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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”. KeywordsLogics for quantum information-Quantum correlations-Entanglement-Correlation models-Epistemic logic-Correlated knowledge
Correlated Knowledge:
An Epistemic-Logic View on Quantum Entanglement
Alexandru Baltagand Sonja Smets
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.
Dept. of Artificial Intelligence and Dept. of Philosophy, University of Groningen and IEG, Oxford University.
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
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
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 KnowledgeLCK. 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 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
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 INof 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
∼}INΣ×Σ for every subsystem IN, we define a general epistemic
frame to be a Kripke frame (or multi-modal frame) (Σ,{I
∼}IN) subject to the following three conditions:
1. all I
are labeled equivalence relations (for every set IN);
2. Information is Monotonic w.r.t. groups: if IJthen J
∼⊆ I
3. Observability Principle: if sN
s0then s=s0.
4. Vacuous Information:s
s0for all s, s0Σ.
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 JI. 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.
Local Information State. Given a general epistemic frame (Σ,{I
∼}IN), we construct the notion of
an I-local state, for each subsystem IN:
sI:= {s0Σ : sI
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
∼}IN), a Σ-proposition is any
subset PΣ. Intuitively, we say that a state ssatisfies the proposition Pif sP. We define a
general epistemic model to be a structure Σ= (Σ,{I
∼}IN,|| .||), 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: PQ:= PQ,PQ:= PQ,¬P:= Σ\P,PQ:= ¬PQ. We also have
the constants >Σ:= Σ and Σ:= . Finally, the “knowledge” operator is defined on Σ-propositions by
putting KIP:= {sΣ : tPfor every tI
General Epistemic Logic (GEL). The language of the logic GEL has the following syntax, starting
from atomic sentences pΩ:
ϕ:= p| ¬ϕ|ϕϕ|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
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
1. KI-Necessitation. From `ϕ, infer `KIϕ
2. Kripke’s Axiom. `KI(ϕψ)(KIϕKIψ)
3. Truthfulness. `KIϕϕ
4. Positive Introspection. `KIϕKIKIϕ
5. Negative Introspection. ` ¬KIϕKI¬KIϕ
6. Monotonicity of Group “Knowledge”. For IJ, we have `KIϕKJϕ
7. Observability. `ϕKNϕ
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.
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 0Ω: TI
T0iff ϕ(KIϕTKIϕT0). For the valuation,
we put kpk:= {T:pT}. Axioms 3,4 and 5 ensure that I
are equivalence relations; axiom 6 ensures
that information is monotonic ( J
∼⊆ I
for IJ); 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:= {TΩ : T
T0}, where T0is some fixed maximally consistent theory
T0Ω such that ϕ0T0. (The existence of T0is ensured by the Lindenbaum Lemma, as usually.) Since
(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
T0for all T , T 00).
So we obtain a general epistemic model 0. 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 TΩ iff it
belongs to T(seen as a theory): T|=ϕiff ϕT. As a consequence, ϕ0holds at state T0in the model
0(since ϕ0T0), 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 TiI
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.e. given by:
s|=DKIϕiff for every state tΣ, if iI 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 JI, we say that the system Iis J-separable in state sif we have sJsI\J=
sI.A system Iis called fully separable in state sif we have TiIsi=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 PΣ. Note that if system Iis fully separable then it is J-separable for all JI, but the
converse is false in general. We call a state I-entangled if it is not I-separable.
Classical Epistemic Frame. A general epistemic frame (Σ,{I
∼}IN) is called classical if all its states
are fully separable, i.e. if it satisfies 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:sΣ}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
s0iff e(s)i=e(s0)ifor all iI.
To show this, put e(s) := (si)iN, where si={s0Σ : 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=sIsN\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
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 INif the mixed states
of subsystem Iare the same in sand s0. Formally:
s0iff trN\I(ρs) = trN\I(ρ0
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 dIUN\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 IN,s s0Σ, 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.
