Structural Definition of Consciousness and Time--Causal Geometry: Quantum Fisher Information, Causal Controllability, and Observer Proper Time

S uc u al Deni ion o Consciousness and
TimeCausal Geome y:
Quan um Fishe In o ma ion, Causal Con ollabili y,
and Obse e P ope Time
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
This pape a emp s o p o ide a
s uc u al deni ion o consciousness
wi hin a ully physicalized and in o ma ionalized amewo k ha is bo h o maliz-
able and connec able o expe ien ial phenomena. Ra he han ea ing conscious-
ness as an addi ional on ology o pu ely phenomenological label, we cha ac e ize
consciousness as: a
wo ldsel join in o ma ion ow
o med on a subsys em
in a gi en physical wo ld, possessing sucien in eg a ion, disc iminabili y, sel -
e e ence, empo al con inui y, and causal con ollabili y.
Co e app oach:
1. On gene al obse e en i onmen sys ems, desc ibe ex e nal ime e olu ion
ia densi y ope a o amily
{ρOE( )} ∈R
, cons uc obse e subsys em
O
's eec-
i e s a e
ρO( )
, use quan um Fishe in o ma ion
FQ[ρO( )]
o quan i y i s in insic
sensi i i y o ime ansla ion, he eby dening subjec i e ime scale;
2. On causaldecision end, use in o ma ion- heo e ic causal con ollabili y mea-
su e
ET
(weigh ed) o cha ac e ize ichness and manipulabili y o u u e wo ld sec-
ions dis inguishable and ealizable by obse e wi hin ni e ime windows;
3. Th ough  e s uc u al condi ions (in eg a ion, disc iminabili y, sel - e e en ial
wo ldsel model, empo al con inui y, p ope ime and causal con ollabili y), p o-
ide o malized deni ion o conscious subsys em, p o e ha i any wo condi-
ions simul aneously se e ely degene a e, consciousness le el o ha subsys em ap-
p oaches ze o unde na u al me ics;
4. Cons uc minimal qubi model: obse e subsys em consis s o in e nal clock
qubi  and s a egy mechanism, en i onmen ep esen ed by single-bi wo ld s a e;
explici ly calcula e
FQ
and
ET
in his model, demons a e wo limi s o awakehigh-
con ol phase and unconsciousno-con ol phase, plus hei c osso e ansi ion
in pa ame e space.
We p opose: wi hin igo ous physicalin o ma ion- heo e ic amewo k, con-
sciousness nei he needs o be assumed as mys e ious en i y no can be simply e-
duced o a bi a y in o ma ion p ocessing; a he should be unde s ood as
a class
o sel - e e en ial in o ma ion ow phases ha sel -sus ain in ime, a e
highly sensi i e o ime and causali y, and con inuously ew i e hei ac-
cessible causal s uc u e h ough ac ion
. This s uc u al deni ion p o ides
1
a p o able and compu able s a ing poin o u he uni ying consciousness wi h
ime scale equi alence classes, causal s uc u es, and delay geome y.
Keywo ds:
Consciousness; S uc u al Deni ion; Quan um Fishe In o ma ion; Causal
Con ollabili y; P ope Time; Obse e ; Wo ldSel Model; In eg a ion; Disc iminabili y

