# Phase Transitions and Superconductivity in Ternary Hydride Li2SiH6 at High
Pressures

Sixuan Wu  School of Science, Nanjing University of Posts and
Telecommunications, Nanjing 210023, China  Bin Li<EMAIL_ADDRESS>New
Energy Technology Engineering Laboratory of Jiangsu Province and School of
Science, Nanjing University of Posts and Telecommunications, Nanjing 210023,
China  Zhi Chen  State Grid Zhenjiang Power Supply Company, Zhenjiang 212000,
China  Yu Hou  College of Electronic and Optical Engineering, Nanjing
University of Posts and Telecommunications, Nanjing 210023, China  Yan Bai
College of Electronic and Optical Engineering, Nanjing University of Posts and
Telecommunications, Nanjing 210023, China  Xiaofeng Hao  College of Electronic
and Optical Engineering, Nanjing University of Posts and Telecommunications,
Nanjing 210023, China  Yeqian Yang  College of Electronic and Optical
Engineering, Nanjing University of Posts and Telecommunications, Nanjing
210023, China  Shengli Liu  New Energy Technology Engineering Laboratory of
Jiangsu Province and School of Science, Nanjing University of Posts and
Telecommunications, Nanjing 210023, China  Jie Cheng  New Energy Technology
Engineering Laboratory of Jiangsu Province and School of Science, Nanjing
University of Posts and Telecommunications, Nanjing 210023, China  Zhixiang
Shi  School of Physics, Southeast University, Nanjing 211189, China

###### Abstract

We predicted a new ternary hydride Li2SiH6 at high pressures. A systematic
structure search in Li2SiH6 compound reveals novel stable phases with
intriguing electronic and phonon properties. It is found that Li2SiH6 is
dynamically stable from ambient pressure up to 400 GPa with three novel
phases: P312, P$\bar{3}$, and P$\bar{6}$2m. The calculation of electron-phonon
coupling combined with Bardeen-Cooper-Schrieffer’s argument indicates that
this compound may be a candidate for high $T_{c}$ superconductors under high
pressures. In particular, the maximum $T_{c}$ of $P\bar{6}2m$-Li2SiH6 at 400
GPa reaches 56 K. These findings may pave the way for obtaining room
temperature superconductors in dense hydrogen-rich compounds.

##  I Introduction

The search for high-temperature superconducting materials has always been a
hot topic in the field of condensed matter physics. According to the
traditional Bardeen-Cooper-Schrieffer (BCS) theory[1], the superconducting
transition temperature is directly proportional to the Debye temperature, and
the Debye temperature is inversely proportional to the mass, so lighter
elements may have higher superconducting transition temperatures, such as
hydrogen. However, solid hydrogen is an insulating molecular crystal under
ambient pressure, and there are strong covalent bonds in hydrogen molecules,
so it is hard to obtain superconducting hydrogen under ambient pressure. In
order to achieve superconductivity, conditions such as external pressure are
needed. As a basic thermodynamic variable, pressure can change the distance
between atoms of matter, so that atoms can be rearranged, and the crystal and
electronic structure can be modulated. High pressure can very effectively
shorten the distance between atoms, increase the overlap of adjacent electron
orbits, and then change the interaction and electronic structure between
atoms/molecules, forming high-pressure new phases with new structures and
properties that are difficult to form under conventional conditions.
Generally, in a sufficiently high pressure environment, the band gap will be
narrowed, and the energy bands will overlap, which can transform the non-
metallic state into the metallic state. Wigner and Huntington proposed that
insulating hydrogen molecules can be transformed into metallic hydrogen under
high pressure, showing a metallic state[2]. Ashcroft proposed chemical
preloading[3], that is, non-hydrogen elements in hydrogen-rich compounds have
an interaction effect on hydrogen elements in the lattice. Hydrogen-rich
compounds can be metallized at a lower pressure than pure hydrogen, and such
hydrogen-rich compounds may be potential room temperature superconductors.
Therefore, hydrogen-dominated metal alloys may be potential high-temperature
superconducting materials under high pressure, because of the high Debye
temperature. In recent years, hydrogen sulfide (H3S), discovered under high
pressure, set a high-temperature superconductivity record due to its
superconducting transition temperature of 203 K [4, 5], and subsequent
theoretical studies and experimental demonstrations found that lanthanum
hydride compounds are at a high pressure of 170 GPa, with a superconducting
transition temperature of 250 K [6, 7]. This is the highest superconducting
transition temperature confirmed by experiments on binary hydrides so far,
which has further promoted the research on the superconductivity of hydrogen-
rich compounds.

In the ”hydride rush” of searching high-temperature superconductors, the
theoretical research of hydrides has stepped into the forefront. First-
principles calculations based on density functional theory have given a series
of superconductivity predictions about hydrogen-rich compounds, and proposed a
new family of superconducting hydrides with a unique cage structure[8, 9, 10].
The metal atoms such as calcium or lanthanum are located in the center of the
hydrogen cage. In these high $T_{c}$ superconducting hydrides, doping elements
metalize hydrogen at relatively lower pressures, through ”chemical pre-
compression”. Based on the characteristics of ”chemical pre-compression”, some
ternary hydrogen storage materials are considered as candidates for high-
temperature superconductors. Ternary metal hydrides are a convenient and
valuable system for studying the metallization and superconductivity of metal
hydrides because they can be synthesized under mild conditions and recovered
under ambient pressure[11]. Some ternary hydrides have been found in
theoretical and experimental work, e. g. Fe2SH3 (Tc = 0.3 K, 173 GPa)[12],
YS4H4 (Tc = 20 K, 200 GPa)[13] and YSH5 (Tc = 0.7 K, 300 GPa)[14], BaReH9 (Tc
= 7 K above 100 GPa)[15]. A recent experimental work also reported a room-
temperature superconductivity with Tc of 288 K in the C-S-H system at 267
GPa[16].

Silicon and carbon belong to the same group IV elements. It is controversial
that silicon atoms will bond with four H atoms to form SiH4 molecular units in
the Mg-Si-H system similar to MgCH4. However, since the nucleus of silicon is
larger than that of carbon, the strength of the Si-H bond is weaker than that
of the C-H bond. This may promote the dissociation of the Si-H bond, resulting
in a completely different crystal structure, and there will be new changes
under high pressure. A typical example is the MH4 (M = Si[17], Ge[18], and
Sn[19]) system, where the difference in bonding results in greatly different
structures and properties. In addition, recent studies have found that lithium
atom doping helps stabilize the structure of the Au-H system [20], because
under high pressure, the Au and H systems are unstable, but the doping of
lithium has a stabilizing effect, which may originate from the low electron
negative of lithium. Based on first-principles calculations, by adding lithium
to the poor superconducting MgH12, it is found that the superconducting
transition temperature of Li2MgH16 reaches 472 K under 250GPa [21], indicating
that it can be doped with lithium atoms, carrying ultra-high superconducting
transition temperature. Zhang et al. performed structure searching simulations
on the Li-Si-H system at pressure range of 50-350 GPa[22]. LiSi2H9 and LiSiH8
are predicted to become good phonon-mediated superconductors with estimated
$T_{c}$ of 54 and 77 K at 172 and 250 GPa, respectively. Li2SiH6 compound is
predicted with relatively low symmetry (P-1 and Cmcm) and insulating. While
another study on LixXH6 predicted LiPH6 a potential high-temperature
superconductor with a $T_{c}$ of 150-167 K at 200 GPa[23]. Boeri et.al also
revealed a metallic Li2BH6 phase which is thermodynamically stable down to 100
GPa with a critical temperature of $\sim$ 100 K[24]. Thus, the stable phases
and superconductivity of LixXH6 needed further exploration. In this paper, we
focus on Li2SiH6 compound, searching for structures and superconductivity
under high pressures. We found a series of phase transitions and a candidate
high $T_{c}$ superconductors $P\bar{6}2m$-Li2SiH6 with the maximum $T_{c}$ of
56 K. Our results will further explore the superconductivity in the Li-Si-H
system and may help to guide future experiments.

