ ∼ ” notation as in “ X ∼ µ ” for expressing that the random v ariable X has probability distribution µ . Theorem 1. L et H b e a Hilb ert sp ac e and Ψ a r andom p oint on S ( H ) (the “initial wave function ”). Supp ose for e ach p oint x in the me asur e sp ac e ( X , F , µ ) , we ar e given an op er ator L ( x ) (“c ol lapse op er ator”) such that Z X µ ( dx ) L † ( x ) L ( x ) = I . (8) Supp ose the r andom p oint X gets chosen with pr ob ability distribution ∥ L ( x )Ψ ∥ 2 µ ( dx ) (9) (“Born distribution ”), and define Ψ ′ = L ( X )Ψ ∥ L ( X )Ψ ∥ (10) (the “final wave function ”). If Ψ ∼ GAP ρ , then the c onditional distribution of Ψ ′ , given X , is GAP ρ ′ ( X ) with ρ ′ ( x ) = L ( x ) ρL † ( x ) tr[ L ( x ) ρL † ( x )] . (11) Pr o of. As a preparation, we collect some form ulas for later use: First, it is w ell known [7, 8] that Z S ( H ) GAP ρ ( dψ ) | ψ ⟩⟨ ψ | = ρ . (12) 3 Next, b y construction, P ( X ∈ dx | Ψ) = ∥ L ( x )Ψ ∥ 2 µ ( dx ) (13) P ( X ∈ dx, Ψ ∈ dψ ) = ∥ L ( x ) ψ ∥ 2 µ ( dx ) GAP ρ ( dψ ) (14) P ( X ∈ dx ) = µ ( dx ) Z S ( H ) GAP ρ ( dψ ) ∥ L ( x ) ψ ∥ 2 (15) = µ ( dx ) Z S ( H ) GAP ρ ( dψ ) tr  | ψ ⟩⟨ ψ | L † ( x ) L ( x )  (16) = µ ( dx ) tr "  Z S ( H ) GAP ρ ( dψ ) | ψ ⟩⟨ ψ |  L † ( x ) L ( x ) # (17) (12) = µ ( dx ) tr  ρL † ( x ) L ( x )  . (18) The quan tity w e are ultimately in terested in is the conditional distribution of Ψ ′ , giv en X ; in formulas, this distribution is, for any B ⊂ S ( H ), P (Ψ ′ ∈ B | X ∈ dx ) = P (Ψ ′ ∈ B , X ∈ dx ) / P ( X ∈ dx ) (19) = P  L ( X )Ψ ∥ L ( X )Ψ ∥ ∈ B , X ∈ dx  / P ( X ∈ dx ) . (20) W rite Ψ = Φ / ∥ Φ ∥ with Φ ∼ GA ρ . Then P  L ( x )Ψ ∥ L ( x )Ψ ∥ ∈ B , X ∈ dx  = P  L ( x )Φ ∥ L ( x )Φ ∥ ∈ B , X ∈ dx  (21) = Z ϕ ∈ H P (Φ ∈ dϕ ) P  L ( x )Φ ∥ L ( x )Φ ∥ ∈ B , X ∈ dx     Φ = ϕ  (22) = Z ϕ ∈ H GA ρ ( dϕ ) 1 B  L ( x ) ϕ ∥ L ( x ) ϕ ∥  P ( X ∈ dx | Φ = ϕ ) (23) = Z ϕ ∈ H GA ρ ( dϕ ) 1 B  L ( x ) ϕ ∥ L ( x ) ϕ ∥  ∥ L ( x ) ϕ ∥ 2 ∥ ϕ ∥ 2 µ ( dx ) (24) [using GA ρ ( dϕ ) = ∥ ϕ ∥ 2 G ρ ( dϕ )] = µ ( dx ) Z ϕ ∈ H G ρ ( dϕ ) 1 B  L ( x ) ϕ ∥ L ( x ) ϕ ∥  ∥ L ( x ) ϕ ∥ 2 . (25) No w substitute ξ = L ( x ) ϕ . Note that if ϕ is Gaussian with mean 0, then for any op erator L , Lϕ is also Gaussian with mean 0, and its cov ariance matrix is E [ L | ϕ ⟩⟨ ϕ | L † ] = L E [ | ϕ ⟩⟨ ϕ | ] L † = LρL † with ρ the co v ariance of ϕ . Thus, ξ ∼ G L ( x ) ρL † ( x ) , and (25) = µ ( dx ) Z ξ ∈ H G L ( x ) ρL † ( x ) ( dξ ) 1 B  ξ ∥ ξ ∥  ∥ ξ ∥ 2 . (26) 4 No w substitute χ = ξ / p tr[ ρL † ( x ) L ( x )], that is, we just rescale ξ b y a fixed constant factor; then χ is also Gaussian, with mean 0 and cov ariance matrix L ( x ) ρL † ( x ) / tr[ ρL † ( x ) L ( x )] = ρ ′ . (27) Th us, (26) = µ ( dx ) Z χ ∈ H G ρ ′ ( dχ ) 1 B ( χ/ ∥ χ ∥ ) ∥ χ ∥ 2 tr[ ρL † ( x ) L ( x )] (28) [using the definition of GA] = µ ( dx ) tr[ ρL † ( x ) L ( x )] Z χ ∈ H GA ρ ′ ( dχ ) 1 B ( χ/ ∥ χ ∥ ) (29) [substituting φ = χ/ ∥ χ ∥ and using the definition of GAP] = µ ( dx ) tr[ ρL † ( x ) L ( x )] Z φ ∈ S ( H ) GAP ρ ′ ( x ) ( dφ ) 1 B ( φ ) (30) = µ ( dx ) tr[ ρL † ( x ) L ( x )] GAP ρ ′ ( x ) ( B ) . (31) Th us, P (Ψ ′ ∈ B | X ∈ dx ) = µ ( dx ) tr[ ρL † ( x ) L ( x )] GAP ρ ′ ( x ) ( B ) µ ( dx ) tr[ ρL † ( x ) L ( x )] (32) = GAP ρ ′ ( x ) ( B ) , (33) whic h is what the theorem claimed. 