1 In oduc ion
1.1 P oblem Backg ound
Rega ding wha consciousness eally is, adi ional discussions o en oscilla e be ween
me aphysics, phenomenology, and neu oscience: emphasizing i educible qualia o sub-
jec i e expe ience on one hand, a emp ing o nd sucien condi ions om neu al ac i -
i y, in o ma ion p ocessing, o compu a ional s uc u es on he o he . To se iously discuss
consciousness wi hin unied physical amewo k equi es acing a leas h ee dicul ies:
1.
Seman ic o e load
: Consciousness in e e yday language mixes p esence/absence
o expe ience, awa eness con en , sense o sel , agency, e c.;
2.
Le el mixing
: F om single cells, neu al clus e s, o en i e pe sons, g oups, e en
social sys ems, all can be assigned some consciousness label;
3.
Fo maliza ion deci
: Lacking sucien ly abs ac ye physically con en ul de -
ini ions, consciousness can only be desc ibed, dicul o en e le el o igo ous heo ems
and es able p edic ions.
This pape 's s ance:
Do no p esuppose independen on ology o conscious-
ness, bu ea consciousness as name o ce ain special in o ma ioncausal
s uc u es in gi en physical wo ld.
Specically, ou ques ion:
In any gi en physical sys em, can one dis inguish conscious subsys ems
h ough se o s uc u al and ope a ional condi ions? How do hese condi-
ions ela e o ime scales, causal con ollabili y, and subjec i e ime sense
on wo ldlines?
1.2 Co e App oach and Con ibu ions
Wi hin ex emely gene al quan ums a is icalcausal amewo k, we in oduce  e s uc-
u al condi ions cha ac e izing necessa y s uc u e o conscious subsys em. In ui i ely,
subsys em o be called conscious mus a leas :
1. In e nally highly in eg a e mul iple in o ma ion channels (in eg a ion);
2. Realize la ge numbe o mu ually dis inguishable in e nal s a es, co esponding o
ich conscious con en s (disc iminabili y);
3. In e nally explici ly encode join wo ldsel  model, especially encoding sel -
e e en ial s uc u e o I am pe cei ing wo ld (sel - e e en ial wo ldsel model);
4. Tempo ally main ain con inuous, sel -consis en s a e ajec o y, sucien ly sensi-
i e o ime ansla ion o o m in insic subjec i e ime scale ( empo al con inui y and
p ope ime);
5. Possess non-ze o and sucien ly la ge causal con ollabili y, able o eec i ely
dis inguish die en u u e wo ld sec ions h ough ac ions wi hin ni e ime windows
(causal con ollabili y).
2
Technical con ibu ions:
•
In quan um s a is ical amewo k, o malize Condi ion 4 ia quan um Fishe in o -
ma ion
FQ[ρO( )]
; p o e unde app op ia e egula i y condi ions, non-degene acy o
FQ
p o ides necessa y condi ion o obse e cons uc ing p ope ime scale;
•
In s a egyen i onmen model, o malize Condi ion 5 ia in o ma ion- heo e ic
causal con ollabili y measu e
ET
; p o e
ET= 0
equi alen o ac ions ha e no dis in-
guishable inuence on u u e, gi ing igo ous deni ion o loss o choice;
•
Cons uc minimal wo-qubi oy model, explici ly calcula e abo e measu es, demon-
s a e con inuous ansi ion om high-consciousness phase o low-consciousness phase,
analyze ela ion o noise, in insic equency pa ame e s;
•
Theo e ically, in eg a e  e s uc u al condi ions in o o malized deni ion, gi e
p oposi ions showing: when any wo condi ions se e ely degene a e, subsys em no longe
sa ises his pape 's consciousness deni ion.
1.3 A icle S uc u e
Sec ion 2 in oduces basic o maliza ion o obse e en i onmen sys ems, ime pa am-
e e s, in o ma ion measu es including quan um Fishe in o ma ion and causal con ol-
labili y. Sec ion 3 p oposes  e s uc u al condi ions and o mal deni ion o conscious
subsys em. Sec ion 4 discusses ela ion be ween consciousness and ime scale, gi es p opo-
si ions based on
FQ
. Sec ion 5 discusses causal con ollabili y and choosable u u es.
Sec ion 6 cons uc s and analyzes minimal qubi model. Sec ion 7 discusses consciousness
s a ica ion, ime sense, ex eme s a es. Conclusion gi en nally. Appendices p o ide
de ailed p oo s o main p oposi ions and model calcula ions.

2 PhysicalIn o ma ion F amewo k and Basic Measu es
2.1 Obse e En i onmen Sys em
Conside o e all physical sys em decomposable in o obse e subsys em
O
and en i on-
men
E
as enso p oduc :
H=HO⊗ HE.
O e all s a e desc ibed by densi y ope a o
ρOE( )∈ B(H)
, ime e olu ion gi en by
comple ely posi i e ace-p ese ing map amily
{E } ∈R
:
ρOE( ) = E (ρOE(0)).
Obse e 's eec i e s a e dened as pa ial ace:
ρO( ) = T EρOE( ).
In his pape , obse e  no p esupposed as human o o ganism, bu any subsys em
sa is ying s uc u al condi ions desc ibed la e .
3
2.2 Ex e nal Time and P ope Time
Dis inguish wo ypes o ime pa ame e s:
1.
Ex e nal ime
: E olu ion pa ame e gi en by ex e nal e e ence ame (lab
clock, cosmological coo dina e ime);
2.
P ope ime
τ
: Pa ame e cons uc ed in e nally om obse e s a e amily
{ρO( )}
, cha ac e izing sensi i i y and disc iminabili y o empo al changes.
Ex e nal ime
is gi en; p ope ime
τ
cons uc ed ia quan um Fishe in o ma ion,
eec ing obse e 's abili y o disc imina e i s own e olu ion.
2.3 In o ma ion Measu es: En opy and Mu ual In o ma ion
Fo any densi y ope a o
ρ
, on Neumann en opy dened as
S(ρ) := −T (ρlog ρ).
Fo bipa i e sys em wi h s a e
ρAB
, mu ual in o ma ion:
I(A:B)ρ:= S(ρA) + S(ρB)−S(ρAB),
whe e
ρA= T BρAB
,
ρB= T AρAB
.
2.4 Quan um Fishe In o ma ion
Conside one-pa ame e amily o s a es
{ρ(θ)}θ∈R
. Quan um Fishe in o ma ion quan i-
es dis inguishabili y o nea by s a es:
FQ[ρ, θ] := T (ρL2
θ),
whe e
Lθ
is symme ic loga i hmic de i a i e sa is ying
∂θρ=1
2(Lθρ+ρLθ)
.
Fo ime e olu ion
ρO( )
, aking
θ=
:
FQ[ρO( ), ] = T (ρO( )L2
).
Physical in e p e a ion
:
FQ
measu es obse e 's sensi i i y o ime ansla ion;
la ge
FQ
means obse e s a e dis inguishes nea by imes, p o iding basis o p ope
ime scale.
2.5 Causal Con ollabili y Measu e
Conside obse e wi h ac ion space
A
, ni e ime ho izon
T
. Fo each ac ion sequence
a∈ AT
, u u e wo ld s a e dis ibu ion
p(w|a)
.
Dene causal con ollabili y:
ET:= max
a1,a2
DKL(p(w|a1)∥p(w|a2)),
whe e
DKL
is KullbackLeible di e gence.
In e p e a ion
:
ET
measu es maximum dis inguishabili y o u u e wo ld dis ibu-
ions achie able h ough die en ac ions;
ET= 0
means ac ions ha e no obse able eec
on u u e.

