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Correlated Knowledge:

An Epistemic-Logic View on Quantum Entanglement

Alexandru Baltag∗and Sonja Smets†

Abstract

In this paper we give a logical analysis of both classical and quantum correlations. We propose a new

logical system to reason about the information carried by a complex system composed of several parts.

Our formalism is based on an extension of epistemic logic with operators for “group knowledge” (the

logic GEL), further extended with atomic sentences describing the results of “joint observations”

(the logic LCK). As models we introduce correlation models, as a generalization of the standard

representation of epistemic models as vector models. We give sound and complete axiomatizations

for our logics, and we use this setting to investigate the relationship between the information carried by

each of the parts of a complex system and the information carried by the whole system. In particular

we distinguish between the “distributed information”, obtainable by simply pooling together all

the information that can be separately observed in any of the parts, and “correlated information”,

obtainable only by doing joint observations of the parts (and pooling together the results). Our

formalism throws a new light on the diﬀerence between classical and quantum information and gives

rise to an informational-logical characterization of the notion of “quantum entanglement”.

1 Introduction

This paper forms part of the recent research trend aimed at connecting the Logical Foundations of

Quantum Physics to fundamental issues in Logic and Theoretical Computer Science, and in particular to

theories of distributed computation and quantum information (see e.g. [2, 16, 3, 5, 6, 7, 8, 20, 18]). Instead

of starting with traditional “quantum logic”, we focus on one of the logical formalisms (namely, Epistemic

Logic) that was successfully used in Theoretical Computer Science. Because of their spatial features,

epistemic logics have traditionally been used to model the ﬂow of classical information in distributed

systems, including secure communication protocols, distributed computation and multi-agent robotic

systems in Artiﬁcial Intelligence. Our aim is to extend this domain of applications to include quantum

information ﬂow. We thus adapt and generalize epistemic logic to reason about multi-partite quantum

systems and quantum correlations.

Our main focus in this paper is the relationship between the information carried by a complex system

and the information carried by each of the parts of the system. As examples of complex systems, we

look at any physical system composed of more elementary subsystems (particles), but also at any group

of “agents” that can perform observations (of their local environment), logical inferences or introspective

reﬂection. The “parts” of a complex system do not have to be independent, on the contrary their behavior

(e.g. the agents’ observations) might happen to be correlated, either due to prior communication or to

other causes (e.g. quantum entanglement). As a result, the pieces of information carried by the parts

might not be independent either. Diﬀerent types of complex systems will manifest diﬀerent types of

correlations or dependencies between the information carried by their parts.

We are interested in modeling these various types of complex systems from an operational point of

view. That is to say, we look for the operational criteria that characterize various types of correlations

between the local parts of a given system. What we show is that there are several ways in which

the information carried by the individual parts of a system can be combined. Depending on how the

∗Computing Laboratory, Oxford University, Oxford UK. Alexandru.Baltag@comlab.ox.ac.uk

†Dept. of Artiﬁcial Intelligence and Dept. of Philosophy, University of Groningen and IEG, Oxford University.

S.J.L.Smets@rug.nl

1

information between the parts is pooled together, the system will exhibit diﬀerent types of correlations.

Our study of informational correlations will allow us to mark the diﬀerence between classical and quantum

correlations.

To formalize these notions, we use a version of epistemic logic, in which the epistemic “agents” are to

be identiﬁed with the elementary parts (basic “subsystems”, or “locations”) of a complex physical system.

As is standard practice in Computer Science, we adopt the “external” view on knowledge advocated by

Halpern et alia [21], which allows for applications of epistemic logic to situations that reach beyond the

traditional study of real agents’ “knowledge”: in this view, any of the localized parts of a complex system

can be seen as a “virtual agent”, while the system itself as well as any of its subsystems can be seen as

a virtual “group of agents”. The “implicit knowledge” that (is not necessarily actually “possessed” by,

but) can be “externally” attributed to such a virtual agent is given simply by the information that is

potentially available at the corresponding location: something is “known” if it is a consequence of features

that can in principle be observed at that location. Implicit knowledge thus gives us the information (about

the overall system) that is carried by a part (of that system). In this sense, we can think of epistemic

logic as a “spatial logic”, meant to express the relationships between (the information available at)

diﬀerent locations in a complex system. Usually, when considering logics with spatial features, one thinks

about a topology or even a metric space, but in this paper we are only interested in the “local” features

indicating which information is carried by which part of the system. The relation between systems and

their subsystems is what is relevant for us, and not any sense of “distance” or nearness between systems.

A natural question raised in the epistemic logic literature is the following: is there a sense in which

one can externally assign knowledge to a group of (virtual) agents? In other words: what is the implicit

knowledge of a group? The relevance of this question to our inquiry comes from the fact that, in our

setting, implicit “group knowledge” would capture the information carried by a complex physical system.

The standard answer to this question in the current literature is given by distributed knowledge: this is

the information obtainable by pooling together (and closing under logical inference) the “knowledge” of

each of the “parts” (agents). According to this view, the implicit knowledge of a group is the same as its

distributed knowledge; in other words, the information carried by a complex system is nothing but the

“sum” of the information carried by its parts.

Our main claim is that, while this standard answer is adequate for classical physics, it fails for quantum

systems. An entangled system carries more information than the sum of its parts. Moreover, there are

also examples of social situations in which this standard answer fails for real-life agents: whenever a

group of agents can cooperate to make joint observations,the implicit knowledge of the group (deﬁned,

as for a single agent, in terms of what follows from the features that can in principle be observed by

the group) will typically go beyond distributed knowledge (which only takes into account the results of

separate, uncorrelated observations by each of the members of the group). Hence, to correctly model

the information implicitly carried by a complex system or group of agents, we propose the notion of

“correlated knowledge”, that takes into account the correlations between the pieces of information carried

by the parts of the system, or the potential results of joint observations that can be performed by the

group.

