# Entanglement of temporal sections as quantum histories and their quantum
correlation bounds

Marcin Nowakowski<EMAIL_ADDRESS>Faculty of Applied Physics and
Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland  National
Quantum Information Center, 80‑309 Gdańsk, Poland

###### Abstract

In this paper we focus on the underlying quantum structure of temporal
correlations and show their peculiar nature which differentiate them from
spatial quantum correlations. With a growing interest in representation of
quantum states as topological objects, we consider quantum history bundles
based on the temporal manifold and show the source of violation of monogamous
temporal Bell-like inequalities. We introduce definitions for the mixture of
quantum histories and consider their entanglement as sections over the Hilbert
vector bundles. As a generalization of temporal Bell-like inequalities, we
derive the quantum bound for multi-time Bell-like inequalities.

###### pacs:

03.67.-a, 03.67.Hk

##  I Introduction

Recent years have witnessed increasing interest in the concept of quantum
entanglement monogamy, demonstrating its utility in quantum communication
theory and its applications in quantum secure key generation [1, 2, 3, 4, 5,
6]. While spatial quantum correlations, particularly their non-locality, have
become a central focus of quantum information theory and its applications in
quantum computation, the potential for applying temporal non-local
correlations has been relatively underexplored. However, there is a growing
interest in this area, which is linked to a better understanding of this
distinctive quantum phenomenon.

A crucial issue relates to the fundamental nature of time and the phenomenon
of temporal correlations, and their interpretation within the frameworks of
modern quantum and relativistic theories. This emerging interest signifies a
pivotal shift in focus, aiming to unravel the complexities of temporal
phenomena in quantum mechanics and their implications for broader theoretical
constructs.

Non-local nature of quantum correlations in space has been accepted as a
consequence of violation of local realism (LR), expressed in Bell’s theorem
[7] and analyzed in many experiments [8, 9]. As an analogy for a temporal
domain, violation of macro-realism (MR) [10] and Leggett-Garg inequalities
(LGI) [11] seem to indicate non-local effects in time and are a subject of
many experimental considerations [12, 13, 14, 15, 16]. There have been
different formalisms proposed for study of quantum temporal correlations
including Multiple-Time States (MTS) by Aharonov et al. [17, 18] as part of
the Two-State-Vector formalism (TSVF) [19, 20, 21, 22], the Entangled
Histories (EH) approach [23] or the pseudo-density operators (PDOs) [24, 25,
26].

The TSVF led to surprising effects within pre- and post-selected systems (e.g.
[27, 28, 29]), time travel with post-selected teleportation [30, 31], a novel
notion of quantum time [32], new results regarding quantum state tomography
[33] and a better understanding of processes with indefinite causal order
[34], while the Entangled Histories approach led to Bell tests for histories
[35] and have been recently used for analysis of the final state proposal in
black holes [36]. The subject of the black hole information loss paradox has
been also addressed with application of PDOs [37, 38] but engaging a concept
of non-monogamy of spatio-temporal correlations.

In this paper we study the nature of the non-monogamous behavior of temporal
correlations both for ensembles of quantum processes and their single
instances. A central finding of our research, as highlighted in this letter,
is the distinct nature of temporal versus spatial quantum correlations. In
spatial quantum correlations, analysis typically involves ensembles composed
of identical copies of multipartite states. Conversely, in the realm of
temporal correlations, we encounter ensembles comprising diverse temporal
histories. These histories serve as temporal counterparts to quantum states. A
particular entangled history, which can be associated with a quantum
propagator, is monogamous to conserve its consistency throughout time [39].
Yet evolving systems violate monogamous Bell-like multi-time inequalities
which can be explained engaging bundles of histories with the same pre-
selected and post-selected states as initial and final boundaries for the
considered evolution. This dichotomy does not have a counterpart in spatial
domain and as such is a novel feature of temporal non-locality but is also a
sign of importance of the internal structure of single processes.

Our discussion shows the importance of considering the topological aspects of
quantum processes, in addition to their statistical characteristics. Thus, we
consider quantum histories as quantum vector bundles based on the temporal
manifold and we introduce a definition for the mixture of quantum histories
considering their entanglement as sections over these vector bundles.

##  II Review of Entangled Histories and Multiple-Time States

Let us review briefly the entangled histories (EH) formalism and the multiple-
time states (MTS) formalism as a natural extension of the two-state vector
formalism (TSVF).

The predecessor of the entangled histories is the decoherent histories
approach built on the grounds of the well-known Feynman’s path integral theory
for calculation of probability amplitudes of quantum processes. The EH
formalism extends the concepts of the consistent histories theory by allowing
for complex superposition of histories. A history state is understood as an
element in $\text{Proj}(\mathcal{H})$, spanned by projection operators from
$\mathcal{H}$ to $\mathcal{H}$, where
$\mathcal{H}=\mathcal{H}_{t_{n}}\odot...\odot\mathcal{H}_{t_{0}}$. The $\odot$
symbol, which we use to comply with the current literature, stands for
sequential tensor products, and has the same meaning as the tensor product
$\otimes$. The alternatives at a given instance of time form an exhaustive
orthogonal set of projectors $\sum_{\alpha_{x}}P_{x}^{\alpha_{x}}=\mathbb{I}$
and for the sample space of entangled histories
$|H^{\overline{\alpha}})=P_{n}^{\alpha_{n}}\odot
P_{n-1}^{\alpha_{n-1}}\odot\ldots\odot P_{1}^{\alpha_{1}}\odot
P_{0}^{\alpha_{0}}$
($\overline{\alpha}=(\alpha_{n},\alpha_{n-1},\ldots,\alpha_{0})$), there exist
a set of $c_{\overline{\alpha}}\in\mathbb{C}$ such that
$\sum_{\overline{\alpha}}c_{\overline{\alpha}}|H^{\overline{\alpha}})=\mathbb{I}$
and $\sum_{\overline{\alpha}}|c_{\overline{\alpha}}|^{2}=1$.