3 Examples W e describ e how the theorem cov ers the examples mentioned b efore. • First, for an ideal quan tum measuremen t of (2), X = sp ectrum( A ) discrete, the σ -algebra F contains all subsets, and µ is the counting measure, so that R X µ ( dx ) means the same as P α . The op erator L ( x ) is giv en by P α , and (8) is satisfied b ecause X α P α = I . (34) The random v alue X is the measuremen t outcome, the distribution (9) is indeed the Born distribution giving w eight ⟨ Ψ | P α | Ψ ⟩ = ∥ P α Ψ ∥ 2 to each eigenv alue α of A , and the collapse form ula (10) reduces to (3). 5 • F or the more general measurements of the form (4), X is still the discrete set of p ossible outcomes, X the actual outcome, L ( x ) means the same as L z , and (9) still reduces to the Born distribution (6), whic h corresp onds to a PO VM with the p ositiv e op erator L † z L z asso ciated to the p ossible outcome z . • F or GR W theory , w e consider as Ψ ′ the w a ve function Ψ τ ∈ L 2 ( R 3 N ) obtained through the sto chastic GR W evolution at time t = τ > 0 and as Ψ the w av e function at t = 0; x is a particular history of collapses, giv en by sp ecifying the n umber n of collapses that o ccur during the time interv al [0 , τ ], the times 0 ≤ t 1 < . . . < t n ≤ τ at which the collapses o ccur, the lo cations x 1 , . . . , x n ∈ R 3 at which the collapses are cen tered, and the labels i 1 , . . . , i n ∈ { 1 , . . . , N } of the particles sub ject to each collapse. Corresp ondingly , X = ∞ [ n =0 X n (35) with X n = n ( t 1 , x 1 , i 1 , . . . , t n , x n , i n ) ∈  [0 , τ ] × R 3 × { 1 ...N }  n : t 1 < . . . < t n o (36) and measure µ ( B ) = ∞ X n =0 µ n ( B ∩ X n ) , (37) µ n ( B n ) = Z τ 0 dt 1 Z τ t 1 dt 2 · · · Z τ t n − 1 dt n ( N λ ) n e − N λt n  1 − e − N λ ( τ − t n )  × × Z R 3 d 3 x 1 · · · Z R 3 d 3 x n N X i 1 ...i n =1 1 N n 1 B n ( t 1 , x 1 , i 1 , . . . , t n , x n , i n ) (38) with λ the collapse rate p er particle (one of the constan ts of the GR W theory). The collapse op erator for x = ( t 1 , x 1 , i 1 , . . . , t n , x n , i n ) is L ( x ) = U τ t n C i n ( x n ) U t n t n − 1 · · · C i 2 ( x 2 ) U t 2 t 1 C i 1 ( x 1 ) U t 1 0 (39) with U t ′ t = e − iH ( t ′ − t ) / ℏ (40) the unitary part of the time evolution from t to t ′ and C i ( x ) the multiplication op erator b y (the square ro ot of ) a Gaussian in the i -th v ariable cen tered at x , C i ( x ) Ψ( q 1 , . . . , q N ) = (2 π σ 2 ) − 3 / 4 exp  − ( q i − x ) 2 4 σ 2  Ψ( q 1 , . . . , q N ) , (41) where σ is the collapse width (another constant of GR W theory). 