For our investigation of group knowledge, we propose a logical system, called “General Epistemic

Logic”(GEL), based on an extension to groups of traditional epistemic logic and on a generalization

of the usual notion of epistemic model. In Section 3, we give a sound and complete axiomatization for

GEL. In subsequent sections, we show that correlated knowledge is useful to capture the non-classicality

of a physical system (or the existence of correlations between the agents’ observational capabilities in

a group). We use the logic GEL to give an informational-logical characterization of the properties of

“separability” and “entanglement”.

In the ﬁnal section of this paper we extend the logic GEL with operators that make explicit the

agents’ observational capabilities, obtaining a “Logic of Correlated Knowledge”LCK. We introduce a

notion of “correlated models”, that generalizes the well-known representation (due to Halpern et alia

[21] of epistemic models as vector models. We give a sound and complete axiomatization for LCK with

respect to correlated models.

2

2 Background Notions

Information. In this paper we are mainly concerned with the “qualitative” (“logical”, or “semantic”)

aspects of information. In contrast to the syntactic (or quantitative) approach to information, which is

primarily concerned with quantitative measures of information (be it in terms of Shannon entropy or

von Neumann entropy), the semantic approach is standard in Logic and Computer Science and focuses

on the meaning or “aboutness” of information. Semantic information can be true or false and is always

“about” something: propositions have a meaning or an “information content”. The main issue in this

semantic approach is to ﬁnd the formal laws governing information ﬂow of a speciﬁc type, so that we

can analyze it, reason about it, verify its correctness etc. There are of course several diﬀerent ways to

model and view “semantic information”. In [13, 14], two compatible views are being discussed: the ﬁrst

looks at “information as range” and the second at “information as correlation”. The ﬁrst is tied to a

possible world model, where an increase in information is related to a decrease of the range of worlds that

are considered possible and vice versa (see [13, 4]). Indeed, “Information as range” takes the point of

view of an “external observer”, whose information-state is represented only via the relations between the

worlds that he or she considers possible. The characteristic feature of the second notion, “information as

correlation”, is the idea that information can manifest itself as a correlation between diﬀerent situations or

their states of aﬀairs. Here, an increase in information is related to a decrease, restriction, or constraint,

on the number of possible correlations between diﬀerent situations. The ideas behind “information as

correlation” trace back to Situation theory [11, 19] and originated as a tool to capture the “aboutness” or

the semantic content of information even in the absence of observers. As a third way to study semantic

information, we mention that it is possible to combine the “information as range” and “information as

correlation” views in one model. As an example we mention here the epistemic constraint logic in [13]

and the interpreted systems in [21]. Also our general epistemic logic in this paper will address issues that

are of concern when we look at “information as range” and at “information as correlation”.

Implicit Knowledge. We adopt the distinction in the CS literature between implicit and explicit

knowledge (see e.g [21]), and use the notion of “information carried by a system” interchangeably with

the notion of “implicit knowledge” (in line also with [8, 10]). Explicit knowledge is the actual information

possessed by a real agent, the information stored in its “database” or its subjective internal state. Explicit

knowledge is not necessarily introspective and might not be closed under logical consequence. In contrast,

implicit knowledge can be visualized as what a virtual agent could in principle come to “know” by

performing observations on that system and deriving logical consequences. It is the potential knowledge

of the agent (what follows implicitly from his possible observations). This is an “external” sense of

knowledge : it is the “knowledge” that “we” externally assign to an agent (or a piece of hardware, a

particle or even just a spatial location), based on what that agent could observe (or what is in principle

observable at that location). Implicit knowledge thus embodies an objective concept of “locally available

(semantic) information”, and has nothing to do with some subjective state internal to the subsystem: it

sums up all information that is potentially available at that location.

As far as implicit knowledge is concerned, there are no problems with positive and negative intro-

spection nor with logical omniscience. Indeed, it seems reasonable to say that if a system carries the

information Pthen it implicitly carries the information that Pis true, and that all consequences of P

are true, and that the system carries the information P. In “agent” terms, this means assuming that

the capabilities of our virtual agent include, not only observations, but also logical inferences and acts of

reﬂection (introspection). So implicit knowledge can be assumed to be always truthful and closed under

logical inference and under (positive and negative) introspection. In other words, implicit knowledge

satisﬁes the axioms of the modal system known as S5.

3 General Epistemic Logic

Consider a complex system composed of nbasic components (or “locations”). We denote each basic

component with a label from a given (ﬁnite) set N={1, ..., n}. Note that the information carried by

this complex system may be distributed throughout space, it can be localized, concentrated at speciﬁc

3

spatial “locations” or “components” of the system. Hence, some information is potentially available only

at some locations, but not at others. We also want to consider (the information) carried by subsystems

composed of several (but not necessarily all) components. To capture these spatial features we introduce

ageneralized “epistemic logic” as a special type of spatial logic, allowing us to have epistemic operators

(“knowledge”) for both individuals (“agents”) and groups (of agents). We use the notion of component

and agent interchangeably: metaphorically, imagine associating to each subsystem a virtual agent that

can “observe” only the state of that subsystem. In the case of a physical system, this sums up all the

information that is obtainable by performing local measurements on that subsystem (and closing under

logical inference). We use sets I⊆Nof labels to denote complex subsystems (or groups of “agents”).