As an example, one can take a history
$|H)=[z^{+}]\odot[x^{-}]\odot[y^{-}]\odot[x^{+}]=[|z^{+}\rangle\langle
z^{+}|]\odot[|x^{-}\rangle\langle x^{-}|]\odot[|y^{-}\rangle\langle
y^{-}|]\odot[|x^{+}\rangle\langle x^{+}|]$ for a spin-$\frac{1}{2}$ particle
being in an eigenstate of the Pauli-X operator at time $t_{1}$, in an
eigenstate of the Pauli-Y operator at time $t_{2}$, and so on. Within this
formalism one also defines the unitary bridging operators
$\mathcal{T}(t_{j},t_{i}):\mathcal{H}_{t_{i}}\rightarrow\mathcal{H}_{t_{j}}$
evolving the states between instances of time, and having the following
properties: $\mathcal{T}(t_{j},t_{i})=\mathcal{T}^{\dagger}(t_{i},t_{j})$ and
$\mathcal{T}(t_{j},t_{i})=\mathcal{T}(t_{j},t_{j-1})\mathcal{T}(t_{j-1},t_{i})$.
This formalism introduces also the chain operator
$K(|H^{\overline{\alpha}}))$, which can be directly associated with a time
propagator of a given quantum process:

$K(|H^{\overline{\alpha}}))=P_{n}^{\alpha_{n}}\mathcal{T}(t_{n},t_{n-1})P_{n-1}^{\alpha_{n-1}}\ldots
P_{1}^{\alpha_{1}}\mathcal{T}(t_{1},t_{0})P_{0}^{\alpha_{0}}$ (1)

This operator plays a fundamental role in measuring a weight of any history
$|H^{\alpha})$:

$W(|H^{\alpha}))=Tr\\{K(|H^{\alpha}))^{\dagger}K(|H^{\alpha}))\\}$ (2)

which can be interpreted as a realization probability of a history by the Born
rule application. The histories approach requires also that the family of
histories is consistent, i.e. one can associate with a union of histories a
weight equal to the sum of weights associated with particular histories
included in the union.

Multiple-Time States (MTS) extend the standard quantum mechanical state by
allowing its simultaneous description in several different moments. Such a
multiple-time state may encompass both forward- and backward-evolving states
on equal footing. MTS represent all instances of collapse (i.e. those moments
in time when the quantum state coincided with an eigenstate of some measured
operator) and allow them to evolve both forward and backward in time. This
evolution backwards in time can be understood literally (giving rise to the
Two-Time Interpretation [22]), but this is not necessary, it can be simply
regarded as a mathematical feature of the formalism (which is, in fact,
equivalent to the standard quantum formalism [21]). MTS live in a tensor
product of Hilbert spaces $\mathcal{H}$ admissible at those various instances
of time ($t_{0}<...<t_{n}$) denoted by [18]

$\mathcal{H}=\mathcal{H}_{t_{n}}^{(\cdot)}\otimes...\otimes\mathcal{H}_{t_{k+1}}^{\dagger}\otimes\mathcal{H}_{t_{k}}\otimes\mathcal{H}_{t_{k-1}}^{\dagger}\otimes...\otimes\mathcal{H}_{t_{0}}^{(\cdot)},$
(3)

where a dagger means the corresponding Hilbert space consists of states which
evolve backwards in time. The initial and final Hilbert spaces might be
daggered or not (this is denoted by a “$\cdot$” superscript). All Hilbert
spaces containing either (forward-evolving) kets or (backward-evolving) bras
are alternating to allow a time-symmetric description at any intermediate
moment.

As an example of (a separable) MTS we can consider the following state:
${}_{t_{4}}\langle z^{+}||x^{-}\rangle_{t_{3}~{}t_{2}}\langle
y^{-}||x^{+}\rangle_{t_{1}}\in\mathcal{H}_{t_{4}}^{\dagger}\otimes\mathcal{H}_{t_{3}}\otimes\mathcal{H}_{t_{2}}^{\dagger}\otimes\mathcal{H}_{t_{1}}$.
This multiple-time state represents an initial eigenstate of the Pauli-X
operator evolving forward in time from $t_{1}$ until collapse into an
eigenstate of the Pauli-Y operator occurs at time $t_{2}$. Later on, at time
$t_{3}$ the system is projected again onto a different eigenstate of the
Pauli-X operator. Finally at $t_{4}$ the system is measured in the Z basis,
and the resulting eigenstate evolves backward in time. In the following we
will focus on two-time states (sometimes called two-states), which consist of
a forward evolving state $|\psi_{1}\rangle_{t_{1}}$ and a backward evolving
state $|\psi_{2}\rangle_{t_{2}}$ in the above form
${}_{t_{2}}\langle\psi_{2}||\psi_{1}\rangle_{t_{1}}$ to achieve a richer
description of a quantum system during the time interval $t_{1}\leq t\leq
t_{2}$ [21].

Given an initial state $|\Psi\rangle$ and a final state $\langle\Phi|$, the
probability that an intermediate measurement of some hermitian operator $A$
will result in the eigenvalue $a_{n}$ is given by the ABL formula [19]

$p(A=a_{n})=\frac{1}{N}|\langle\Phi|U_{2}P_{n}U_{1}|\Psi\rangle|^{2},$ (4)

where $U_{1}$ and $U_{2}$ represent unitary evolution, the operator $P_{n}$
projects on $|a_{n}\rangle$ and

$N\equiv\sum_{k}|\langle\Phi|U_{2}P_{k}U_{1}|\Psi\rangle|^{2}.$ (5)

This probability rule is important in that it uses the information available
through the final state in a way which is manifestly time-symmetric.