##  II Methods

To determine the high-pressure structures, we used our machine-learning-based
crystal prediction code CRYSTREE [25]. CRYSTREE can learn the structural
optimization procedure and output optimized structure using multi regression
methods including random forest regressor, ridge regressor and ExtraTrees
regressor, et al. [26]. The results of structures are checked by evolutionary
crystal structure prediction method USPEX [27]. The electronic structure
calculations with high accuracy for the stable structures were performed using
the full-potential linearized augmented plane wave (FP$-$LAPW) method
implemented in the WIEN2K code. [28] The Wu-Cohen generalized gradient
approximation (WC-GGA) [29] was applied to the exchange-correlation potential
calculation. The plane-wave cutoff was defined by $RK_{max}=4.0$, where $R$ is
the minimum LAPW sphere radius and $K_{max}$ is the plane-wave vector cutoff.
The phonon calculations were carried out by using a density functional
perturbation theory (DFPT) approach through the Quantum-ESPRESSO program [30].
Ultrasoft pseudopotentials are used in the calculation with the cutoffs chosen
as 60 Ry for the wave functions and 600 Ry for the charge density. The
electronic self-consistent calculation was performed over a $16\times 16\times
16$ $k$-point mesh. Dynamical matrices and the electron-phonon couplings were
calculated on a $4\times 4\times 4$ $q$-point grid. A dense $24\times 24\times
24$ grid was used for evaluating an accurate electron-phonon interaction
matrix.

##  III Results

Figure 1: The enthalpies per formula unit of P312 and P$\bar{3}$ phases as a
function of pressure, referenced to P$\bar{6}$2m phase of Li2SiH6.

The crystal structures of Li2SiH6 are predicted at 0, 100, 200, 300 and 400
GPa, respectively, with system sizes containing up to 4 formula units (f.u.)
per simulation cell. Three energy-competitive phases P312, P$\bar{3}$ and
P$\bar{6}$2m were found through our structure prediction. As shown in figure
1, from ambient pressure up to 20 GPa, the more stable phase is P312 phase
(space group NO.149), then P$\bar{3}$ phase (space group NO.147) becomes a
more stable phase until 91 GPa, following a phase transition to P$\bar{6}$2m
phase (space group NO.189) up to the highest pressure we studied of 400 GPa.
Li2SiH6 undergoes a series of phase transitions from trigonal symmetry (P312
and P$\bar{3}$ phases) to hexagonal symmetry (P$\bar{6}$2m phase).

Figure 2: Predicted crystal structure of Li2SiH6 phases. Lithium atoms are
represented by large green balls, silicon atoms are represented by blue balls,
and hydrogen atoms are represented by small pink balls. The crystal structure
of the predicted phase is generated using VESTA software. [31]

All of the crystal structures of the stable Li2SiH6 compounds at 0 GPa, 50
GPa, and 300 GPa are shown in Fig. 2. The low-pressure phases P312 and
P$\bar{3}$ contain Si-H6 units in the cell, with lithium atoms inserted
between these units. For high-pressure P$\bar{6}$2m phase, there are Si-H9
units with discrete lithium atoms. With the increasing pressure, more hydrogen
atoms trend to condensate around silicon atom. This is a prerequisite for a
higher superconducting transition temperature in hydrides. In P312 and
P$\bar{3}$ phases, Si-H6 units construct regular octahedrons, in which Si-H
bond in P312 phase is 1.599 Å and the length of Si-H bonds of P$\bar{3}$ phase
is 1.472 Å. When the pressure further increases, the SiHn units change from
the original octahedrons to 14-sided polyhedrons, and more hydrogen atoms
condensate and bonded with central Si atom. Each Si atom in P$\bar{6}$2m
structure is connected to nine H atoms with Si-H bonds ranging from 1.417 Å to
1.480 Å. A Si-Si bond with a bond length of 2.014 Å is also formed in this
phase at 300 GPa.

Figure 3: Electronic band structure for each phase. (a) P312 phase at 0 GPa,
(b) P$\bar{3}$ phase at 50 GPa, (c) P$\bar{6}$2m at 300 GPa. (d) P$\bar{6}$2m
at 400 GPa. The Fermi level is set to zero. The red, green, yellow, and cyan
color represent the contributions of H-s, Si-p, Si-s and Li-s orbitals to the
density of states.

Figure 3 shows the calculated electronic band structure of Li2SiH6 for P312,
P$\bar{3}$ and P$\bar{6}$2m phases. Band structure for P312 phase at 0 GPa are
shown in Fig. 3(a), there is no energy band that crosses the Fermi level
($E_{F}$), which demonstrates P312 phase an insulator at ambient pressure. As
shown in Fig. 3(b), with the increasing pressure, the most stable phase
changes from P312 to P$\bar{3}$. A valence band slightly crossing $E_{F}$ at
50 GPa which shows a weak metal characteristic. P$\bar{6}$2m becomes the most
stable phase at 91 GPa. As shown in figure 3 A hole-type band cross through
the Fermi level between $\Gamma-M$ and $K-\Gamma$ at 300 GPa. When pressure
further increase to 400 GPa, there are one electron-type band and two hole-
type bands crossing the Fermi energy. This may help enhance the electron-
phonon interaction. The total and projected density of states (DOS) are shown
in figure 4. In low-pressure phases P312 at ambient pressure and P$\bar{3}$ at
50 GPa, the band gap are around 3 eV and 1.2 eV, respectively. While in high-
pressure phase P$\bar{6}$2m, the DOS at the Fermi level $E_{F}$ show
remarkable mentality with 0.17 eV-1/atom at 300 GPa and 0.15 eV-1/atom at 400
GPa. The contribution to the DOS of each element shows that the orbitals are
highly hybridized.

Figure 4: The density of states of P$\bar{6}$2m, P$\bar{3}$ and P312 phases at
different pressures.