The largest group is the “whole world” N, while the smallest groups are singletons {i}consisting of an

individual agent. Our logic will have an “information” (or “implicit knowledge”) operator KIfor each

subsystem I. As mentioned above, knowledge (“episteme”) is used here only in the implicit, external

sense, as “information that is in principle available” (via local observations) at a given location. So one

may think of the proposition KIPas saying that the subsystem, or group, I(potentially) carries the

information that Pis the case. For groups {i}of one individual agent, we use the simpliﬁed notation

Kiinstead of K{i}. The use of KI-operators to capture qualitative spatial features of complex systems,

extends our previous approach in [8] which in its turn is based on [7, 5] and inspired by [16, 2].

Observational Equivalence. Let sand s0be two possible states of the world (or “possible worlds”), if

the implicit information carried by a system Iis the same in these two states we write sI

∼s0,and say

that the states are “observationally equivalent”, or “indistinguishable”, for system I. Again, we simply

write si

∼s0when I={i}. Observational equivalence for Imeans in agent’s terms that the virtual

agent or group of agents (associated to) Ican make exactly the same observations (at location I) in two

states of the world. We will use the relation of observational equivalence to give an interpretation to the

KI-modalities in our logic:

General Epistemic Frame. For a given set Nof basic components, a set of states (or “possible worlds”)

Σ, a family of binary relations {I

∼}I⊆N⊆Σ×Σ for every subsystem I⊆N, we deﬁne a general epistemic

frame to be a Kripke frame (or multi-modal frame) (Σ,{I

∼}I⊆N) subject to the following three conditions:

1. all I

∼are labeled equivalence relations (for every set I⊆N);

2. Information is Monotonic w.r.t. groups: if I⊆Jthen J

∼⊆ I

∼;

3. Observability Principle: if sN

∼s0then s=s0.

4. Vacuous Information:s∅

∼s0for all s, s0∈Σ.

We read sI

∼s0as: subsystem I cannot distinguish state sfrom s0(via any local observations, performable

on I). In other words, I

∼gives us a notion of ‘‘observational equivalence” relative to subsystem I. The

above conditions seem natural for this interpretation. The ﬁrst condition says that the relation is reﬂexive,

transitive and symmetric. The second condition assumes that every observation that can be performed

by an agent of a group is in principle available to the whole group. In other words, the members of a

group have the capacity to share information among themselves. In terms of systems we then say that if a

subsystem carries the information that Pthen the whole system carries the information that P.In logical

terms this captures the fact that information behaves monotonically with respect to group inclusion. And

relationally, it says that the states of the world that are observationally equivalent for a system/group I

are also observationally equivalent for any subgroup J⊆I. The third condition says that if two possible

states of the world are indistinguishable with respect to the “whole world” Nthen they are the same.

We call this the “observability” principle, as it identiﬁes states of the world that diﬀer in ways that are

not observable even by the whole world. Finally, the last condition says that the empty group has no

non-trivial information whatsoever: being unable to make any observations, the empty group cannot

distinguish between any two states.

4

Local Information State. Given a general epistemic frame (Σ,{I

∼}I⊆N), we construct the notion of

an I-local state, for each subsystem I⊆N:

sI:= {s0∈Σ : sI

∼s0}.

Here, sIcaptures the “agent i’s local state of information”, while sIdoes the same for groups.

Σ-Propositions and Models. Given a general epistemic frame (Σ,{I

∼}I⊆N), a Σ-proposition is any

subset P⊆Σ. Intuitively, we say that a state ssatisﬁes the proposition Pif s∈P. We deﬁne a

general epistemic model to be a structure Σ= (Σ,{I

∼}I⊆N,|| .||), consisting of a general epistemic frame

together with a valuation map || .||: Ω →P(Σ), mapping every element of a given set of atomic sentences

into Σ-propositions. We use the standard notation for satisfaction of atomic sentences in a given state

of model Σ denoted by s|=por s∈|| p||. For every model Σ, we have the usual Boolean operators

on Σ-propositions: P∧Q:= P∩Q,P∨Q:= P∪Q,¬P:= Σ\P,P⇒Q:= ¬P∨Q. We also have

the constants >Σ:= Σ and ⊥Σ:= ∅. Finally, the “knowledge” operator is deﬁned on Σ-propositions by

putting KIP:= {s∈Σ : t∈Pfor every tI

∼s}.

General Epistemic Logic (GEL). The language of the logic GEL has the following syntax, starting

from atomic sentences p∈Ω:

ϕ:= p| ¬ϕ|ϕ∧ϕ|KIϕ

The semantics is given by an interpretation map associating to each sentence ϕof GEL a proposition

|| ϕ||. Equivalently, we extend the satisfaction relation s|=ϕfrom atomic sentences to arbitrary formulas

ϕ. The deﬁnition is by induction in terms of the obvious compositional clauses. In particular, for the KI

modality we set:

s|=KIϕiﬀ t|=ϕfor all states tI

∼s.

So KIϕis true in a given state s, or a system carries the information that ϕis the case in state s, if and

only if ϕholds in all states of the world that are observationally equivalent for Ito s. We would like to

stress the fact that this captures the idea that “information is based on potential observations”.

Proof System. In addition to the rules and axioms of propositional logic, the proof system of GEL

includes:

1. KI-Necessitation. From `ϕ, infer `KIϕ

2. Kripke’s Axiom. `KI(ϕ⇒ψ)⇒(KIϕ⇒KIψ)

3. Truthfulness. `KIϕ⇒ϕ

4. Positive Introspection. `KIϕ⇒KIKIϕ

5. Negative Introspection. ` ¬KIϕ⇒KI¬KIϕ

6. Monotonicity of Group “Knowledge”. For I⊆J, we have `KIϕ⇒KJϕ

7. Observability. `ϕ⇒KNϕ

Using standard results in Modal Correspondence theory (see e.g. [15]), it is easy to show the following:

Theorem 1. The above proof system is sound and complete with respect to general epistemic frames.