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
sKIP, 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
Properties. In addition to the properties of the KIoperator in GEL, we add:
If sis I-separable, then sI
s0iff sI=s0
If Iis fully separable then we have: sI
s0iff si
s0for all iI. 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” DKiIof 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 , z0H1.
(The equivalence of e.g. the second and the third clause can be shown by finding a 2-remote evolution
U1Id2such 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
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.
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 = |0iN=|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)iNO1× · · · × 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)iI:os}.So the observational equivalence is then
given by: sI
tiff {(oi)iI:os}={(oi)iI:ot}.
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)iI) := X{s(o) : oO1× · ·· × Onsuch that oi=eifor all iI}.
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 AR), but
it is required to satisfy the condition: P{PAk:kK}=P(SkKAk) whenever {Ak:kK}are
pairwise disjoint. In this case, (R, P) will be called a result structure.
Correlation Models Given a result structure Rand a tuple ~
O= (Oi)iNof sets of possible observations,
acorrelation model over (R, P,~
O) is given by a set Σ ⊆ {s:sis a function : O1× · · · OnR}of maps
assigning results to (global) joint observations o= (oi)iN. So global states will then be functions from
O1× · · · × Oninto R. We put OI:= ×iIOi={(oi)iI:oiOifor every iI}. As before, in a global
state s, the state sIof a subsystem Iwill be a map from sI:OIR, given by:
sI((ei)iI):=X{s(o) : oO1× · ·· × Onsuch that oi=eifor all iI}.
To put this more succinctly, for every tuple e= (ei)iIOIof I-observations, let
e:= {o= (oi)iNO1× · · · On:oi=eifor all iI}.
Then we can define, for every eOI:
sI(e) = X{s(o) : oe}.
Correlated Knowledge Correlation models are general epistemic models, in which we take our obser-
vational equivalence to be identity of the corresponding local states:
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.
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 Σ1Σ2⊗ · · · ⊗ Σnare correlation models, in which the sets of observa-
tions Oiare given by the (state spaces associated to) Hilbert spaces Σi. Joint observations (oi)iI
are interpreted as projectors onto the corresponding state in NiIΣi. The result structure is the
interval R= [0,1] with renormalized addition. The “result” of a joint observation (oi)iImade on
a state sis interpreted as the probability that the outcome of a local measurement (in any basis that
includes o=iIoi) of the I-subsystem of a (global system in) state swill be o. It is well-know
that any quantum state sNiNΣiis uniquely characterized (up to multiplication by a non-zero
scalar) by the function mapping any fully separable state o=o1⊗ · · · onΣ1× · · · × 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|ϕψ|KIϕ
where rRand o= (oi)iIOIis a I-tuple of observations, for any subset INof agents. (Recall
that OI:= ×iIOi.) 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.
Notation For any group IN, any set EOIof I-observations and any E-tuple of results r=
(re)eERE, one for each observation eE, put
Er:= ^
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 IN, the following axioms:
8. Observations always yield results: for every IN, we have
9. Observations have unique results: i.e. for r6=p,oOI, we have
or⇒ ¬op
10. Groups know the results of their (joint) observations : for oOI,rR, we have
11. Group knowledge is correlated knowledge (i.e. is based on joint observations): for every tuple (ro)oOI
of results, one for each possible joint observation o= (oi)iIOIby group I, we have
12. Result Composition Axiom : for every tuple e= (ei)iIOIof I-observations and every tuple
r= (ro)oe, one for each global observation oe, put Pr:= P{ro:oe}; then we have
(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,~
Proof. Soundness is trivial: axioms 8 and 9 hold because each state sinduces a function sI:OIR;
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:= {T0:orT}). As in Theorem 1, we show that this is a general epistemic model and that
it satisfies a “Truth Lemma” (T|=ϕiff ϕT), and hence that ϕ0holds at T0. Second, we can define a
map T7→ sT, associating to each theory T0some function sT:O1× · ·· × OnR, given for each
oO1× · · · × Onby: sT(o) = riff orT. 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:T0} ⊆ {s:
sis a function : O1× · · · OnR}, by putting sT
sT0iff TI
T0and kpk={sT:pT}(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
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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The intrinsic informational content that characterizes quantum computational logics has naturally inspired some interesting and intriguing epistemic problems. We investigate the possibility of a quantum computational semantics for a first-order language that can express sentences like “Alice knows that everybody knows that she is pretty”. The basic question is: to what extent is it possible to interpret quantiers and epistemic operators as special examples of Hilbert-space operations? These logical operators turn out to have a similar logical behavior, giving rise to a “reversibility-breaking”. Unlike logical connectives, quantiers and epistemic operators can be represented as special operations that are generally irreversible (as happens in the case of measurement-procedures). An interesting feature of the epistemic quantum semantics is the failure of the unrealistic phenomenon of logical omniscience: Alice might know a given sentence without knowing all its logical consequences.