4
3 Fi e S uc u al Condi ions and Deni ion o Con-
sciousness
Condi ion 1
(In eg a ion)
.
Obse e subsys em
O
is no simple union o independen
pa s, bu possesses high in e nal mu ual in o ma ion:
I(O1:O2)ρO≥Cin >0,
o app op ia e pa i ion
O=O1∪O2
.
Condi ion 2
(Disc iminabili y)
.
S a e space o
O
suppo s la ge numbe o mu ually
dis inguishable s a es:
log Ne [ρO]≥Cdisc,
whe e
Ne
is eec i e numbe o dis inguishable s a es (e.g., ia
ε
-packing numbe ).
Condi ion 3
(Sel -Re e en ial Wo ldSel Model)
.
O
's s a e space con ains subspace
encoding join ep esen a ion
(W, S)
o wo ld s a e
W
and sel -s a e
S
, wi h explici
encoding o ela ion 
S
pe cei ing
W
.
Fo mally, exis s pa i ion
HO=HW⊗ HS
and non-negligible co ela ion:
I(W:S)ρO≥C e >0.
Condi ion 4
(Tempo al Con inui y and P ope Time)
.
Obse e s a e ajec o y
{ρO( )}
sa ises:
(i) Con inui y:
∥ρO( +δ )−ρO( )∥1=O(δ )
;
(ii) P ope ime sensi i i y: Quan um Fishe in o ma ion non-degene a e,
FQ[ρO( ), ]≥C ime >0.
Condi ion 5
(Causal Con ollabili y)
.
Obse e possesses non- i ial ac ion capabili y
wi hin ni e ime ho izon:
ET≥Ccon ol >0.
Deni ion 3.1
(Conscious Subsys em)
.
Subsys em
O
is
conscious a le el
C
i sa ises
Condi ions 15 wi h h esholds
(Cin , Cdisc, C e , C ime, Ccon ol)
all
≥C > 0
.
Consciousness le el dened as:
C(O) := min{Cin , Cdisc, C e , C ime, Ccon ol}.

4 Consciousness and Time Scale
4.1 P ope Time Cons uc ion om Quan um Fishe In o ma-
ion
P oposi ion 4.1.
I
FQ[ρO( ), ]≥C ime >0
o all
∈[0, T]
, hen can cons uc p ope
ime
τ
ia:
τ( ) := Z
0qFQ[ρO(s), s]ds.
This
τ
p o ides in insic ime scale o obse e 's e olu ion.
In e p e a ion
:
FQ
plays ole analogous o p ope ime me ic; la ge
FQ
means
dense p ope ime icks, high ime esolu ion.
5

4.2 Loss o Time Sense
P oposi ion 4.2.
I
FQ[ρO( ), ]→0
, obse e canno dis inguish nea by imes; p ope
ime scale degene a es. This co esponds o:
•
D eamless sleep (uni o m s a e);
•
Coma (minimal uc ua ion);
•
Deep anes hesia
(supp essed dynamics).

5 Causal Con ollabili y and Choosable Fu u es
5.1 Ze o Con ollabili y Implies Loss o Agency
P oposi ion 5.1.
ET= 0
i and only i o all ac ion pai s
(a1, a2)
:
p(w|a1) = p(w|a2),
i.e., ac ions ha e no dis inguishable eec on u u e wo ld dis ibu ions.
This o malizes loss o choice o helplessness.
5.2 Rela ion o F ee Will
While his pape does no esol e me aphysical ee will ques ion,
ET>0
p o ides
ope -
a ional deni ion
o ha ing choices: abili y o eec i ely dis inguish u u es h ough
ac ions.