Proof. Soundness is trivial: rule 1 and axiom 2 hold in any Kripke model; axioms 3,4,5 hold in any

model in which I

∼are equivalence relations; axiom 6 holds in any model satisfying the condition that

“information is monotonic” (i.e. J

∼⊆ I

∼); axiom 7 holds in any model satisfying the Observability Principle.

For completeness, let ϕ0be a sentence that is consistent (with respect to the above proof system). We

need to show that ϕ0is satisﬁable in some general epistemic model. For this, we use the standard

“canonical model” construction familiar from Modal Logic [15], in which states are “theories” (=sets of

sentences in the language of GEL) that are maximally consistent with respect to the above proof system.

5

Let Ω be the set of all such maximally consistent theories. We deﬁne equivalence relations I

∼on Ω, by

putting for every two theories T , T 0∈Ω: TI

∼T0iﬀ ∀ϕ(KIϕ∈T⇐⇒ KIϕ∈T0). For the valuation,

we put kpk:= {T:p∈T}. Axioms 3,4 and 5 ensure that I

∼are equivalence relations; axiom 6 ensures

that information is monotonic ( J

∼⊆ I

∼for I⊆J); axiom 7 ensures that the Observability Principle holds.

However, the “Vacuous Information” condition does not hold in Ω. Still, we can ensure it by restricting

our model Ω to the set Ω0:= {T∈Ω : T∅

∼T0}, where T0is some ﬁxed maximally consistent theory

T0∈Ω such that ϕ0∈T0. (The existence of T0is ensured by the Lindenbaum Lemma, as usually.) Since

I

∼⊆ ∅

∼(by monotonicity of information), this restriction preserves all the properties of I

∼. In addition, it

obviously ensures that the “Vacuous Information” condition holds (since T∅

∼T0

∅

∼T0for all T , T 0∈Ω0).

So we obtain a general epistemic model Ω0. Finally, using all our axioms (including rule 1 and axiom 2),

we can prove in the usual way a “Truth Lemma”, stating that a sentence ϕholds at a state T∈Ω iﬀ it

belongs to T(seen as a theory): T|=ϕiﬀ ϕ∈T. As a consequence, ϕ0holds at state T0in the model

Ω0(since ϕ0∈T0), and hence ϕ0is satisﬁable. a

4 Distributed Knowledge

The notion of distributed knowledge was ﬁrst introduced in the work of Halpern and Moses [22]. In [21]

the authors say that a group has distributed knowledge of ϕif roughly speaking the agents’ combined

knowledge implies ϕ. Or “distributed knowledge can be viewed as what a wise man - one who has complete

knowledge of what every member of the group knows - would know.” [21, p3]. The intuition is that, in

addition to making individual observations, the agents of the group can “combine” their knowledge, by

sharing all they know: they can announce to the group the knowledge obtained by each member on the

basis of their separate observations.

Similar as for implicit knowledge, we introduce a modal operator for the distributed knowledge of a

group I, denoted by DKI. This operator DKIcan be deﬁned using a “distributed observational equiva-

lence” relation, which is given by the intersection Ti∈I

i

∼of all the individual observational equivalence

relations. The idea is that two states are indistinguishable for the group if and only if they are indistin-

guishable for all the members of the group. Distributed knowledge DKIis simply deﬁned as the Kripke

modality for Ti

i

∼, i.e. given by:

s|=DKIϕiﬀ for every state t∈Σ, if ∀i∈I s i

∼tthen t|=ϕ.

In [21], the authors identify implicit knowledge of a group Iwith distributed knowledge DKI. We

doubt however that this identiﬁcation holds in all situations. Recall that implicit knowledge was explained

as what the agent/group could come to know based on potential observations. So the question is: what

are, in general, the observational capabilities of a group? Looking at the deﬁnition, we see that the use

of the intersection of individual observational equivalencies makes sense only if we assume that a group’s

observations are nothing but observations done by either of the members of the group. But this is in

general a highly unreasonable assumption : joint observations by a group are not in general the same as

independent observations by each of the members of the group.

In a group whose implicit knowledge is the same as distributed knowledge, each agent can only share

with the group the end-result of all her separate, independent observations, but agents are not allowed to

correlate (the results of) their observations, in a simultaneous or sequential manner. But according to us,

the natural notion of (implicit) group knowledge is something diﬀerent, something that could be called

“correlated knowledge”: this is what the group could come to know by performing joint (correlated)

observations and sharing the results.

Separability. For system J⊆I, we say that the system Iis J-separable in state sif we have sJ∩sI\J=

sI.A system Iis called fully separable in state sif we have Ti∈Isi=sI.Full separability means that

group I’s knowledge in state sis the same as its distributed knowledge, i.e. s|=KIPiﬀ s|=DKIP,

for all sets P⊆Σ. Note that if system Iis fully separable then it is J-separable for all J⊆I, but the

converse is false in general. We call a state I-entangled if it is not I-separable.

6

Classical Epistemic Frame. A general epistemic frame (Σ,{I

∼}I⊆N) is called classical if all its states

are fully separable, i.e. if it satisﬁes I

∼=Ti∈I

i

∼for all systems I. So a frame is classical if and only if any

group’s “knowledge” in any state (coincides with its distributed knowledge, and hence) can be obtained by

pooling together the information of each of its components:KI=DKI

Classically, a group cannot distinguish two states if and only if no member of the group can distinguish

them. Note that not only classical or macroscopic systems will satisfy the conditions of a classical

epistemic frame. Indeed, one may also encounter classical epistemic frames in the quantum world. This

happens when the subsystems are separated.