1 1 Alternativ ely , we would obtain an equiv alent theory if we remov e the factor ( N λ ) n e − N λt n  1 − e − N λ ( τ − t n )  from (38) and include its square ro ot in the definition (39) of L ( x ). 6 Then (9) is exactly the joint distribution of all collapse even ts (e.g., [18, (5.27)]) according to GR W theory , and Ψ ′ as in (10) is indeed equal to Ψ τ . • F or CSL theory , Ψ is again the wa ve function at t = 0 and Ψ ′ the one at t = τ ; x is the “noise field” ξ ( x , t ) for all x ∈ R 3 , t ∈ [0 , τ ], and µ the distribution of white noise in R 3 × [0 , τ ] with mean 0 and correlation E  ξ ( x , t ) ξ ( x ′ , t ′ )  = γ δ 3 ( x − x ′ ) δ ( t − t ′ ) , (42) γ > 0 a constan t; L ( x ) ψ is the solution at time τ of the (Stratono vic h-type) equation [1, (8.6)] d dt ψ ( t ) = " − i ℏ H + Z R 3 d 3 x N ( x ) ξ ( x , t ) − γ Z R 3 d 3 x N 2 ( x ) # ψ ( t ) (43) starting from ψ (0) = ψ with N ( x ) the smeared-out particle n umber densit y op er- ator, N ( x ) ψ ( q 1 , . . . , q N ) = N X i =1 (2 π σ 2 ) − 3 / 2 exp  − ( q i − x ) 2 2 σ 2  ψ ( q 1 , . . . , q N ) . (44) In other words, L ( x ) is the time-ordered exp onen tial of the square brack et in (43); X is the actual realization of the noise field; its distribution (9) is traditionally called the “co oked” probability , while µ is called the “ra w” probabilit y [1, (7.40)]; the known fact that the “co ok ed” distribution is alw a ys normalized pro ves (8); Ψ ′ as given by (10) then agrees with what is called the “ph ysical state v ector” [1, (7.42-3)]. Remark 1. If we do not wan t to condition on x then, in order to obtain the (uncondi- tional) distribution of Ψ ′ , w e need to av erage ov er x : P (Ψ ′ ∈ B ) = Z x ∈ X P ( X ∈ dx ) P (Ψ ′ ∈ B | X = x ) (45) = Z x ∈ X µ ( dx ) tr[ ρL ( x ) L † ( x )] GAP ρ ′ ( x ) ( B ) . (46) In particular, the distribution of Ψ τ in GR W theory is a mixture of GAP measures. Remark 2. Since for any probability distribution π on S ( H ), the asso ciated density matrix is ρ π = Z ψ ∈ S ( H ) π ( dψ ) | ψ ⟩⟨ ψ | , (47) and since this formula is linear in π , it follo ws that for any mixture of distributions ov er the unit sphere, the densit y matrix is the corresp onding mixture of the density matrices 7 of the con tributing distributions. Applying this to (46), we obtain that ρ P (Ψ ′ ∈ · ) = Z x ∈ X µ ( dx ) tr[ ρL ( x ) L † ( x )] ρ ′ ( x ) (48) (11) = Z x ∈ X µ ( dx ) L ( x ) ρL † ( x ) , (49) whic h is the post-collapse densit y matrix. In particular for GR W theory , this is the densit y matrix of Ψ τ , i.e., the solution of the GR W master equation [1, (6.8)]. References [1] A. Bassi and G.C. Ghirardi: Dynamical reduction mo dels. Physics R ep orts 379 : 257–426 (2003) http://arxiv.org/abs/quant- ph/0302164 [2] J.S. Bell: Are there quan tum jumps? In Schr¨ odinger. Centenary Celebr ation of a Polymath, Cam bridge Universit y Press (1987). Reprinted as chapter 22 of [3]. [3] J.S. Bell: Sp e akable and unsp e akable in quantum me chanics . Cambridge Univ ersity Press (1987) [4] G.C. Ghirardi