Conference Paper
We propose an epistemic strategy logic with future and past time operators, called \(\text {SLKP}\), for Strategy Logic with Knowledge of the Past. With \(\text {SLKP}\) we can model mutually observed moves/actions in strategic contexts. In a semantic game, agents may completely or partially observe other agents’ moves, their moves may depend on their knowledge of other players’ strategies, and their knowledge may depend on the history of their own or other’s moves. The logic \(\text {SLKP}\) also allows us to describe temporal properties involving past, future, and composed tenses such as future perfect or counterfactual assertions. We illustrate SLKP by formalising the quantum cryptography protocol BB84, with the purpose to initiate an integrated epistemic and strategic treatment of agent interactions in quantum systems.
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In this paper we analyze classical and quantum correlations using the tools of epistemic logic. Our main contribution consists of two new logical systems. The first one is called General Epistemic Logic (GEL), it extends traditional epistemic logic with operators that allow us to reason about the information carried by a complex system composed of several parts. The second system is called the Logic of Correlated Knowledge (LCK), which extends GEL with sentences that describe the observational capabilities of an agent. On the semantic side we introduce correlation models, as a generalization of the “interpreted systems” semantics. We use this setting to investigate several types of informational correlations (e.g. distributed information, quantum correlated information) that complex systems can exhibit. We also provide an informational-logical characterization of the notion of “quantum entanglement”.
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We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general category-theoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set FRel.
This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Researchers in areas ranging from economics to computational linguistics have since realised its worth. The book is for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter. The development is mathematical; prior acquaintance with first-order logic and its semantics is assumed, and familiarity with the basic mathematical notions of set theory is required. The authors focus on the use of modal languages as tools to analyze the properties of relational structures, including their algorithmic and algebraic aspects, and applications to issues in logic and computer science such as completeness, computability and complexity are considered. Three appendices supply basic background information and numerous exercises are provided. Ideal for anyone wanting to learn modern modal logic.
It seems to be difficult to give a description of the whole universe at once, so we are always obliged to consider subsystems of a bigger physical system. We all know that quantum mechanics is not compatible with this situation. A lot of problems of interpretation are due to this fact. Some physicists do not bother about this situation because for them an isolated system is an idealization of the real situation and the theory need not be able to describe such an idealization. I think the procedure of considering subsystems and describing them as isolated systems is much more fundamental. One can ask for example the following question: “Should we ever have found mathematics without considering our fingers as isolated systems?” One could protest by saying that to do quantum logic one does not need mathematics. But then it is possible to ask the following question: “Should we ever have agreed to treat two symbols a and b written on the blackboard as being two symbols and not a mass of chalk dust if we should never have seen and understood two cars as two cars, two houses as two houses, etc…”. So the situation seems to be that we use a language to describe nature and this language is built up with the notion of isolated systems. There have been several attempts to axiomatize this language (e.g. logic, set theory).
The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic. Preprint available at