The Vector Model Representation of Classical Epistemic Models. Classical epistemic frames

(models) can be given a “normal form” representation as vector frames (models):

Fact ([12]) Let, for each i,Σi:= {si:s∈Σ}be the set of all i-local states. Every classical epistemic

frame Σcan be canonically embedded into the Cartesian product Σ1×Σ2× · · · Σn,via some embedding

esatisfying

sI

∼s0iﬀ e(s)i=e(s0)ifor all i∈I.

To show this, put e(s) := (si)i∈N, where si={s0∈Σ : si

∼s0}is the i-local state of s, as deﬁned

in the previous section. This vector representation corresponds to another well-known way to model

epistemic logic: the “interpreted systems” representation, in the style of Halpern et alia [21], in which

the global states are simply taken to be tuples of local states, with identity of the i-th components as the

indistinguishability relation i

∼. But note that such a representation is not possible in the case of general

(non-classical) epistemic frames !

Surprisingly enough, the same validities hold in the class of classical epistemic models as in the class

of general epistemic models. This is shown by the following result:

Proposition 1. The proof system of GEL (given in the previous section) is sound and complete with

respect to classical epistemic models as well.

The proof is in Halpern et alia [21]: essentially, this is their proof of completeness for epistemic logic

extended with distributed knowledge operators.

In conclusion, the language of general epistemic logic GEL cannot distinguish between general epis-

temic models and the classical ones. In order to be able to distinguish them, we will have to extend the

language of GEL to a richer “logic of correlated knowledge” LCK (Section 6).

5 Quantum “Knowledge”

A single quantum system can be represented by a state space Σ consisting of rays 1in a Hilbert space H.

A quantum system composed of Nsubsystems Σ1, ..., Σnis represented by the state space Σ1⊗ ·· · ⊗ Σn

corresponding to the tensor product H1⊗ · · · Hn.Note that this tensor product is much richer than the

Cartesian product Σ1× · · · Σn: when the state of a system sis entangled, then it cannot be decomposed as

a tuple of local states. So we cannot think of a composed quantum system as a classical epistemic frame.

Nevertheless, we will show that they can still be thought of (non-classical) general epistemic frames, in a

natural way.

To see this, we consider the following question: what is the “state” of an entangled subsystem I? For

instance, what is the state of component 1 in the binary system |00i+|11i? As we saw, the answer for

I-separable states s=sI⊗sN\Iis simply given by the local state sI. But we also saw that one cannot

talk in any meaningful way about the I-local state of an i-entangled system. To proceed, we ﬁrst give

the standard QM deﬁnition in terms of density operators, then we justify its usefulness for our purposes

by looking at the results of local observations.

1i.e. vector identiﬁed up to a multiplication with a non-zero scalar

7

The State of a Subsystem. If a global system is in state s(thus having an associated density operator

ρs), then Quantum Mechanics describes the state s(I)of any of its subsystems I(possibly entangled with

its environment N\Iin the state s) by the density operator

s(I):= trN\I(ρs).

In other words, the “state” of subsystem Iis obtained by taking the partial trace trN\I(with respect to

the subsystem’s environment N\I) of (the density operator associated to) the global state s.

Notice that, when the subsystem Iis entangled with its environment N\I, the above description

does not really give us a “state” in the sense of this paper (i.e. a pure state), but a “mixed state”.

Nevertheless, as an abstract description, it can still give us an indistinguishability relation I

∼on global

states: the speciﬁc deﬁnition of the “state” of a subsystem is not relevant for us in itself, but only the

resulting notion of “identity of states” of the given subsystem. This leads us to the following deﬁnition:

Observational Equivalence in Quantum Systems. Two quantum states s, s0of a global quantum

system Nare observationally equivalent (“indistinguishable ”) for a subsystem I⊆Nif the mixed states

of subsystem Iare the same in sand s0. Formally:

sI

∼s0iﬀ trN\I(ρs) = trN\I(ρ0

s).

This indirect deﬁnition using density operators may look ad-hoc and unnatural, but it can be justiﬁed

in terms of what an observer can learn about an entangled subsystem Iby observing only that subsystem

(so by performing local measurements on I). Indeed, it is known that a mixed state corresponds to a

probability measure over pure states, but what is not always well-appreciated is the meaning of the mixed

state s(I)describing a (possibly entangled) subsystem:

Quantum I-equivalence via local observations. Provided that we have an unlimited supply of

identical I-entangled systems in the same (entangled) global state s, imagine that the virtual agent

associated to Ican perform all possible local measurements (in various bases) on (various copies of )

subsystem I. The agent can also repeat the same tests on diﬀerent copies and observe the frequency of

each result. After many tests, he can approximate the probability of every given result, for each possible

local measurement. The list of all these probabilities (for each result of each type of measurement) gives

us the “information carried by subsystem I”, or the “information obtainable by local observations at

location I”. Two global states s, s0are I-indistinguishable if all these probabilities are the same in sand

s0, i.e. if the two states behave the same way under I-local measurements.

Quantum I-equivalence via remote evolutions. A third way to deﬁne observational equivalence is

via invariance under changes that do not aﬀect the information carried by subsystem I(see also [8]):

An evolution (unitary map) Uis said to be I-remote (or “remote from I”) if it corresponds to

applying only a local unitary map on the subsystem N\I(the “non-I” part of the system, also known

as I’s “environment”): i.e., if Uis of the form I dI⊗UN\I, where I dIis the identity map on subsystem

Iand UN\Iis a unitary map on the subsystem N\I. In other words, Uis I-remote if it is N\I-local.