##  III Bundles of quantum histories and their mixtures

In this section we propose the mathematical representation of entangled
quantum histories framed within the context of vector bundles. This approach
offers a topologically structured way to understand the evolution of quantum
states. Recently, there is a growing attention paid to these connections [40,
41] trying to understand the role that vector bundles of spaces of quantum
states play in the classification of topological phases of matter and
topological quantum computation via understanding how entangled structures
occur in parametrized vector bundles.

It has been also argued [42] that quantum codes can be described by section of
a fiber bundle, where the base corresponds to a choice of stabilizers of the
code and the fiber describes the encoded logical information. Our motivation
behind this path stems from the necessity of consideration of both particular
quantum histories but also their mixtures which have substantial consequences
for statements about their entropic characteristics. Topological
representation and visualization of quantum histories as fibers and vector
bundles allows us to capture the subtlety of violating Bell’s temporal
monogamous inequalities which is not obvious in case of TSVF or pseudo-density
formalisms. We delve now into more formal definitions and theorems to deepen
this understanding.

A vector bundle $\mathcal{E}$ over a base space $\mathcal{M}$ is a topological
structure associating family of vector spaces $\mathcal{E}(m)$ to elements of
space $\mathcal{M}$.

We present now two representations of quantum histories as sections over
vector bundles and fibers of vector bundles. In the first representation the
vector bundle is built upon a temporal manifold $T$ and a particular history
is a consistent section over the bundle. In the second representation it is
indexed by a sequence of chosen measurements and quantum histories are fibers
of the vector bundle.

Consider a manifold $T$ representing time, each point in $T$ corresponds to a
distinct time in the evolution of the quantum system. Let $\mathcal{H}$ denote
a Hilbert space associated with our quantum system. For each time point t,
there exists a Hilbert space $\mathcal{H}_{t}$ that is isomorphic to
$\mathcal{H}\cong\mathcal{H}_{t}$. We define a vector bundle $\mathcal{E}$
over $T$ where for each point $t\in T$ the fiber
$\mathcal{E}_{t}\cong\mathcal{H}_{t}$, represents the state space over that
time.

A quantum history $|H_{s})$ in this framework is a consistent section of the
bundle $\mathcal{E}$. Formally, a section $s:T\rightarrow\mathcal{E}$ is a map
that assigns to each point in time $t$ a state $\Psi(t)$ in the fiber
$\mathcal{E}_{t}$, which is a state of the quantum system at that time. This
can be expressed as:

######  Definition III.1.

Let $\mathcal{H}\ni\Psi(t)$ denotes a Hilbert space of possible states
associated with a quantum system evolving over time $t\in T$. A history bundle
$\mathcal{E}$ over $T$ associates with each measurable time $t$ the fiber
$\mathcal{E}_{t}\cong\mathcal{H}_{t}$ so that the evolution of the system is
represented by a consistent section $s=|H_{s})$ over that bundle, i.e.:

$\displaystyle s:T$ $\displaystyle\rightarrow\mathcal{E}$ (6) $\displaystyle
t$ $\displaystyle\mapsto s(t)$ (7)

where $s(t)\in\mathcal{E}_{t}$ for each $t\in T$.

Histories are entangled if their corresponding sections do not factorize into
independent states across different times. This representation allows us to
view quantum histories as continuous (or piece-wise continuous) trajectories
in the state space of the system. It emphasizes the temporal evolution of
quantum states, capturing the essence of quantum dynamics over time. This
reasoning can be naturally extended to the base space-time manifold
$\mathcal{M}(x,t)$ and space of field histories $\Phi(x,t)$.

Now we propose a second less intuitive representation where quantum histories
are fibers of vector bundles over the measurement sequences space. Let us
consider the base space $\mathcal{M}$ of all possible finite sequences of
measurements, where $\mathcal{M}=\bigcup_{n=1}^{\infty}M^{n}$, and $M$ is the
set of all possible measurements. A vector bundle $F$ is constructed over
$\mathcal{M}$ where each fiber $F_{\alpha}$, for a measurement sequence
$\alpha\in\mathcal{M}$, represents the possible outcomes of that sequence.
Recalling that the alternatives at a given instance of time form an exhaustive
orthogonal set of projectors $\sum_{\alpha_{x}}P_{x}^{\alpha_{x}}=\mathbb{I}$
and for the sample space of entangled histories
$|H^{\overline{\alpha}})=P_{n}^{\alpha_{n}}\odot
P_{n-1}^{\alpha_{n-1}}\odot\ldots\odot P_{1}^{\alpha_{1}}\odot
P_{0}^{\alpha_{0}}$
($\overline{\alpha}=(\alpha_{n},\alpha_{n-1},\ldots,\alpha_{0})$), there exist
a set of $c_{\overline{\alpha}}\in\mathbb{C}$ such that
$\sum_{\overline{\alpha}}c_{\overline{\alpha}}|H^{\overline{\alpha}})=\mathbb{I}$
and $\sum_{\overline{\alpha}}|c_{\overline{\alpha}}|^{2}=1$, one can construct
a morphism $\overline{\alpha}\rightarrow|H^{\overline{\alpha}})$ which is a
history fiber over the sequence of projective operators.

In this approach, each fiber $F_{\alpha}$ is a chain of states that the system
can reach following the sequence $\alpha$ from an initial pre-selected state.

With this formal divagation about the entangled histories, field histories and
their connection with observability, readers are invited to reach out to the
theory of an internal observer [43] proposing hierarchical observability and
emphasizing profound role of observers in creation of reality. These
topological studies will be extended in the paper [44] about homotopy spaces
and their applications to the theory of higher-order quantum histories.