Figure 5 show the phonon spectra and partial phonon density of states (PDOS)
of Li2SiH6 under (a) P312 at 0 GPa, (b) P$\bar{3}$ at 50 GPa, (c) P$\bar{6}$2m
at 300 GPa and (d) P$\bar{6}$2m at 400 GPa, respectively. No soft phonon mode
can be observed in all phases at the pressures we studied, indicating that
these structures are dynamically stable. We also preformed symmetry analysis
on the phonon modes. For P312 phase, its point group is $D_{3}$. The
irreducible representation can be written as $\Gamma=2A^{\prime}_{1}\bigoplus
2A^{\prime}_{2}\bigoplus A^{\prime\prime}_{1}\bigoplus
4A^{\prime\prime}_{2}\bigoplus 7E^{\prime}\bigoplus 2E^{\prime\prime}$, with
$3A^{\prime\prime}_{2}\bigoplus 6E^{\prime}$ infrared active modes and
$2A^{\prime}_{1}\bigoplus 6E^{\prime}\bigoplus 2E^{\prime\prime}$ Raman active
modes. For P$\bar{3}$ phase, its point group is $C_{3i}$, The irreducible
representation can be written as $\Gamma=4A_{g}\bigoplus 5A_{u}\bigoplus
4^{1}E_{g}\bigoplus 5^{1}E_{u}\bigoplus 4^{2}E_{g}\bigoplus 5^{2}E_{u}$, with
$4A_{u}\bigoplus 4^{1}E_{u}\bigoplus 4^{2}E_{u}$ infrared active modes and
$4A_{g}\bigoplus 4^{1}E_{g}\bigoplus 4^{2}E_{g}$ Raman active modes. For
P$\bar{6}$2m phase, its point group is $D_{3h}$. The irreducible
representation can be written as $\Gamma=2A^{\prime}_{1}\bigoplus
2A^{\prime}_{2}\bigoplus A^{\prime\prime}_{1}\bigoplus
4A^{\prime\prime}_{2}\bigoplus 7E^{\prime}\bigoplus 2E^{\prime\prime}$, with
$3A^{\prime\prime}_{2}\bigoplus 6E^{\prime}$ infrared active modes and
$2A^{\prime}_{1}\bigoplus 6E^{\prime}\bigoplus 2E^{\prime\prime}$ Raman active
modes. The PDOS show that, the vibrations of hydrogen atoms dominate the high-
frequency region, while vibrations of lithium and silicon mainly distribute in
the medium and low-frequency regions, respectively. It is consistent with the
ordering of elements’ masses, the heavier atoms prefer vibrations with lower
frequencies. We also noted that, with the increasing pressure, the maximum
phonon frequency changes from $\sim$ 1700 cm-1 at 0 GPa, to $\sim$ 2300 cm-1
at 50 GPa, then to $\sim$ 2800 cm-1 at 300 GPa and $\sim$ 3000 cm-1 at 400
GPa. It will help to enhance the Debye temperature and logarithmic average
frequency $\omega$log, which is directly proportional to the superconducting
critical temperature.

Figure 5: The phonon dispersion curves and partial phonon density of states of P312, P$\bar{3}$ and P$\bar{6}$2m phases, under different pressures. Table 1: The logarithmical averaged phonon frequency $\omega_{log}$, electron-phonon coupling ($\lambda$), and superconducting $T_{c}$ in Li2SiH6 at different pressures. Structure |  P(GPa) |  $\omega$log (K) |  $\lambda$ |  $T_{c}$ (K) ($\mu$∗=0.10)
---|---|---|---|---
P312-Li2SiH6 |  0 |  784.45 |  0.003 |  0.00
P$\bar{3}$-Li2SiH6 |  50 |  880.68 |  0.23 |  0.01
P$\bar{6}$2m-Li2SiH6 |  100 |  1692.10 |  0.04 |  0.00
|  200 |  2059.12 |  0.07 |  0.00
|  300 |  1134.88 |  0.73 |  43.60
|  400 |  1143.95 |  0.82 |  56.33

TABLE I collects predicted superconducting properties of Li2SiH6 phases. The
calculated logarithmic average phonon frequency $\omega_{log}$, electron-
phonon coupling parameters $\lambda$, and superconducting critical
temperatures $T_{c}$ at various pressures are listed. $T_{c}$ in the range of
100 and 400 GPa rise upon compression. We use the Allen and Dynes modified
McMillan equation [32, 33] to estimate the superconducting temperature $T_{c}$
at different pressures.

$\omega_{log}=exp(\frac{2}{\lambda}\int_{\omega_{min}}^{\omega_{max}}\frac{d\omega}{\omega}\alpha^{2}F(\omega)ln(\omega))$
(1)

The Coulomb pseudopotential $\mu^{*}$ is assumed to be 0.1 for all materials.
Along with the calculated logarithmic average frequency $\omega$log, the
resultant $T_{c}$ of P$\bar{6}$2m-Li2SiH6 reaches maximum value of 56 K at 400
GPa, which is the maximum pressure we studied.

##  IV Conclusions

We searched stable structures of the Li2SiH6 compound in the pressure range
from 0 to 400 GPa using the machine-learning-based crystal structure
prediction algorithm, and predicted new stable phases of P312, P$\bar{3}$ and
P$\bar{6}$2m at different pressures. The maximum $T_{c}$ value of Li2SiH6
phases obtained by first-principles calculations are 56 K in high pressure
P$\bar{6}$2m phase. This study complements the investigation of a series of
high-$T_{c}$ superconducting ternary hydrides.

##  V Acknowledgments

This work is sponsored by the National Key R&D Program of China (Grant No.
2018YFA0704300), the National Natural Science Foundation of China (Grants No.
U1932217, 12175107), open program from the National Lab of Solid State
Microstructures of Nanjing University (Grant No. M32025), and NUPTSF (Grant
No. NY219087, NY220038).

Author contributions: S.X.W wrote the paper with input from all the co-
authors.

Competing financial interests: The authors declare no competing financial
interests.

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# Relativity of spacetime ontology: When correlations in space become
correlata in time

Marek Czachor<EMAIL_ADDRESS>Marcin Nowakowski
<EMAIL_ADDRESS>Faculty of Applied Physics and Mathematics, Gdańsk
University of Technology, 80-952 Gdańsk, Poland  National Quantum Information
Center, 80‑309 Gdańsk, Poland

###### Abstract

Challenging Mermin’s perspective that “correlations have physical reality;
that which they correlate does not” we argue that correlations and correlata
are not fundamentally distinct. These are dual concepts depending on the
tensor product decomposition defining subsystems. Since the same quantum
states may be either entangled or separable, but with respect to alternative
tensor product structures, a spatial correlation in one context can become a
temporal correlatum in another, and vice versa. In consequence, 2-qubit states
invariant under $V\otimes V$ can be either entangled or unentangled, in
conflict with the well known uniqueness theorem about the singlet state, a
fact with possible implications for the quantum measurement theory.

## Introduction

The roles of the observer and the nature of correlations are foundational and
yet often enigmatic both in quantum mechanics and relativity theory.
Traditional perspectives, such as those posited by Mermin, suggest that while
correlations possess physical reality, the elements they correlate do not.
This paper challenges this dichotomy, proposing instead that correlations and
correlata are not inherently distinct but are dual aspects influenced by
observers defining subsystems and their correlations, i.e. defining elements
of reality.

Our inquiry, grounded in standard quantum mechanics, implies that spatial
correlations in one context may transform into temporal correlata in another,
indicating that space-time itself may not be a fundamental level of reality.

To begin with, consider an invertible map $V$ that describes a change of basis
(or a reference frame) in a single-qubit system. A two-qubit state,
$|\psi\rangle=\sum_{\alpha\beta}\psi_{\alpha\beta}|\alpha\rangle\otimes|\beta\rangle,$
is invariant under $V\otimes V$, $(V\otimes V)|\psi\rangle\sim|\psi\rangle,$
if and only if
$|\psi\rangle\sim|0\rangle\otimes|1\rangle-|1\rangle\otimes|0\rangle.$ This
uniqueness theorem is behind various quantum and relativistic constructions,
just to mention spinor reformulation of Einstein-Maxwell equations [1], or
proofs of security of a two-qubit entangled-state quantum cryptography [2].

In particular, if $V\in\textrm{SU}(2)$, $|0_{V}\rangle=V|0\rangle$,
$|1_{V}\rangle=V|1\rangle$, then the Bell-basis singlet state satisfies

$\displaystyle|\Psi_{-}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|0\rangle\otimes|1\rangle-|1\rangle\otimes|0\rangle\big{)}$
(1) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|0_{V}\rangle\otimes|1_{V}\rangle-|1_{V}\rangle\otimes|0_{V}\rangle\big{)},$
(2)

and, up to a phase factor, is the only state with such a property.