and A. Bassi: Collapse Theories. In E.N. Zalta and U. No delman (ed.s), Stanfor d Encyclop e dia of Philosophy , published online b y Stanford Univ er- sit y (Winter 2025 edition) http://plato.stanford.edu/entries/qm- collapse/ [5] G.C. Ghirardi, P . Pearle, and A.Rimini: Marko v pro cesses in Hilb ert space and con tinuous spontaneous localization of systems of identical particles. Physic al R e- view A (3) 42 : 78–89 (1990) [6] G.C. Ghirardi, A. Rimini, and T. W eb er: Unified dynamics for microscopic and macroscopic systems. Physic al R eview D 34 : 470–491 (1986) [7] S. Goldstein, J. Leb owitz, R. T umulk a, and N. Zangh ` ı: On the distribution of the w av e function for systems in thermal equilibrium. Journal of Statistic al Physics 125(5-6) : 1193–1221 (2006) http://arxiv.org/abs/quant- ph/0309021 [8] S. Goldstein, J. Leb o witz, C. Mastro donato, R. T umulk a, and N. Zangh ` ı: Univ er- sal probabilit y distribution for the wa ve function of a quan tum system en tangled with its en vironmen t. Communic ations in Mathematic al Physics 342(3) : 965–988 (2015) [9] C. Igelspac her, R. T umulk a, and C. V ogel: Grand-canonical typicalit y . Preprint (2026) [10] R. Jozsa, D. Robb, and W.K. W o otters: Lo wer b ound for accessible information in quan tum mechanics. Physic al R eview A 49(2) : 668–677 (1994) 8 [11] D.K. Mark, F. Surace, A. Elb en, A.L. Shaw, J. Choi, G. Refael, M. Endres, and S. Choi: A maxim um en trop y principle in deep thermalization and in Hilb ert-space ergo dicit y . Physic al R eview X 14 : 041051 (2024) 11970 [12] M. McGinley and T. Sch uster: The Scro oge ensem ble in man y-b o dy quan tum systems. Preprin t (2025) [13] W.-K. Mok, T. Haug, W.W. Ho, and J. Preskill: Nature is stingy: Universalit y of Scro oge ensembles in quantum man y-b o dy systems. Preprin t (2026) http:// [14] P . Pearle: Com bining sto chastic dynamical state-vector reduction with sponta- neous lo calization. Physic al R eview A 39 : 2277–2289 (1989) [15] P . Reimann: Typicalit y of pure states randomly sampled according to the Gaussian adjusted pro jected measure. Journal of Statistic al Physics 132(5) : 921–935 (2008) [16] S. T eufel, R. T um ulk a, and C. V ogel: Canonical t ypicality for other ensembles than micro-canonical. Annales Henri Poinc ar´ e 26 : 1477–1518 (2025) http://arxiv. org/abs/2307.15624 [17] R. T umulk a: Thermal equilibrium distribution in infinite-dimensional Hilb ert spaces. R ep orts on Mathematic al Physics 86(3) : 303–313 (2020) http://arxiv. org/abs/2004.14226 [18] R. T um ulk a: F oundations of Quantum Me chanics . Heidelb erg: Springer (2022) [19] R. T um ulk a and N. Zangh ` ı: Smo othness of wa v e functions in thermal equilibrium. Journal of Mathematic al Physics 46(11) : 112104 (2005) math- ph/0509028 [20] C. V ogel: Long-time b eha vior of t ypical pure states from thermal equilibrium ensem bles. Journal of Mathematic al Physics 66 : 103304 (2025) http://arxiv. org/abs/2412.16666 9