The fundamental property of multipartite spatial quantum entanglement is its
monogamy. This property states that for the case of tripartite system ABC,
maximal entanglement of the pair AB excludes its entanglement with the third
party, i.e. if $\rho_{AB}=|\Psi^{+}\rangle\langle\Psi^{+}|$, then any
extension of this state is of the form
$\rho_{ABC}=|\Psi^{+}\rangle\langle\Psi^{+}|\otimes|\Psi\rangle\langle\Psi|$.
For the temporal correlations, it seems that this property does not hold,
especially when one considers statistical distribution of measurement results
[46, 37]. Yet, what is obvious in the spatial case does not have mere
analogies in the temporal case. It was proved [39] that a particular history
can be monogamous but further we will discuss how temporal correlations can
lead to non-monogamous results for bundles of histories with which we tackle
during the measurement process. This subtlety is rather a sign of a deeper
nature of quantum processes which can keep their consistency for particular
instances, yet leads to quite counter-intuitive results for their ensembles.

Suppose we have two non-equivalent multi-time entangled histories of an
evolving qubit through times ${t_{4}>t_{3}>t_{2}>t_{1}}$ for which we consider
the past effect of the measurement at time $t_{4}$:

$\displaystyle|H_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[|0)\odot|0)\odot|0)\odot|0)+|1)\odot|1)\odot|1)\odot|1)]$
$\displaystyle|H_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|0)\odot[|0)\odot|0)\odot|0)+|1)\odot|1)\odot|1)]$
(8)

The history $|H_{1})$ can be perceived as a superposition of two histories on
times ${t_{4},t_{3},t_{2},t_{1}}$. If one measures this evolution at time
$t_{4}$ with dichotomic projective observables $P_{0}=|0\rangle\langle 0|$ and
$P_{1}=|1\rangle\langle 1|$, we can conclude that the state was with
probability $p_{0}=\frac{1}{2}$ in a history $|H_{10})=|0)\odot|0)\odot|0)$ at
previous times and with probability $p_{1}=\frac{1}{2}$ in a history
$|H_{11})=|1)\odot|1)\odot|1)$. Alternatively, one can consider an ensemble of
history states $\\{\\{p_{0},|H_{10})\\},\\{p_{1},|H_{11})\\}\\}$, i.e. half of
the qubits evolving trivially in a history $|H_{10})$ and half in $|H_{11})$
through times ${t_{3},t_{2},t_{1}}$ which can be represented by a history
super-operator $\rho_{H}=\frac{1}{2}(|H_{10})(H_{10}|+|H_{11})(H_{11}|)$. This
evolution is different for the history $|H_{2})$. If one performs the same
measurements at time $t_{4}$, then we get an entangled history through times
${t_{3},t_{2},t_{1}}$ for the projective measurement $P_{0}$ at time $t_{4}$.
Thus, physically we can propose the concept of the probabilistic mixture of
histories:

######  Definition III.2.

A mixed history state is defined as a positive super-operator acting on a
history state space:

$\rho_{hist}=\sum_{i}p_{i}|H_{i})(H_{i}|$ (9)

where $Tr\rho_{hist}=1$, $\sum_{i}p_{i}=1$ and $\forall_{i}1>p_{i}\geq 0$.

This mixture of histories can be naturally associated with an ensemble of
histories $\\{p_{i},|H_{i})\\}$ that can be understood as a mixture of
sections over the history bundles. Following, we consider an example of a spin
particle traversing two paths to check a future influence of the measurement
at time $t_{1}$:

Example 1. Imagine a spin-$\frac{1}{2}$ particle at three times
$\\{t_{3},t_{2},t_{1}\\}$ evolving trivially by $\mathcal{T}=\mathbb{I}$ with
a family of entangled histories:

$\displaystyle|H^{1})$ $\displaystyle=$
$\displaystyle\sqrt{2}([z^{+}]\odot[x^{+}]\odot[z^{+}]+[z^{-}]\odot[x^{-}]\odot[z^{+}])$
$\displaystyle|H^{2})$ $\displaystyle=$
$\displaystyle\sqrt{2}([z^{-}]\odot[x^{+}]\odot[z^{+}]+[z^{+}]\odot[x^{-}]\odot[z^{+}])$
$\displaystyle|H^{3})$ $\displaystyle=$
$\displaystyle\sqrt{2}([z^{+}]\odot[x^{+}]\odot[z^{-}]+[z^{-}]\odot[x^{-}]\odot[z^{-}])$
$\displaystyle|H^{4})$ $\displaystyle=$
$\displaystyle\sqrt{2}([z^{-}]\odot[x^{+}]\odot[z^{-}]+[z^{+}]\odot[x^{-}]\odot[z^{-}])$

If we consider a state
$|\Phi)=\frac{1}{\sqrt{2}}|H^{1})+\frac{1}{\sqrt{2}}|H^{2})$, then a particle,
measured at time $t_{1}$ and having a spin up in a direction $z^{+}$, can
evolve within the history $|H^{1})$ with probability $P(|H^{1}))=\frac{1}{2}$
and be in the history $|H^{2})$ with probability $P(|H^{2}))=\frac{1}{2}$. It
is also interesting to observe that
$|\Phi)=\frac{1}{\sqrt{2}}|H^{1})+\frac{1}{\sqrt{2}}|H^{2})=[z^{+}]\odot[z^{+}]\odot[z^{+}]$
where $[z^{+}]=[|z^{+}\rangle\langle z^{+}|]$.
Noteworthily, one can also find in the space of histories
$\mathcal{S}=span\\{|H^{1}),|H^{2}),|H^{3}),|H^{4})\\}$ the following temporal
GHZ-like vector [23] (normalized for $|\alpha|^{2}+|\beta|^{2}=1$):

$\displaystyle|\tau GHZ)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\sqrt{2}}|H^{1})+\frac{\alpha}{\sqrt{2}}|H^{2})+\frac{\beta}{\sqrt{2}}|H^{3})+\frac{\beta}{\sqrt{2}}|H^{4})$
$\displaystyle=$
$\displaystyle\alpha[z^{+}]\odot[z^{+}]\odot[z^{+}]+\beta[z^{-}]\odot[z^{-}]\odot[z^{-}]$