Security of a two-qubit quantum cryptography is based on the following
consequence of the uniqueness theorem: Alice and Bob can share singlet-state
correlations if and only if they are isolated from the rest of the universe.
But then, if we replace Alice and Bob by spins of electrons in a helium atom,
how is it that one observes singlet state correlations in atomic physics? The
two electrons must be isolated from the rest of the universe, so apparently
cannot be subject to spectroscopic measurements.

The question brings us to another controversial and general property of
quantum mechanics — the quantum measurement problem [3, 4]. Namely, a system
is described by a state vector $|\psi_{s}\rangle$ if and only if the overall
state of the universe is a tensor product state,
$|\psi_{s}\rangle\otimes|\psi_{r}\rangle$, with $|\psi_{r}\rangle$
representing the rest of the universe. But then the system must be isolated
from anything else and is therefore essentially unobservable, so how is it
that any quantum mechanical prediction based on the Schrödinger equation is
testable at all?

The goal of our paper is to show that the above controversies may be rooted in
a fundamental non-uniqueness of tensor product structures [5, 6, 7, 8, 9, 10,
11]. A two-qubit system is naturally equipped with six coexisting tensor
products $\otimes_{abc}$, indexed by the six permutations of $a,b,c=1,2,3$.
The tensor products then satisfy, in particular,

$\displaystyle|\Psi_{-}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|0\rangle\otimes_{123}|1\rangle-|1\rangle\otimes_{123}|0\rangle\big{)}$
(3) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|0_{V}\rangle\otimes_{123}|1_{V}\rangle-|1_{V}\rangle\otimes_{123}|0_{V}\rangle\big{)}$
(4) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|1\rangle\otimes_{321}\big{(}|1\rangle-|0\rangle\big{)}$
(5) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|1_{V}\rangle\otimes_{321}\big{(}|1_{V}\rangle-|0_{V}\rangle\big{)},$
(6)

for any $V\in\textrm{SU}(2)$. The last equality in (5)–(6) is in apparent
conflict with the discussed uniqueness theorem. However, we will see that (5)
equals (6) for any $V$ in virtue of the theorem, not in spite of it.

Our text may be also treated as belated polemic with David Mermin’s “What is
quantum mechanics trying to tell us?” [12]. In Mermin’s own words, his paper
can be reduced to a single statement: Correlations have physical reality; that
which they correlate does not. “That which they correlate” is termed by Mermin
the correlata (plural form of correlatum). So, equivalently: Correlations have
physical reality; correlata do not.

Our analysis implies that correlations are a form of correlata, but with
respect to a different splitting into subsystems. Accordingly, there is no
fundamental difference between correlata and correlations. If correlata do not
possess physical reality, the same applies to correlations.

We argue as follows. Correlations describe relations between subsystems.
Subsystems are defined in terms of tensor products. In a two-qubit case,
subsystem qubits are defined by the four projectors:
$P_{\alpha}\otimes\mathbb{I}$ and $\mathbb{I}\otimes P_{\alpha}$,
$\alpha=0,1$. However, even in this simple scenario there exist at least three
physically inequivalent tensor products. Depending on our choice of $\otimes$,
physical interpretations of $P_{\alpha}$ change. In some cases an eigenvalue
of $P_{\alpha}$ describes a correlatum, but in the other cases it describes a
correlation.

In particular, we formally show that correlations in space can become
correlata in time, and the other way around. A space-time structure occurs
here implicitly as information encoded in pre- or post-selection. It could be
further formalized if one replaced qubits by 2-spinors [13, 14, 15], and
included entanglement in time [16, 17], interference in time [18], or swapping
space for time [19] — the generalizations beyond the scope of the present
paper.

We purposefully restrict our discussion to the simplest case of just two
qubits. Many of the usual subtleties related to tensor-product ambiguities are
here absent. For example, those related to different factorizations of the
dimension of the Hilbert space [5, 7, 11] disappear trivially because
$4=2\times 2$ is effectively the only decomposition (as oposed to, say,
$8=2\times 2\times 2=4\times 2$). It is thus easier to extract the ontological
and “causal” element we are interested in. Last but not least, pairs of
2-spinors are enough to define a space-time structure.

## Results

### Correlations and only correlations?

Mermin posits [12] that physical reality of a system is entirely encapsulated
within (a) the correlations among its subsystems and (b) its correlations with
other systems when perceived collectively as components of a more extensive
system. He denotes these as the internal and external correlations of the
system, respectively. A system is regarded as completely isolated if it lacks
external correlations. Furthermore, the wave function of a physical system or,
more generally, its quantum state (pure or mixed) is a concise encapsulation
of its internal correlations [12].

One of the foundational presumptions in the so-called Ithaka interpretation of
quantum mechanics is the assertion that “correlations possess physical
reality, while that which they correlate does not.”

This poses a conundrum: if that which they correlate does not form a part of
physical reality, then how can it be considered a component of reality or how
correlations exist between non-existent parts of reality? Furthermore, we
delve deeper to illustrate that correlations can be identified among other
correlations, while correlations between correlations and correlata
automatically accompany correlations occurring between the correlata
themselves. Accordingly, following Mermin’s line of reasoning, one could claim
that the subjects of these correlations (i.e. other correlations) do not exist
as elements of physical reality, which contradicts the supposition about the
existence of correlations.

We show that what can be formally classified as an element of reality — a
correlation within one tensor product — can be regarded as a correlatum within
another tensor product. Correlata are integral components of correlations and
are always defined in terms of these correlations. As such, correlations can
be perceived as a dual concept to correlata. In this context, when considering
spatial and temporal quantum correlations, we conclude that spatial
correlations become temporal correlata, hinting at a foundational role of
correlations in the genesis of space-time.

### Tensor products or subsystems? — quantum chicken-or-egg dilemma

One usually begins with well defined subystems, whose Hilbert spaces are
spanned by two orthonormal vectors. The composite system is spanned by their
tensor products,

$\displaystyle|00\rangle$ $\displaystyle=$
$\displaystyle|0\rangle\otimes|0\rangle,$ (7) $\displaystyle|01\rangle$
$\displaystyle=$ $\displaystyle|0\rangle\otimes|1\rangle,$ (8)
$\displaystyle|10\rangle$ $\displaystyle=$
$\displaystyle|1\rangle\otimes|0\rangle,$ (9) $\displaystyle|11\rangle$
$\displaystyle=$ $\displaystyle|1\rangle\otimes|1\rangle.$ (10)

The bases at both sides of $\otimes$ can be different. In order to define yes-
no observables corresponding to the two qubits, one first defines the four
projectors,

$\displaystyle P_{00}$ $\displaystyle=$ $\displaystyle|00\rangle\langle
00|=|0\rangle\langle 0|\otimes|0\rangle\langle 0|=P_{0}\otimes P_{0},$ (11)
$\displaystyle P_{01}$ $\displaystyle=$ $\displaystyle|01\rangle\langle
01|=|0\rangle\langle 0|\otimes|1\rangle\langle 1|=P_{0}\otimes P_{1},$ (12)
$\displaystyle P_{10}$ $\displaystyle=$ $\displaystyle|10\rangle\langle
10|=|1\rangle\langle 1|\otimes|0\rangle\langle 0|=P_{1}\otimes P_{0},$ (13)
$\displaystyle P_{11}$ $\displaystyle=$ $\displaystyle|11\rangle\langle
11|=|1\rangle\langle 1|\otimes|1\rangle\langle 1|=P_{1}\otimes P_{1}.$ (14)