Intuitively, I-remote evolutions should not aﬀect the “state” of subsystem I; hence, we could deﬁne

the “state” of Ias what is left invariant by all I-remote evolutions. As a consequence, two states will be

I-indistinguishable if they diﬀer only by some I-remote evolution.

The following result shows the equivalence of these three ways of deﬁning observational equivalence:

Proposition 2. For I⊆N,s s0∈Σ, the following are equivalent:

(1). trN\I(ρs) = trN\I(ρs0);

(2). for every I-local measurement, the probability of obtaining any given result is the same in state s

as in state s0;

(3). s0=U(s)for some I-remote unitary map.

8

So we can deﬁne the quantum equivalence relation sI

∼s0, and hence our notion of implicit knowledge

KI, by any of the clauses given above. Using for example the third clause we obtain that KIPholds

at siﬀ, for all I-remote evolutions U,Pholds at U(s) . If Pis implicitly known by Iin state s, i.e. if

s∈KIP, then we say that the subsystem Icarries the information that P.

Aquantum epistemic frame is a (state space Σ associated to a) Hilbert space endowed with the quantum

I-equivalence relations I

∼(as deﬁned above) for every subsystem I.

Proposition 3. Quantum epistemic frames are (i.e. satisfy all the postulates of) general epistemic

frames.

Properties. In addition to the properties of the KIoperator in GEL, we add:

•If sis I-separable, then sI

∼s0iﬀ sI=s0

I

•If Iis fully separable then we have: sI

∼s0iﬀ si

∼s0for all i∈I. As a consequence, the quantum

“group” knowledge KIof a fully separated system Iis the same as the “distributed knowledge” DKI

of the “group” I.

•In general (for non-fully separated systems I), the previous statement is false: the information KI

carried by a quantum (sub)system Iis not the “sum” DKi∈Iof the information carried by its

i-component systems. In other words: quantum epistemic frames are “non-classical”.

Example. For instance, in a Bell state when the information stored in two subsystems is correlated

according to the identity rule, the agents associated to these subsystems will never recover fully the infor-

mation possessed by the global system if they cannot correlate the results of their individual observations.

Indeed, the following observation shows that in the Bell state |00i+|11icomposed of two entangled

qubits 1 and 2, the two subsystems 1 and 2 are in the same mixed state:

Proposition 4. The following are equivalent:

1. s1

∼ |00i+|11i

2. s2

∼ |00i+|11i

3. s=|z0i+|z01i, for some orthogonal vectors z , z0∈H1.

(The equivalence of e.g. the second and the third clause can be shown by ﬁnding a 2-remote evolution

U1⊗Id2such that U1(|0i) = zand U1(|1i) = z0, and using the deﬁnition of I-equivalence in terms of

remote evolutions.)

This proposition tells us that, in the state |00i+|11i, the two subsystems carry exactly the same

information. So pooling together their “knowledge” will not lead to any increase of information: the

“distributed knowledge” of the group {1,2}is the same as the implicit knowledge of each of the qubits

1 and 2. In contrast, the “group knowledge” of {1,2}is much stronger: s{1,2}

∼ |00i+|11iis equivalent

to s=|00i+|11i. The group “knows” its own state, so it knows (that the qubits are in) the Bell state

|00i+|11i.

Informational Characterizations of Separability and Entanglement Recall that we already gave

an “epistemic” characterization of entanglement and separability in general epistemic frames:

A state sis I-separable iﬀ I’s knowledge in state sis the same as its distributed knowledge.

In the special case of quantum systems, this gives us the standard Quantum Mechanical notion of separa-

bility 2. But this characterization cannot be expressed in the language of epistemic logic since it involves

a second-order quantiﬁer over all subsets Pof the state space: it requires that, for every such subset, s

satisﬁes KIPiﬀ it satisﬁes DKIP.

2In QM, a separable quantum state refers to a non-entangled state. For instance the global state of a bi-partite system

is separable if it belongs to the Cartesian product H1×H2of the two corresponding Hilbert spaces; in this case, each of

the subsystems is in a well-deﬁned (pure) local state.

9

However, in line with the ideas presented in [8] we can use epistemic logic to give “informational

characterizations” of separability and entanglement in a quantum system, provided we are given only

one (logical constant denoting a) fully separable state. Indeed, let w=w0⊗ · · · wnbe some (ﬁxed) fully

separable state; for example, we may take w= 0 = |0i⊗N=|0i⊗|0i · · · |0i. Then we have:

Two subsystems Iand Jare entangled in a (global) state siﬀ ssatisﬁes KJKI¬wor (equiv-

alently, KIKJ¬w. The state sis I-entangled iﬀ the subsystem Iand N\Iare entangled in

s, i.e. if ssatisﬁes KIKN\Is. The system is separable if it is not entangled: ¬KIKN\I¬w.

To summarize: two physical systems are entangled if and only if they potentially carry (non-trivial)

information about each other (assuming no prior communication).

Example. For n= 2, we consider the set Σ(1) of all 1-separable (=2-separable=fully separable) global

states, as our model. Then given that the system is in state |00i, subsystem 1 is in state |0iand “implicitly

knows” his own state. 1 implicitly knows also that it is not possible that the whole system is in state |10i.

Hence, |00i |=K1¬|10i. Subsystem 1 does not implicitly know the local state of the other component

when they are fully separable. So subsystem 1 does not implicitly know that the global state is not |01i

or in other words: |00i |=¬K1¬|01i. Subsystem 2 implicitly knows that the global state cannot be |11i,

however subsystem 1 doesn’t know that the local state of subsystem 2 is not |1i. Hence subsystem 1

does not know that 2 excludes state |11i, that is to say |00i |=¬K1K2¬|11i.