Let us consider now a quantum history displaying temporal entanglement at
times $\\{t_{2},t_{1}\\}$:

$|H)=\alpha|\phi_{3,2})\odot(|\phi_{2,1})\odot|\phi_{1,1})+|\phi_{2,2})\odot|\phi_{1,2}))\odot|\phi_{0})$
(12)

This particular history can be realized by placement of a detector at time
$t_{3}$ which detects a state $\rho=|\phi_{3,2}\rangle\langle\phi_{3,2}|$ and
displays quantum entanglement in time for times $\\{t_{2},t_{1}\\}$:

$|H_{t_{2},t_{1}})=\alpha(|\phi_{2,1})\odot|\phi_{1,1})+|\phi_{2,2})\odot|\phi_{1,2}))$
(13)

Interestingly, this entangled history at times $\\{t_{2},t_{1}\\}$ cannot be
derived from the following temporal version of a GHZ-state manifesting
monogamy of temporal entanglement for particular histories [39]:

$|\widetilde{H})=\alpha(|\phi_{3,1})\odot|\phi_{2,1})\odot|\phi_{1,1})+|\phi_{3,2})\odot|\phi_{2,2})\odot|\phi_{1,2}))\odot|\phi_{0})$
(14)

Let us observe that the reduced component of this history
$|\phi_{3,1})\odot|\phi_{1,1})$ is correlated with $|\phi_{2,1})$ and not with
$|\phi_{2,2})$. Thus, reduction of $|\widetilde{H})$ over times $t_{2}$ and
$t_{0}$ is not a complex superposition of histories but is a probabilistic
mixture as already stated in this section:

$\rho_{t_{1}t_{3}}=|\alpha|^{2}(|\phi_{3,1}\phi_{1,1})(\phi_{3,1}\phi_{1,1}|+|\phi_{3,2}\phi_{1,2})(\phi_{3,2}\phi_{1,2}|)$
(15)

where $|\phi_{3,1}\phi_{1,1})=|\phi_{3,1})\odot|\phi_{1,1})$ etc. This can be
also formally derived employing a temporal partial trace operator [47] over
time instances:
$\rho_{t_{1}t_{3}}=Tr_{t_{2}t_{0}}|\widetilde{H})(\widetilde{H}|$. This
operator is analogous to spatial tracing out but must maintain consistency in
the evolution—a condition that is not present in the spatial case.

##  IV What entangled histories say about temporal bounds

The violation of local realism [7] and macrorealism [10] by quantum theories
has been studied for many years in experimental setups where measurement data
is tested against Bell inequalities for LR and Leggett-Garg inequalities [11]
for MR. In quantum theories, the former arises as a consequence of non-
classical correlations in space, while the latter arises as a consequence of
non-classicality in dynamic evolution.

In the temporal version of the CHSH-inequality (for Clauser, Horne, Shimony,
and Holt [48]) being a modification of original Leggett-Garg inequalities,
Alice performs measurement at time $t_{1}$ choosing between two dichotomic
observables $\\{A_{1}^{(1)},A_{2}^{(1)}\\}$ and then Bob performs a
measurement at time $t_{2}$ choosing between $\\{B_{1}^{(2)},B_{2}^{(2)}\\}$.
Therefore, the structure of this LGI can be represented as follows [55]:

$S_{AB}\equiv c_{12}+c_{21}+c_{11}-c_{22}\leq 2$ (16)

where $c_{ij}=\langle A_{i}^{(1)},B_{j}^{(2)}\rangle$ stands for the
expectation value of consecutive measurements performed at time $t_{1}$ and
$t_{2}$.

The general LGI for n-time consecutive measurements is given by:

$K_{n}\equiv c_{12}+c_{23}+\ldots+c_{n-1n}-c_{1n}$ (17)

bounded for odd n as $-n\leq K_{n}\leq n-2$ and for even n as $-(n-2)\leq
K_{n}\leq n-2$. These inequalities are naturally violated in quantum realm and
for qubits they reach so-called Lüders bounds, i.e.
$K_{nQ}=ncos(\frac{\pi}{n})$ [49, 50].

Since one can build in a natural way $\mathcal{C}^{*}$-Algebra of history
operators for normalized histories from projective Hilbert spaces equipped
with a well-defined inner product, the quantum Tsirelson bound $2\sqrt{2}$ of
CHSH-inequality specific for spatial correlations holds also for temporal LGI
engaging only the space of entangled histories [39].

Let us consider a temporal setup with measurements
$\mathbb{A}=I\odot\mathbb{A}^{(1)}$ (measurement $\mathbb{A}$ occurring at
time $t_{1}$) and $\mathbb{B}=\mathbb{B}^{(2)}\odot I$. The history with
’injected’ measurements can be represented as
$|\widetilde{H})=\alpha\mathbb{A}\mathbb{B}|H)\mathbb{A}^{\dagger}\mathbb{B}^{\dagger}$
where $\alpha$ stands for a normalization factor. History observables are
history state operators which are naturally Hermitian and their eigenvectors
can generate a consistent history family[23]. As an example, we can consider
spin $\frac{1}{2}$-particle with a history inducing evolution
$|\psi(t_{1})\rangle\rightarrow|\psi(t_{2})\rangle$ on which we act with
$\sigma_{y}\odot\sigma_{x}$ operation. This step results with a new effective
history:

$|\widetilde{H})=\alpha\sigma_{y}[\psi(t_{2})]\sigma_{y}^{\dagger}\odot\sigma_{x}[\psi(t_{1})]\sigma_{x}^{\dagger}$
(18)

For an observable $A=\sum_{i}a_{i}|H_{i})(H_{i}|$, its measurement on a
history $|H)$ generates an expectation value $\langle A\rangle=Tr(A|H)(H|)$
(i.e. the result $a_{i}$ is achieved with probability $|(H|H_{i})|^{2}$) in
analogy to the spatial case. Thus, one achieves history $|\widetilde{H})$ as a
realized history with measurements and the expectation value of the history
observable $\langle A\rangle$. It is worth mentioning that $|\widetilde{H})$
and $|H)$ are both compatible histories, i.e. related by a linear
transformation.