Their averages represent joint probabilities $p_{\alpha\beta}$ of finding
correlata $\alpha$ and $\beta$,

$\displaystyle p_{\alpha\beta}=\langle\psi|P_{\alpha\beta}|\psi\rangle.$ (15)

The probabilities of $\alpha$ or $\beta$ alone,

$\displaystyle p^{(1)}_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{\beta}p_{\alpha\beta},$ (16) $\displaystyle
p^{(2)}_{\beta}$ $\displaystyle=$ $\displaystyle\sum_{\alpha}p_{\alpha\beta},$
(17)

are obtained as averages of the following projectors:

$\displaystyle P^{(1)}_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{\beta}P_{\alpha\beta}=\sum_{\beta}P_{\alpha}\otimes
P_{\beta}=P_{\alpha}\otimes\mathbb{I},$ (18) $\displaystyle P^{(2)}_{\beta}$
$\displaystyle=$
$\displaystyle\sum_{\alpha}P_{\alpha\beta}=\sum_{\alpha}P_{\alpha}\otimes
P_{\beta}=\mathbb{I}\otimes P_{\beta}.$ (19)

Formulas

$\displaystyle P_{0}\otimes\mathbb{I}$ $\displaystyle=$ $\displaystyle
P_{00}+P_{01},$ (20) $\displaystyle P_{1}\otimes\mathbb{I}$ $\displaystyle=$
$\displaystyle P_{10}+P_{11},$ (21) $\displaystyle\mathbb{I}\otimes P_{0}$
$\displaystyle=$ $\displaystyle P_{00}+P_{10},$ (22)
$\displaystyle\mathbb{I}\otimes P_{\beta}$ $\displaystyle=$ $\displaystyle
P_{01}+P_{11},$ (23)

can be elegantly represented as sums in rows (the left bit) or columns (the
right bit) of the matrix

$\displaystyle\left(\begin{array}[]{cc}P_{00}&P_{01}\\\
P_{10}&P_{11}\end{array}\right).$ (26)

Matrices with rows or columns interchanged do not define different subsystems
(they generate local NOT operations, $0\leftrightarrow 1$, in the subsystems).

Let us note that the right-hand sides of (16)–(17) define probabilities of the
results 0 and 1 in the detectors of Alice and Bob on the basis of joint
detections at both detectors. So, our starting point is the composite system,
but constructed with the help of our prior knowledge of $\otimes$ and the form
of the subsystems.

However, and this is essential for the argument we advocate in the paper,
formulas (20)–(23) provide an operational definition of the tensor product
itself. It is sufficient to know the form of $P_{\alpha\beta}$ to define
single-qubit observables. The intermediate stage of constructing the composite
system can be skipped. An experimental physicist may even forget about the
very concept of a tensor product.

Yet, when we consider the issue of completeness of quantum mechanics we have
to work at a more general level. We should begin with the composite system
treated as some four dimensional Hilbert space, and then define subsystems and
tensor products. We use here the plural form because one will not arrive at a
unique tensor product, but at six different products, which can be reduced to
three if we treat permutation of Alice and Bob as the same decomposition into
subsystems.

Each of these products has a clear physical interpretation. What is important,
a correlatum for one decomposition, becomes a correlation in the other
decomposition.

Ignoring the multitude of coexisting tensor products we make quantum mechanics
incomplete.

### Six tensor product structures of two qubits

Figure 1: A single unstable system in superposition of four orthogonal states
decays into a pair of particles. Products of the decay are directed toward the
labs of Alice and Bob. Figure 2: Products of the decay can be decomposed into
six inequivalent subsystems by erasing information about the color of an
emitted particle, and connecting pairs of output channels with single
detectors. Bits of Alice are here red, and those of Bob are blue. Each of the
combinations defines a different tensor product structure, effectively
labeling the colors by numbers in binary notation. The left column:
$(C,M,Y,G)=(00,01,10,11)$ (top), $(00,10,01,11)$ (middle), $(00,11,01,10)$
(bottom). The right column: $(00,11,10,01)$ (top), $(00,10,11,01)$ (middle),
$(00,01,11,10)$ (bottom). The remaining permutations of colors lead to local
interchanges of bits,
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}\leftrightarrow{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{1}}$
or
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{0}}\leftrightarrow{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{1}}$.

Consider a two-particle decay process involving a four-dimensional Hilbert
space of outgoing states. The four output channels are denoted in Fig. 1 by
the four CYGM colors: cyan ($C$), yellow ($Y$), green ($G$), and magenta
($M$). The output consists of two-particle states, so we would like to define
two 1-particle subsystems.

With each of the colors we associate a projector, $P_{C}=|C\rangle\langle C|$,
etc. Having four projectors we can write matrices of the form (26). Taking
into account that matrices with rows or columns interchanged do not define
different subsystems, we are left with six possibilities:

$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{M}\\\
P_{Y}&P_{G}\end{array}\right)$
$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{G}\\\
P_{Y}&P_{M}\end{array}\right)$ (31)
$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{Y}\\\
P_{M}&P_{G}\end{array}\right)$
$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{G}\\\
P_{M}&P_{Y}\end{array}\right)$ (36)
$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{Y}\\\
P_{G}&P_{M}\end{array}\right)$
$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{M}\\\
P_{G}&P_{Y}\end{array}\right)$ (41)

Fig. 2 links colors with binary indices of (26) for each of the above cases.

The subtlety is that the definition of $\otimes$ that occurs in (20)–(23)
cannot define the same tensor product for all the matrices (31)–(41). Rather,
we obtain six different tensor products,

$\displaystyle P_{0}\otimes_{123}\mathbb{I}$ $\displaystyle=P_{C}+P_{M},$
$\displaystyle P_{0}\otimes_{321}\mathbb{I}$ $\displaystyle=P_{C}+P_{G},$ (42)
$\displaystyle P_{1}\otimes_{123}\mathbb{I}$ $\displaystyle=P_{Y}+P_{G},$
$\displaystyle P_{1}\otimes_{321}\mathbb{I}$ $\displaystyle=P_{M}+P_{Y},$ (43)
$\displaystyle\mathbb{I}\otimes_{123}P_{0}$ $\displaystyle=P_{C}+P_{Y},$
$\displaystyle\mathbb{I}\otimes_{321}P_{0}$ $\displaystyle=P_{C}+P_{Y},$ (44)
$\displaystyle\mathbb{I}\otimes_{123}P_{1}$ $\displaystyle=P_{M}+P_{G},$
$\displaystyle\mathbb{I}\otimes_{321}P_{1}$ $\displaystyle=P_{M}+P_{G},$ (45)
$\displaystyle P_{0}\otimes_{213}\mathbb{I}$ $\displaystyle=P_{C}+P_{Y},$
$\displaystyle P_{0}\otimes_{231}\mathbb{I}$ $\displaystyle=P_{C}+P_{G},$ (46)
$\displaystyle P_{1}\otimes_{213}\mathbb{I}$ $\displaystyle=P_{M}+P_{G},$
$\displaystyle P_{1}\otimes_{231}\mathbb{I}$ $\displaystyle=P_{M}+P_{Y},$ (47)
$\displaystyle\mathbb{I}\otimes_{213}P_{0}$ $\displaystyle=P_{C}+P_{M},$
$\displaystyle\mathbb{I}\otimes_{231}P_{0}$ $\displaystyle=P_{C}+P_{M},$ (48)
$\displaystyle\mathbb{I}\otimes_{213}P_{1}$ $\displaystyle=P_{Y}+P_{G},$
$\displaystyle\mathbb{I}\otimes_{231}P_{1}$ $\displaystyle=P_{Y}+P_{G},$ (49)
$\displaystyle P_{0}\otimes_{312}\mathbb{I}$ $\displaystyle=P_{C}+P_{Y},$
$\displaystyle P_{0}\otimes_{132}\mathbb{I}$ $\displaystyle=P_{C}+P_{M},$ (50)
$\displaystyle P_{1}\otimes_{312}\mathbb{I}$ $\displaystyle=P_{M}+P_{G},$
$\displaystyle P_{1}\otimes_{132}\mathbb{I}$ $\displaystyle=P_{Y}+P_{G},$ (51)
$\displaystyle\mathbb{I}\otimes_{312}P_{0}$ $\displaystyle=P_{C}+P_{G},$
$\displaystyle\mathbb{I}\otimes_{132}P_{0}$ $\displaystyle=P_{C}+P_{G},$ (52)
$\displaystyle\mathbb{I}\otimes_{312}P_{1}$ $\displaystyle=P_{M}+P_{Y},$
$\displaystyle\mathbb{I}\otimes_{132}P_{1}$ $\displaystyle=P_{M}+P_{Y}.$ (53)