6 Correlated Knowledge

In addition to our presentation so far, our complex systems can be modeled more accurately if we add

structure to our general epistemic frames and enrich the language of the logic. In this section we will

only provide a brief sketch of how this can be brought about, leaving the further details to be explored

in future work. The idea is to capture explicitly the “observational capabilities” of the individual agents

and of the groups. This allows us to generalize the concrete semantics given by “interpreted systems”

(i.e. the vector model representation of classical epistemic models) to a type of general epistemic frames

that we call correlation models.

We generalize vector models in three stages: the ﬁrst type of models we consider are relation-based

models. In these models, the states are relations between the agents’ possible observations. Given sets

O1, . . . , Onof possible observations for each agent, a joint observation will be a tuple of observations

o=~o = (oi)i∈N∈O1× · · · × On. A state of the world can be characterized by the joint observations

that can be performed on it, so a state is a set of such tuples namely a relation. A model will have as its

state space any set Σ ⊆ P(O1× · · · × On). The state sIof a subsystem Iof a global system in state s

will be naturally given by the projection: si={(oi)i∈I:o∈s}.So the observational equivalence is then

given by: sI

∼tiﬀ {(oi)i∈I:o∈s}={(oi)i∈I:o∈t}.

The second, wider stage of generalization is given by multi-set models. Instead of sets of tuples of

observations as in the relational models, we now consider multi-sets. Working with multisets has the

advantage that we can model the case when agents record the frequencies of their observations. Now the

states are multi-sets of joint observations, i.e. functions sfrom tuples of observations from O1× · · · × On

into natural numbers. The state sIof a subsystem Iin a global state swill be naturally given by

sI((ei)i∈I) := X{s(o) : o∈O1× · ·· × Onsuch that oi=eifor all i∈I}.

The third type of models we consider are correlation models. We generalize natural numbers to an

abstract set Rof possible observational results, together with some abstract operation P:P(R)→R

of composing results. This operation may be partial (i.e. deﬁned only for some subsets A⊆R), but

it is required to satisfy the condition: P{PAk:k∈K}=P(Sk∈KAk) whenever {Ak:k∈K}are

pairwise disjoint. In this case, (R, P) will be called a result structure.

10

Correlation Models Given a result structure Rand a tuple ~

O= (Oi)i∈Nof sets of possible observations,

acorrelation model over (R, P,~

O) is given by a set Σ ⊆ {s:sis a function : O1× · · · On→R}of maps

assigning results to (global) joint observations o= (oi)i∈N. So global states will then be functions from

O1× · · · × Oninto R. We put OI:= ×i∈IOi={(oi)i∈I:oi∈Oifor every i∈I}. As before, in a global

state s, the state sIof a subsystem Iwill be a map from sI:OI→R, given by:

sI((ei)i∈I):=X{s(o) : o∈O1× · ·· × Onsuch that oi=eifor all i∈I}.

To put this more succinctly, for every tuple e= (ei)i∈I∈OIof I-observations, let

e:= {o= (oi)i∈N∈O1× · · · On:oi=eifor all i∈I}.

Then we can deﬁne, for every e∈OI:

sI(e) = X{s(o) : o∈e}.

Correlated Knowledge Correlation models are general epistemic models, in which we take our obser-

vational equivalence to be identity of the corresponding local states:

sI

∼tiﬀ sI=tI.

The “group knowledge” KIin a correlation model will be called correlated knowledge.

It is easy to see that, in general, correlation models are not necessarily classical (as epistemic frames).

Hence, correlated knowledge is in general diﬀerent from distributed knowledge.

Examples:

•The relation-based models mentioned ﬁrst can be recovered as special cases of correlation models,

if we take R={0,1}and logical disjunction as the composition operation.

•Epistemic vector models can be seen as special cases of relation-based models (in which every state

is a singleton consisting of only one joint observation), and hence they also are correlation models.

•The multi-sets models are also correlation models, with Rbeing the set of natural numbers, and

addition as the composition operation.

•Quantum epistemic systems Σ1⊗Σ2⊗ · · · ⊗ Σnare correlation models, in which the sets of observa-

tions Oiare given by the (state spaces associated to) Hilbert spaces Σi. Joint observations (oi)i∈I

are interpreted as projectors onto the corresponding state in Ni∈IΣi. The result structure is the

interval R= [0,1] with renormalized addition. The “result” of a joint observation (oi)i∈Imade on

a state sis interpreted as the probability that the outcome of a local measurement (in any basis that

includes o=⊗i∈Ioi) of the I-subsystem of a (global system in) state swill be o. It is well-know

that any quantum state s∈Ni∈NΣiis uniquely characterized (up to multiplication by a non-zero

scalar) by the function mapping any fully separable state o=o1⊗ · · · on∈Σ1× · · · × Σnto the

probability |< s, o > |2of scollapsing to o(after a measurement in a basis that includes o).3

The Logic of Correlated Knowledge. We extend the general epistemic logic GEL with atomic

sentences describing the results of possible joint observations by groups of agents, obtaining the logic of

correlated knowledge LCK:

ϕ::= p|or| ¬ϕ|ϕ∧ψ|KIϕ

where r∈Rand o= (oi)i∈I∈OIis a I-tuple of observations, for any subset I⊆Nof agents. (Recall

that OI:= ×i∈IOi.) The semantics of oris naturally given by: s|=oriﬀ sI(o) = r.