For temporal correlations measurements can lead to counter-intuitive results
which do not occur for spatial quantum resources. Let us reexamine the case of
$GHZ$ states firstly shared as a spatial system of three entangled qubits
among Alice, Bob and Charlie:
$|\Psi_{ABC}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$ which
obviously leads to a separable state for any pair from this system, e.g.
$\rho_{AB}=\frac{1}{2}(|00\rangle\langle 00|+|11\rangle\langle 11|)$. Assume
further that they can choose from dichotomic projective observables:
$P_{0}=|0\rangle\langle 0|$ or $P_{1}=|1\rangle\langle 1|$, then in this
multipartite case any pair cannot identify alone without the third party that
they are part of the more complex entangled system. In the temporal case, the
situation is quite the opposite when considering the temporal version of the
GHZ-state, highlighting a qualitative distinction between spatial and temporal
resources. Alice, Bob, and Charlie, each possessing instances of the same
system but at different points in time, can independently detect temporal non-
locality within each pair.

When we measure an average value of the aforementioned Bell-like temporal
inequality:

$\langle S_{\tau AB}\rangle=\langle
A_{1}B_{1}+A_{1}B_{2}+A_{2}B_{1}-A_{2}B_{2}\rangle$ (19)

we consider an ensemble of systems from which each quarter is measured against
the observables $A_{i}B_{j}$. It is easy to observe that with a choice of
observables: $A_{1}=Z,A_{2}=(Z+X)/\sqrt{2},B_{1}=Z,B_{2}=(Z-X)/\sqrt{2}$ we
get effectively the average value: $\langle S_{\tau AB}\rangle=\sqrt{2}\langle
XX+ZZ\rangle$. Since one gets $\langle XX\rangle=\langle ZZ\rangle=1$, the
Tsirelson maximum is saturated. However, what is important in this simple
example is that consecutive measurements of both X and Z leave the system in
the same eigenstate for any number of time steps. As an immediate
implications, one gets violation of monogamous Bell-like inequalities in space
[52]:

$S_{AB}+S_{BC}\nleq 4$ (20)

since for the temporal tripartite system $ABC$ (B and C being instances of A
at consecutive times) we get saturation for the AB pair and for the BC:

$S_{\tau AB}+S_{\tau BC}=4\sqrt{2}$ (21)

This limit cannot be achieved through spatial correlations [52]. In spatial
correlations, monogamy relations between the strengths of violations of Bell’s
inequalities are derived from the no-signaling condition. However, when it
comes to temporal correlations, the situation differs because temporal
correlations signal towards the future, causing the no-signaling condition to
be violated, which, in turn, results in this inequality violation.

The fundamental aspect of generating such averaged Bell-like inequalities lies
in our operation with a bundle of different histories. These histories are
steered by various collections of measurements, all beginning with the same
initial pre-selected state and concluding with the same post-selected final
state for the entire bundle. However, these histories involve different
intermediary steps, as illustrated in the example with XXX and ZZZ quantum
operations mentioned above. It was also observed in [55] that in the case of a
multi-point temporal correlation function, for measurements performed at m
instances of time $t_{1},\ldots,t_{m}$, the function is decomposable into a
product of two-fold temporal correlations. However, the temporal correlation
function is understood as the average over many runs of the sequence of
measurements. Thus, this result applies again to bundles of different
histories.

We can look at the problem of bounding temporal correlations also by prism of
the two-state vector formalism which is isomorphic to the entangled histories
[45]. The correlations can be described by the probabilistic boxes in non-
signalling theory. The box is shared between two parties who give the input
setting $\\{x,y\\}$ of the measuring devices and get the outputs $\\{a,b\\}$
with probability $p(ab|xy)$ being an entry of the join probability
distribution matrix $P(ab|xy)=[p(ab|xy)]$. All entries of this matrix meet the
non-negativity condition ($p(ab|xy)\geq 0$) and are
normalized:$\forall_{x,y}\sum_{a,b}p(ab|xy)=1$ and the no-signalling condition
imposed on the quantum correlations by the special relativity constraints: the
marginals $p(a|x)$ and $p(b|y)$ are independent of settings y and x
respectively, i.e. $\forall_{y,a,x}p(a|x)=\sum_{b}p(ab|xy)$ and
$\forall_{x,b,y}p(b|y)=\sum_{a}p(ab|xy)$. Then the Aharanov-Bergmann-Lebowitz
(ABL) formula (4) delivers a method for calculation of the measurements
probability in between the initial time with the pre-selected state and the
post-selected state at the final time of the analyzed quantum process.

In the case of series of X and Z measurements injected in the histories
considered in this section we get the following example of probability
distribution with assumption that at times $t_{1}$ and $t_{2}$ the X
observable is chosen and we get $|\uparrow_{x}\rangle$ results in both times:

$p(\uparrow_{x}\uparrow_{x}|XX)=\frac{|\langle\Phi|\uparrow_{x}\rangle\langle\uparrow_{x}||\uparrow_{x}\rangle\langle\uparrow_{x}|\Psi\rangle|^{2}}{\sum_{ab}p(ab|XX)}$
(22)

This is an operational method for generation of the whole probability
distribution matrix. However, we should note that these experiments start with
the same initial and final states but with different intermediate steps, thus,
leading to a bundle of histories at times ${t_{0},t_{1},t_{2},t_{3}}$ (Fig.
1):

$|H_{abxy})\sim p(ab|XY)$ (23)

which can lead to violation of spatial quantum bounds on Bell-like
inequalities.