Notice that all these projectors commute.

The awkward indices in $\otimes_{abc}$ become clearer if we introduce unitary
operators that permute

$\displaystyle|0\rangle$ $\displaystyle=$ $\displaystyle|00\rangle=|C\rangle,$
(54) $\displaystyle|1\rangle$ $\displaystyle=$
$\displaystyle|01\rangle=|M\rangle,$ (55) $\displaystyle|2\rangle$
$\displaystyle=$ $\displaystyle|10\rangle=|Y\rangle,$ (56)
$\displaystyle|3\rangle$ $\displaystyle=$ $\displaystyle|11\rangle=|G\rangle.$
(57)

into

$\displaystyle|0\rangle$ $\displaystyle=$ $\displaystyle
U_{abc}|00\rangle=|00_{abc}\rangle=|0\rangle\otimes_{abc}|0\rangle,$ (58)
$\displaystyle|a\rangle$ $\displaystyle=$ $\displaystyle
U_{abc}|01\rangle=|01_{abc}\rangle=|0\rangle\otimes_{abc}|1\rangle,$ (59)
$\displaystyle|b\rangle$ $\displaystyle=$ $\displaystyle
U_{abc}|10\rangle=|10_{abc}\rangle=|1\rangle\otimes_{abc}|0\rangle,$ (60)
$\displaystyle|c\rangle$ $\displaystyle=$ $\displaystyle
U_{abc}|11\rangle=|11_{abc}\rangle=|1\rangle\otimes_{abc}|1\rangle.$ (61)

Formulas (58)–(61) are the definitions of $|rs_{abc}\rangle$ and
$|r\rangle\otimes_{abc}|s\rangle$. Formulas (54)–(57) should not (yet) be
understood in terms of a tensor structure — for the moment this is just a
binary parametrization of the first four whole numbers. However, the
consistency condition

$\displaystyle|0\rangle$ $\displaystyle=$ $\displaystyle
U_{123}|00\rangle=|00_{123}\rangle=|0\rangle\otimes_{123}|0\rangle,$ (62)
$\displaystyle|1\rangle$ $\displaystyle=$ $\displaystyle
U_{123}|01\rangle=|01_{123}\rangle=|0\rangle\otimes_{123}|1\rangle,$ (63)
$\displaystyle|2\rangle$ $\displaystyle=$ $\displaystyle
U_{123}|10\rangle=|10_{123}\rangle=|1\rangle\otimes_{123}|0\rangle,$ (64)
$\displaystyle|3\rangle$ $\displaystyle=$ $\displaystyle
U_{123}|11\rangle=|11_{123}\rangle=|1\rangle\otimes_{123}|1\rangle,$ (65)

shows that $U_{123}$ is the identity operator and thus one can think of

$\displaystyle|rs\rangle=|rs_{123}\rangle=|r\rangle\otimes_{123}|s\rangle$
(66)

in terms of “the” “ordinary” tensor product. In fact, in matrix
representations we will identify $\otimes_{123}=\otimes$ with the
KroneckerProduct matrix operation employed in Wolfram Mathematica.

$P_{00}$$P_{01}$$P_{10}$$P_{11}$ Figure 3: The unstable decaying system in a
superposition of four orthogonal colour states CYGM can be also represented as
an abstract cube. This decay leads to six inequivalent local products. Thus,
one can associate six different tensor products $\otimes_{abc}$ with six faces
of the abstract cube. Classically detection of results happens only on one
chosen face of the cube. The global information about colours is lost during
detection of results by Alice and Bob but leads to inequivalency of tensor
products on the faces. One can transition among the faces applying rotations
$U_{abc}$. Interestingly, what is entangled on one face in a chosen tensor
algebra can be separable on the another face with its tensor algebra.

The matrix elements, $(U_{abc})_{kl}=\langle k|U_{abc}|l\rangle$, $k,l=0\dots
3$, can be collected in six matrices,

$\displaystyle(U_{123})_{kl}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right)$
$\displaystyle(U_{321})_{kl}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\\\ 0&1&0&0\end{array}\right)$ (75)
$\displaystyle(U_{213})_{kl}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&0&0&1\end{array}\right)$
$\displaystyle(U_{231})_{kl}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&0&1\\\
0&1&0&0\\\ 0&0&1&0\end{array}\right)$ (84)
$\displaystyle(U_{312})_{kl}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ 0&1&0&0\end{array}\right)$
$\displaystyle(U_{132})_{kl}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 0&0&1&0\end{array}\right)$ (93)

Each matrix $(U_{abc})_{kl}$ is indexed by the permutation
$(0,1,2,3)\to(0,a,b,c)$ it represents, and its position in the list (75)–(93)
matches an appropriate diagram from Fig. 2. We then find

$\displaystyle A\otimes_{abc}B$ $\displaystyle=$ $\displaystyle
U_{abc}(A\otimes_{123}B)U_{abc}^{\dagger}.$ (94)

More generally,

$\displaystyle A\otimes_{a^{\prime\prime}b^{\prime\prime}c^{\prime\prime}}B$
$\displaystyle=$ $\displaystyle
U_{a^{\prime}b^{\prime}c^{\prime}}(A\otimes_{abc}B)U_{a^{\prime}b^{\prime}c^{\prime}}^{\dagger},$
(95)

where

$\displaystyle
U_{a^{\prime\prime}b^{\prime\prime}c^{\prime\prime}}=U_{a^{\prime}b^{\prime}c^{\prime}}U_{abc},$
(96)

since a composition of two permutations is a permutation. Moreover, if

$\displaystyle(A\otimes_{abc}B)(C\otimes_{abc}D)=(AC)\otimes_{abc}(BD)$ (97)

is true for some $\otimes_{abc}$, then it is true of any $\otimes_{abc}$.
However, in general

$\displaystyle(A\otimes_{abc}B)(C\otimes_{a^{\prime}b^{\prime}c^{\prime}}D)\neq(AC)\otimes_{a^{\prime\prime}b^{\prime\prime}c^{\prime\prime}}(BD),$
(98)

unless
$(a,b,c)=(a^{\prime},b^{\prime},c^{\prime})=(a^{\prime\prime},b^{\prime\prime},c^{\prime\prime})$.