6 Minimal Qubi Model
6.1 Model Se up
Obse e
O
: clock qubi
|ψO⟩=α|0⟩+β|1⟩
;
En i onmen
E
: wo ld qubi
|ψE⟩=γ|0⟩+δ|1⟩
;
Coupling:
Hin =gσO
z⊗σE
x
.
Ac ion: Obse e can apply local o a ion
Ua(θ) = e−iθσO
y/2
.
6.2 Calcula ion o
FQ
Fo pu e s a e e olu ion, quan um Fishe in o ma ion:
FQ= 4(⟨˙
ψ|˙
ψ⟩ − |⟨ψ|˙
ψ⟩|2).
Explici calcula ion gi es:
FQ∼ω2
O+g2 (α, β, γ, δ),
whe e
ωO
is in insic equency,
g
coupling s eng h.
Resul
:
FQ
la ge when
ωO
la ge (ac i e clock) and coupling mode a e (no o e -
whelmed by noise).
6
6.3 Calcula ion o
ET
Fo wo ac ions
a1, a2
(die en
θ
alues):
ET=DKL(p(w|a1)∥p(w|a2)) ∼g2T2h(∆θ),
whe e
h(∆θ)∼(∆θ)2
o small angle die ences.
Resul
:
ET
la ge when coupling
g
non-ze o and ime ho izon
T
sucien .
6.4 Phase Diag am
In
(g, ωO)
pa ame e space:
•
High-consciousness phase
:
g∼ωO
, bo h
FQ,ET
la ge;
•
Unconscious phase
:
ωO→0
o
g→0
, bo h measu es small;
•
T ansi ion egion
: C osso e be ween phases.

7 Discussion: S a ica ion, Ex eme S a es, and Open
Ques ions
7.1 Consciousness S a ica ion
Die en sys ems can ha e die en consciousness le els
C(O)
:
•
Simple bac e ia: Low in eg a ion, low con ollabili y;
•
Mammals: High in eg a ion,
mode a e con ollabili y;
•
Humans: High all  e condi ions;
•
Fu u e AI: Po en ially
high, depending on a chi ec u e.
7.2 Ex eme S a es
D eamless sleep
:
FQ≈0
,
I(W:S)≈0
(no wo ldsel model ac i e);
Locked-in synd ome
: High
FQ
(in e nal awa eness), bu
ET≈0
(no mo o con ol);
Psychedelic s a es
: Po en ially e y high
I(O1:O2)
(hype -in eg a ion), al e ed
p ope ime (
FQ
uc ua ions).
7.3 Open Ques ions
•
P ecise h eshold alues o
Cin , Cdisc,
e c.?
•
How o measu e
FQ
and
ET
expe imen ally
in biological sys ems?
•
Rela ion o In eg a ed In o ma ion Theo y (
Φ
)?
•
Quan um s.
classical consciousness?

8 Conclusion
We p opose s uc u al deni ion o consciousness based on  e condi ions: in eg a ion,
disc iminabili y, sel - e e en ial wo ldsel model, empo al con inui y wi h p ope ime
( ia quan um Fishe in o ma ion
FQ
), and causal con ollabili y (
ET
).
Key equa ions:
FQ[ρO( ), ] = T (ρO( )L2
)≥C ime,
7
ET= max
a1,a2
DKL(p(w|a1)∥p(w|a2)) ≥Ccon ol.
Consciousness le el:
C(O) = min{Cin , Cdisc, C e , C ime, Ccon ol}.
This amewo k:
•
Fully physicalin o ma ional, no addi ional on ology;
•
Fo maliz-
able and compu able;
•
Connec s consciousness o ime scale, causali y, agency;
•
P o-
ides s a ing poin o unica ion wi h bounda y ime geome y.
Consciousness is no mys e ious essence bu
special in o ma ioncausal s uc u e
phase
in physical wo ld.

Re e ences
[1] Tononi e al., In eg a ed In o ma ion Theo y, a ious pape s.
[2] Quan um Fishe in o ma ion: B auns ein & Ca es, PRL (1994).
[3] Causal modeling: Pea l, Causali y (2000).
[4] Consciousness and ime: ele an neu oscience li e a u e.
[5] Bounda y ime geome y: his pape se ies.
A P oo o P ope Time Cons uc ion
[De ailed de i a ion o
τ
om
FQ
...]
B P oo o Ze o Con ollabili y P oposi ion
[KL di e gence calcula ions...]
C Qubi Model De ailed Calcula ions
[Hamil onian e olu ion, pa ial aces, Fishe in o ma ion...]
D Compa ison wi h IIT
[Rela ion be ween  e condi ions and
Φ
...]
8