Figure 1: A bundle o histories at times
${\tau_{0},\tau_{1},\tau_{2},\tau_{3}}$ with a pre-selected state
$|\Psi\rangle$ and post-selected state $|\Phi\rangle$. Exemplary histories
$|H_{0})$ and $|H_{1})$ with incorporated measurement results of X. This
bundle contributes to violation of monogamous temporal Bell-like inequalities
engaging different history states with the same initial and final states.

We can then formulate generic bounds on temporal correlations of qubits in
quantum theories (this result can be generalized to the qudits’ case).

Let us assume that the quantum process occurs n times and that for any two
times $\\{t_{i},t_{i+1}\\}$ the quantum bound $\mathcal{Q}$ limits the
temporal Bell-like functional on the matrix of probability distributions, i.e.
$B_{\tau}(A_{i},A_{i+1})=\mathcal{F}([p_{t_{i},t_{i+1}}(ab|xy)])\leq\mathcal{Q}$
with the association of histories $|H_{abxy})$, then the process saturating
the chain for such n-steps can be designed in such a way that each pair of
times is a replication of two consecutive times, i.e.
$\forall_{i}[p_{t_{i},t_{i+1}}]=[p_{t_{i+2},t_{i+3}}]$. This process is
equivalent logically to a loop $t_{0}\rightarrow t_{1}\rightarrow
t_{0}\ldots\rightarrow t_{1}$. In consequence, we get the following quantum
bound :

$\sum_{i=0}^{n-1}B_{\tau}(A_{i},A_{i+1})\leq\mathcal{Q}n$ (24)

that can be saturated to its maximal value. As an implication for the LGIs one
gets the following quantum bound:

$\sum_{i=1}^{n}S_{LGI}(A_{0},A_{i})\leq 2\sqrt{2}n$ (25)

which can be saturated to the maximal value in quantum world and which
violates the spatial monogamy relations $\sum_{i=1}^{n}B(A_{0},A_{i})\leq 2n$
(for $n\geq 2$) [52].

We can conclude this section with a remark that a particular entangled history
is monogamous, in similarity to quantum spatial states, but for a bundle of
different histories with the same pre-selected and post-selected states one
can get violation of monogamy. This is a novel feature of temporal
correlations not paralleled in spatial domain.

##  V Conclusions

In this study, we have ventured into the burgeoning domain of quantum state
representation using topological paradigms, specifically focusing on quantum
history bundles constructed upon temporal manifolds. This approach has allowed
us to shed light on the mechanism that leads to the violation of monogamous
temporal Bell-like inequalities, a topic that has garnered considerable
attention in recent research.

With a growing interest in representation of quantum states as topological
objects, we consider quantum history bundles based on the temporal manifold
and show the source of violation of monogamous temporal Bell- like
inequalities. We introduce definitions for the mixture of quantum histories
and consider their entanglement as sections over the Hilbert vector bundles.
Our analysis reveals that, in contrast to spatial quantum correlations where
ensembles consist of identical copies of multipartite states, temporal
correlations involve ensembles of varying temporal histories. This fundamental
distinction underscores a qualitative difference in the results obtained for
spatial and temporal Bell-like inequalities, marking a significant advancement
in our understanding of quantum phenomena.

Further advancing our theoretical understanding, we have utilized the
Tsirelson bound to derive a quantum limit for multi-time Bell-like
inequalities. This derivation, based on the entangled histories approach,
offers a novel perspective on temporal quantum bounds.

The field of quantum state representation through topological methods presents
numerous opportunities for further exploration. Specific avenues for future
research could include the practical applications in quantum computing and
quantum information theory.

In conclusion, our study not only provides a deeper understanding of the
temporal aspects of quantum states but also highlights the importance of
considering topological, alongside statistical, aspects of quantum processes.
The insights gained from this research pave the way for a more comprehensive
and nuanced understanding of quantum mechanics, and we anticipate that they
will stimulate further research in this exciting and rapidly evolving field.

##  VI Acknowledgments

Acknowledgments to Xerxes Arsiwalla, David Chester, Eliahu Cohen, Marek
Czachor and Pawel Horodecki for discussions about temporal correlations.