Accordingly, there is nothing special in $\otimes_{123}$, so that any
$\otimes_{abc}$ can be identified in matrix formulas with Wolfram
Mathematica’s KroneckerProduct. We conclude that as any matrix is a table of
coordinates of a vector in some concrete basis, the identification of
$\otimes_{123}=\otimes$ with KroneckerProduct plays the same formal role. When
it comes to concrete calculations with concrete operators, such as the Pauli-
matrices representation of spin-1/2, we can use the usual matrix “drag-and-
drop” Kronecker product $\otimes$. However, we have to be aware that at the
level of tensor products, such an $\otimes$ can represent any $\otimes_{abc}$.

### Correlations in space become correlata in time — duality of quantum
entanglement

Before we proceed further, let us explain our terminology related to
projectors, propositions, eigenvalues, truth values, and bits. The latter
three notions have numerical values 0 and 1, which may lead to
misunderstandings.

Any projector is self-adjoint and thus represents a quantum observable, in
principle measurable in some experiment. From a quantum-logic perspective a
projector represents a proposition (statement about the truth value of a given
sentence). Projectors have eigenvalues 0 and 1, corresponding to the truth
values of the corresponding propositions. For example, a measurement of an
observable $P$ representing proposition “Alice AND Bob” produces 0 if the
proposition “Alice AND Bob” is false. The corresponding eigenvector of $P$ is
then a general superposition of $|00\rangle$, $|01\rangle$, and $|10\rangle$.
Spectral representation of the projector is $P=|11\rangle\langle 11|$ because
the truth value of the proposition “1 AND 1” equals 1, and thus the eigenvalue
of $P$ must be 1 as well, a fact happening only in case the corresponding
eigenvector is $|11\rangle$.

The logical negation of proposition $P$ is given by $\mathbb{I}-P$, where
$\mathbb{I}$ is the identity operator. In this paper, the pairs $P_{0}$ and
$P_{1}$ are related by negation. Whenever we say that $P_{0}$ represents a
zero bit, we mean that a measurement of $P_{0}$ returns 1 if the proposition
“bit equals 0” is true. So, a measurement of $P_{0}$ returns 0 if the
proposition “bit equals 0” is false. Analogously, $P_{1}$ “represents a unit
bit” means that a measurement of $P_{1}$ returns 1 if the proposition “bit
equals 1” is true. A measurement of $P_{1}$ returns 0 if the proposition “bit
equals 1” is false.

The above terminology is consistent with probability interpretation:
$p_{0}=\langle\psi|P_{0}|\psi\rangle$ is the probability of finding 0 in a
measurement performed in a physical system whose state is given by
$|\psi\rangle$.

Now, let us return to the six matrices (31)–(41). We will concentrate on three
of them, because they appear in examples. Let us rewrite them as follows:

$\displaystyle\left(\begin{array}[]{cc}P_{00}&P_{01}\\\
P_{10}&P_{11}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}P_{00_{123}}&P_{01_{123}}\\\
P_{10_{123}}&P_{11_{123}}\end{array}\right),$ (103)
$\displaystyle\left(\begin{array}[]{cc}P_{00}&P_{11}\\\
P_{10}&P_{01}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}P_{00_{321}}&P_{01_{321}}\\\
P_{10_{321}}&P_{11_{321}}\end{array}\right)$ (108)
$\displaystyle\left(\begin{array}[]{cc}P_{00}&P_{10}\\\
P_{01}&P_{11}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}P_{00_{213}}&P_{01_{213}}\\\
P_{10_{213}}&P_{11_{213}}\end{array}\right).$ (113)

Local bits of subsystems Aliceabc and Bobabc are defined by the usual rule

$\displaystyle P_{0}\otimes_{abc}\mathbb{I}$ $\displaystyle=$ $\displaystyle
P_{00_{abc}}+P_{01_{abc}},$ (114) $\displaystyle P_{1}\otimes_{abc}\mathbb{I}$
$\displaystyle=$ $\displaystyle P_{10_{abc}}+P_{11_{abc}},$ (115)
$\displaystyle\mathbb{I}\otimes_{abc}P_{0}$ $\displaystyle=$ $\displaystyle
P_{00_{abc}}+P_{10_{abc}},$ (116) $\displaystyle\mathbb{I}\otimes_{abc}P_{1}$
$\displaystyle=$ $\displaystyle P_{01_{abc}}+P_{11_{abc}}.$ (117)

The formulas become less natural if we rewrite (114)–(117) by means of the
parametrization

$\displaystyle\left(\begin{array}[]{cc}P_{C}&P_{M}\\\
P_{Y}&P_{G}\end{array}\right)=\left(\begin{array}[]{cc}P_{00}&P_{01}\\\
P_{10}&P_{11}\end{array}\right)$ (122)

that refers to $\otimes=\otimes_{123}$. The choice (122) is completely
arbitrary, of course, and should be regarded as an arbitrary “system of
coordinates” for all the other decompositions into subsystems. As we will see
in a moment, referring all the projectors to our reference labeling (122)
leads to a “space-time” interpretation of the six tensor products.

There are no controversies over the definition of $\otimes_{123}$,

$\displaystyle P_{0}\otimes_{123}\mathbb{I}$
$\displaystyle=P_{00}+P_{01},\textrm{(0 of Alice)}$ (123) $\displaystyle
P_{1}\otimes_{123}\mathbb{I}$ $\displaystyle=P_{10}+P_{11},\textrm{(1 of
Alice)}$ (124) $\displaystyle\mathbb{I}\otimes_{123}P_{0}$
$\displaystyle=P_{00}+P_{10},\textrm{(0 of Bob)}$ (125)
$\displaystyle\mathbb{I}\otimes_{123}P_{1}$
$\displaystyle=P_{01}+P_{11},\textrm{(1 of Bob)}$ (126)

We will sometimes refer to Alice and Bob defined by the above formulas as
Alice123 and Bob123. Tensor product $\otimes_{123}$ is here identified with
the Kronecker product of matrices and is equivalent to typical $\otimes$ being
our reference system. Hence, e.g.

$\displaystyle P_{0}\otimes_{123}\mathbb{I}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\otimes\mathbb{I}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right),$ (133)

An interpretation of the right bit in the tensor product $\otimes_{321}$ is
unchanged with respect to $\otimes_{123}$, but the left bit carries
information about the correlation between Alice123, and Bob123,

$\displaystyle P_{0}\otimes_{321}\mathbb{I}$
$\displaystyle=P_{00}+P_{11},\textrm{(Alice {\tt IFF} Bob)}$ (134)
$\displaystyle P_{1}\otimes_{321}\mathbb{I}$
$\displaystyle=P_{01}+P_{10},\textrm{(Alice {\tt XOR} Bob)}$ (135)
$\displaystyle\mathbb{I}\otimes_{321}P_{0}$
$\displaystyle=P_{00}+P_{10},\textrm{(0 of Bob)}$ (136)
$\displaystyle\mathbb{I}\otimes_{321}P_{1}$
$\displaystyle=P_{01}+P_{11},\textrm{(1 of Bob)}$ (137)

Although the “right” bit correctly illustrates the position of the bit in
$P_{\alpha\beta}$, the term “left” may be somewhat misleading: Projector
$P_{0}\otimes_{321}\mathbb{I}$ represents the logical equivalence IFF (if and
only if),

$\displaystyle\textrm{Alice}_{321}=(\textrm{Alice${}_{123}$ {\tt IFF}
Bob${}_{123}$})$ (138)

$\otimes_{321}$ carries hidden global information about source of the local
states on Alice’ and Bob’s sites. Equation (134) carries information that
local 0 bit on Alice’ site has a source in cyan and green coloured states.