3A diﬀerent type of relational models for a generalized version of QM is proposed in [17]. Note that our models

are “relational” in the sense that quantum “states” correspond in our settings to relations (in relation-based models) or

functions (in correlation models). In contrast, in the categorical approach of [17], relations (between ﬁnite sets) play the

role of morphisms, i.e. they are the analogue of linear maps (between Hilbert spaces) in QM.

11

Notation For any group I⊆N, any set E⊆OIof I-observations and any E-tuple of results r=

(re)e∈E∈RE, one for each observation e∈E, put

Er:= ^

e∈E

ere.

Proof System. Fix a ﬁnite set N={1, . . . , n}of agents, a ﬁnite result structure (R, P) and a tuple

of ﬁnite observation sets ~

O= (O1, . . . , On). The proof system of LCK over (R, P,~

O) includes the rules

and axioms 1-7 of the logic GEL, and in addition, for every I⊆N, the following axioms:

8. Observations always yield results: for every I⊆N, we have

^

o∈OI

_

r∈R

or

9. Observations have unique results: i.e. for r6=p,o∈OI, we have

or⇒ ¬op

10. Groups know the results of their (joint) observations : for o∈OI,r∈R, we have

or⇒KIor

11. Group knowledge is correlated knowledge (i.e. is based on joint observations): for every tuple (ro)o∈OI

of results, one for each possible joint observation o= (oi)i∈I∈OIby group I, we have

(Or

I∧KIϕ)⇒K∅(Or

I⇒ϕ)

12. Result Composition Axiom : for every tuple e= (ei)i∈I∈OIof I-observations and every tuple

r= (ro)o∈e, one for each global observation o∈e, put Pr:= P{ro:o∈e}; then we have

er⇒er

(In axioms 11 and 12 we used the notation Erfrom above: ﬁrst with E=OI; then with E=e,I=N.)

Theorem 2. (Soundness and Completeness for LCK) For every ﬁnite set N={1, . . . , n}of

agents, every ﬁnite result structure (R, P)and every tuple of ﬁnite observation sets ~

O= (O1, . . . , On),

the above proof system is sound and complete with respect to correlation models over (R, P,~

O).

Proof. Soundness is trivial: axioms 8 and 9 hold because each state sinduces a function sI:OI→R;

axioms 10 and 11 hold because of the deﬁnition of indistinguishability in correlation models in terms

of the sIfunctions: sI

∼tiﬀ sI=tI; axiom 12 holds due to the speciﬁc deﬁnition of the map sI(in

terms of the map sand of the composition operation P) in correlation models. For completeness, let

ϕ0be a consistent sentence. First, we introduce the restricted canonical model Ω0, constructed as in

the proof of Theorem 1 (with the same deﬁnition of valuation extended to the atomic constants or, i.e.

kork:= {T∈Ω0:or∈T}). As in Theorem 1, we show that this is a general epistemic model and that

it satisﬁes a “Truth Lemma” (T|=ϕiﬀ ϕ∈T), and hence that ϕ0holds at T0. Second, we can deﬁne a

map T7→ sT, associating to each theory T∈Ω0some function sT:O1× · ·· × On→R, given for each

o∈O1× · · · × Onby: sT(o) = riﬀ or∈T. Axioms 8 and 9 (with I=N={1, . . . , n}) ensure that the

map T7→ sTis well-deﬁned, and axiom 11 (with I=N, hence with KIϕmeaning the same as ϕ) ensures

that this map is injective. Hence, we can “identify” theories Twith the corresponding functions sT, or

in other words we can “lift” the epistemic model structure from Ω0to a subset {sT:T∈Ω0} ⊆ {s:

sis a function : O1× · · · On→R}, by putting sT

I

∼sT0iﬀ TI

∼T0and kpk={sT:p∈T}(and similarly

for the atomic constants or). We can easily see that this is a correlation model : axioms 10 and 11 ensure

that the indistinguishability condition for correlation models (sI

∼tiﬀ sI=tI) is satisﬁed, and axiom 12

ensures that our speciﬁc deﬁnition (in terms of the map sand of the composition operation P) of the

map sIin correlation models is satisﬁed. As general epistemic models, Ω0and this correlation model are

in fact isomorphic, hence ϕ0is satisﬁed in (state sT0of) this correlation model. a

12

7 Concluding Remarks

In this paper we modeled complex systems (from classical to quantum) using the setting of General

Epistemic Frames and we introduced a particular type of such frames called Correlation Models. Our

aim has been to throw new light on the diﬀerence between classical and quantum information and to

gain a better understanding of quantum correlations. As such this paper should be of particular interest

to quantum logicians, looking for an abstract formal logical setting that can naturally accommodate

entanglement. Note that entanglement posed a problem to the lattice-theoretic approach of traditional

Quantum Logic (see e.g. [1, 23]), which opened the road to the quest for possible solutions and alternative

settings. Our paper might also be of interest to quantum information theorists who want to abstract away

from Hilbert space models, looking for higher levels of abstraction similar to those successfully developed

for classical computing. We note that a shorter version of this paper was presented during the 6th QPL

workshop on Quantum Physics and Logic held at Oxford University in 2009 [9].

Acknowledgements During the writing of this paper, S. Smets’ research was supported by a post-

doctoral fellowhip of the Flemish Fund for Scientiﬁc Research. S. Smets thanks the Flemish Academic

Centre for Science and the Arts (VLAC) in Brussels for supporting part of this work by granting her a

four-month VLAC-fellowship in 2008. During the writing of this paper, A. Baltag’s research was partially

supported by the Netherlands Organisation for Scientiﬁc Research (NWO), grant number B 62-635, which

is herewith gratefully acknowledged. We would also like to kindly thank the anonymous referee for very

useful comments.

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