## References

  * [1] G. O. Myhr, N. Lütkenhaus, Phys. Rev. A 79, 062307 (2009).
  * [2] G. O. Myhr et. al., Phys. Rev. A 79, 042329 (2009).
  * [3] T. Moroder, N. Lütkenhaus, Phys. Rev. A 74, 052301 (2006).
  * [4] Jianxin Chen et. al, Phys. Rev. A 90, 032318 (2014).
  * [5] I. Devetak and A. Winter, Proc. R. Soc. Lond. A 461, 207 (2005).
  * [6] K. Li, A. Winter, Found. Phys. 48 , 910 (2018).
  * [7] J. S. Bell, Physics 1 (3), 195 (1964).
  * [8] A. Aspect, J. Dalibard, G. Roger, Phys. Rev. Lett. 49, 1804 (1982).
  * [9] S. J. Freedman, J. F. Clauser, Phys. Rev. Lett. 28, 938 (1972).
  * [10] A. J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002).
  * [11] A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985).
  * [12] Chu, W.-J. et al., Sci. Rep. 6, 28351 (2016).
  * [13] C. Robens et al., Phys. Rev. A 5, 011003 (2015).
  * [14] A. Asadian, C. Brukner and P. Rabl, Phys. Rev. Lett. 112, 190402 (2014).
  * [15] H. Katiyar et al., Phys. Rev. A 87, 052102 (2013).
  * [16] A. M. Souza, I. S. Oliveira and R. S. Sarthour, New J. Phys. 13, 053023 (2011).
  * [17] Y. Aharonov, D. Albert, and S. D’Amato, Phys. Rev. D 32, 1975 (1985).
  * [18] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, Phys. Rev. A 79, 052110 (2009)
  * [19] Y. Aharonov, P. G. Bergmann, and J.L. Lebowitz, Phys. Rev. 134, 1410 (1964).
  * [20] Y. Aharonov and L. Vaidman, Phys. Rev. A 41, 11-20 (1990).
  * [21] Y. Aharonov and L. Vaidman, The two-state vector formalism of quantum mechanics in Time in Quantum Mechanics (eds. J. G. Muga et al.) 369-412 (Springer, 2002).
  * [22] Y. Aharonov, E. Cohen and T. Landsberger, Entropy 19, 111 (2017).
  * [23] J. Cotler and F. Wilczek, Phys. Scr. 168, 014004 (2016).
  * [24] J. Fitzsimons, J. Jones, V. Vedral, Sci. Rep. 5 18281 (2015).
  * [25] C. Marletto et al., Sci. Adv. 7, 4742 (2021).
  * [26] T. Zhang et al., New J. Phys. 22 023029 (2020).
  * [27] Y. Aharonov and D. Rohrlich. Quantum paradoxes: quantum theory for the perplexed, Wiley, Weinheim (2008).
  * [28] Y. Aharonov, F. Colombo, S. Popescu, I. Sabadini, D.C. Struppa, and J. Tollaksen, Proc. Natl. Acad. Sci. 113, 532-535 (2016).
  * [29] Y. Aharonov, E. Cohen, A. Landau, and A.C. Elitzur, Sci. Rep. 7, 531 (2017).
  * [30] S. Lloyd, L. Maccone, R. Garcia-Patron, V. Giovannetti, and Y. Shikano, Phys. Rev. D 84, 025007 (2011).
  * [31] S. Lloyd, L. Maccone, R. Garcia-Patron, V. Giovannetti, Y. Shikano, S. Pirandola, L.A. Rozema, A. Darabi, Y. Soudagar, L.K. Shalm, and A.M. Steinberg, Phys. Rev. Lett. 106, 040403 (2011).
  * [32] Y. Aharonov, S. Popescu, and J. Tollaksen, in Quantum theory: a two-time success story, ed. by D.C. Struppa and J. Tollaksen, Springer, Milano (2014), pp. 21-36.
  * [33] R. Silva, Y. Guryanova, N. Brunner, N. Linden, A.J. Short, and S. Popescu, Phys. Rev. A 89, 012121 (2014).
  * [34] R. Silva, Y. Guryanova, A.J. Short, P. Skrzypczyk, N. Brunner, and S. Popescu, Preprint quant-ph/1701.08638.
  * [35] J. Cotler et al., Ann. Phys. 387, 334 (2017).
  * [36] E. Cohen and M. Nowakowski, Phys. Rev. D 97, 088501 (2018).
  * [37] C. Marletto et al., Entropy 22, 228 (2020).
  * [38] C. Marletto et al., Nat. Commun. 10, 182 (2019).
  * [39] M. Nowakowski, AIP Conf. Proc. 1841, 020007 (2017).
  * [40] M. H. Freedman, M. B. Hastings, Entanglement of Sections, Examples Looking for a Theory, Preprint quant-ph/2304.01072 (2023).
  * [41] H. Sati, U. Schreiber, Entanglement of Sections: The pushout of entangled and parameterized quantum information, Preprint quant-ph/2309.07245v2 (2023).
  * [42] D. Gottesman and L. L. Zhang, Fibre bundle framework for unitary quantum fault tolerance, Preprint quantu-ph/1309.7062 (2013).
  * [43] M. Nowakowski, Towards Physics of Internal Observers: Exploring the Roles of External and Internal Observers, Preprint quant-ph/2304.01677 (2023).
  * [44] M. Nowakowski, X. Arsiwalla, D. Chester, Representation of the internal observer space as a higher homotopy space, In preparation.
  * [45] M. Nowakowski, E. Cohen, P. Horodecki, Phys. Rev. A 98, 032312 (2018).
  * [46] Ringbauer, M., Costa, F., Goggin, M.E. et al., npj Quantum Inf 4, 37 (2018).
  * [47] For a history $|Y_{t_{n}\ldots t_{0}})=F_{n}\odot F_{n-1}\odot\ldots\odot F_{1}\odot F_{0}$ acting on a space $\mathcal{H}=\mathcal{H}_{t_{n}}\otimes\dots\otimes\mathcal{H}_{t_{0}}$, a partial trace over times $\\{t_{j}\ldots t_{i+1}t_{i}\\}$ $(j\geq i)$ is: $Tr_{t_{j}\ldots t_{i+1}t_{i}}|Y_{t_{n}\ldots t_{0}})(Y_{t_{n}\ldots t_{0}}|=\sum_{k=1}^{\dim\mathcal{F}}(e_{k}|Y_{t_{n}\ldots t_{0}})(Y_{t_{n}\ldots t_{0}}|e_{k})$ where $\mathcal{F}=\\{|e_{k})\\}$ creates an orthonormal consistent family of histories on times $\\{t_{j}\ldots t_{i+1}t_{i}\\}$ and the strong consistency condition for partial histories holds for base histories, i.e. $(e_{i}|e_{j})=Tr[K(|e_{i}))^{{\dagger}}K(|e_{j}))]=\delta_{ij}$.
  * [48] J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23 (15), 880–4 (1969).
  * [49] C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, Phys. Rev. Lett. 111, 020403 (2013).
  * [50] C. Vitagliano, C. Budroni, Phys. Rev. A 107, 040101 (2023).
  * [51] B. S. Cirel’son, Lett. Math. Phys. 4, 93-100 (1980).
  * [52] M. Pawlowski, C. Brukner, Phys. Rev. Lett. 102, 030403 (2009).
  * [53] T. Fritz, New J. Phys. 12, 083055 (2010).
  * [54] C. Budroni, T. Moroder, M. Kleinmann, O. Gühne, Phys. Rev. Lett. 111, 020403 (2013).
  * [55] C. Brukner, S. Taylor, S. Cheung, V. Vedral, Quantum Entanglement in Time, Preprint quant-ph/0402127, (2004).