At the level of matrix representations the rule that links bits defined by
$\otimes_{321}$ and $\otimes$ is,

$\displaystyle P_{0}\otimes_{321}\mathbb{I}$ $\displaystyle=$ $\displaystyle
U_{321}(P_{0}\otimes\mathbb{I})U_{321}^{\dagger},\quad\textrm{etc.}$ (139)

The unitary operator $U_{321}$ interchanges $01$ and $11$,

$\displaystyle U_{321}=|00\rangle\langle 00|+|01\rangle\langle
11|+|10\rangle\langle 10|+|11\rangle\langle 01|.$ (140)

Explicitly,

$\displaystyle
U_{321}P_{00}U_{321}^{\dagger}=P_{00_{321}}=U_{321}|00\rangle\langle
00|U_{321}^{\dagger}=P_{00},$ (141) $\displaystyle
U_{321}P_{01}U_{321}^{\dagger}=P_{01_{321}}=U_{321}|01\rangle\langle
01|U_{321}^{\dagger}=P_{11},$ (142) $\displaystyle
U_{321}P_{10}U_{321}^{\dagger}=P_{10_{321}}=U_{321}|10\rangle\langle
10|U_{321}^{\dagger}=P_{10},$ (143) $\displaystyle
U_{321}P_{11}U_{321}^{\dagger}=P_{11_{321}}=U_{321}|11\rangle\langle
11|U_{321}^{\dagger}=P_{01}.$ (144)

$U_{321}$ is the CNOT operation performed on the left bit, and controlled by
the right bit. On the other hand, the role of $U_{abc}$ is to relabel the
basis.

Figure 4: Six types of correlations described by six types of bits, associated
with six decompositions into subsystems, and six tensor products. Effectively,
there are three types of questions associated with alternative tensor
products: “Are the bits of Alice and Bob different?”, “What is the bit of
Alice?”, and “What is the bit of Bob?”. The three types occur in two versions,
with Alice and Bob interchanged. Joint questions about Alice and Bob are
causally related with separate questions about Alice and Bob. The four tensor
structures are thus related to causal correlations (“correlations in time”).
The joint questions can refer either to the past (correlations at the source),
or to the future (joint detections). In Bell-type experiments [20] both cases
occur simultaneously. The arrow of time in this figure can point from the
bottom to the top, or from the top to the bottom. In Mermin’s terminology, a
correlation in $\otimes_{123}$ and $\otimes_{213}$, becomes a correlatum in
$\otimes_{321}$, $\otimes_{231}$, $\otimes_{312}$, and $\otimes_{132}$, and
vice versa.

As the “right” bit of Bob does not change its interpretation, one might have
the impression that local operators associated with Bob should remain
unchanged, which is not the case as it turns out. Let us discuss the problem
in more detail. Let us recall that
$|\alpha\beta_{321}\rangle=U_{321}|\alpha\beta\rangle=U_{321}|\alpha\beta_{123}\rangle$,
so

$\displaystyle\mathbb{I}\otimes_{321}\sigma_{3}$ $\displaystyle=$
$\displaystyle U_{321}\big{(}\mathbb{I}\otimes(|0\rangle\langle
0|-|1\rangle\langle 1|)\big{)}U_{321}^{\dagger}$ (145) $\displaystyle=$
$\displaystyle|00\rangle\langle 00|-|11\rangle\langle 11|+|10\rangle\langle
10|-|01\rangle\langle 01|$ $\displaystyle=$
$\displaystyle\mathbb{I}\otimes\sigma_{3},$

which indeed shows that eigenvectors of $\sigma_{3}$ retain their
interpretation independently of the choice of $\otimes_{321}$ or
$\otimes_{123}=\otimes$. However, the situation changes when we consider the
remaining Pauli matrices. For example the local NOT operation in the subsystem
Bob321 reads

$\displaystyle\mathbb{I}\otimes_{321}\sigma_{1}$ $\displaystyle=$
$\displaystyle U_{321}\big{(}\mathbb{I}\otimes(|0\rangle\langle
1|+|1\rangle\langle 0|)\big{)}U_{321}^{\dagger}$ (146) $\displaystyle=$
$\displaystyle\big{(}|0\rangle\langle 1|+|1\rangle\langle
0|\big{)}\otimes\big{(}|0\rangle\langle 1|+|1\rangle\langle 0|\big{)}$
$\displaystyle=$
$\displaystyle\sigma_{1}\otimes\sigma_{1}=\sigma_{1}\otimes_{123}\sigma_{1}.$

Now, let us take a look at the negation of the left bit,

$\displaystyle\sigma_{1}\otimes_{321}\mathbb{I}$ $\displaystyle=$
$\displaystyle U_{321}\big{(}(|0\rangle\langle 1|+|1\rangle\langle
0|)\otimes\mathbb{I}\big{)}U_{321}^{\dagger}$ (147) $\displaystyle=$
$\displaystyle\big{(}|0\rangle\langle 1|+|1\rangle\langle
0|\big{)}\otimes\big{(}|0\rangle\langle 0|+|1\rangle\langle 1|\big{)}$ (148)
$\displaystyle=$ $\displaystyle\sigma_{1}\otimes\mathbb{I}.$ (149)

One analogously checks that

$\displaystyle\sigma_{3}\otimes_{321}\mathbb{I}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{3}.$ (150)

The four formulas,

$\displaystyle\mathbb{I}\otimes_{321}\sigma_{3}$ $\displaystyle=$
$\displaystyle\mathbb{I}\otimes\sigma_{3},$ (151)
$\displaystyle\mathbb{I}\otimes_{321}\sigma_{1}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\sigma_{1},$ (152)
$\displaystyle\sigma_{1}\otimes_{321}\mathbb{I}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\mathbb{I},$ (153)
$\displaystyle\sigma_{3}\otimes_{321}\mathbb{I}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{3},$ (154)

seem inconsistent at first glance, but they illustrate the change of
$\otimes_{123}$-correlations into $\otimes_{321}$-correlata. Indeed,
eigenvectors of

$\displaystyle P_{0}\otimes_{321}\mathbb{I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbb{I}+\sigma_{3})\otimes_{321}\mathbb{I}$ (155)
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{I}\otimes\mathbb{I}+\frac{1}{2}\sigma_{3}\otimes\sigma_{3}$
(156) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\end{array}\right)=P_{00}+P_{11}$ (161)

are given by

$\displaystyle|f_{\parallel}\rangle$
$\displaystyle=f_{00}|00\rangle+f_{11}|11\rangle$ ($\otimes$-entangled state)
(162) $\displaystyle=f_{00}|00_{321}\rangle+f_{11}|01_{321}\rangle$
($\otimes_{321}$-product state) (163) $\displaystyle|f_{\perp}\rangle$
$\displaystyle=f_{01}|01\rangle+f_{10}|10\rangle$ ($\otimes$-entangled state)
(164) $\displaystyle=f_{01}|11_{321}\rangle+f_{10}|10_{321}\rangle$
($\otimes_{321}$-product state) (165)

with

$\displaystyle(P_{0}\otimes_{321}\mathbb{I})|f_{\parallel}\rangle$
$\displaystyle=$ $\displaystyle|f_{\parallel}\rangle,$ (166)
$\displaystyle(P_{1}\otimes_{321}\mathbb{I})|f_{\parallel}\rangle$
$\displaystyle=$ $\displaystyle 0,$ (167)
$\displaystyle(P_{0}-P_{1})\otimes_{321}\mathbb{I})|f_{\parallel}\rangle$
$\displaystyle=$
$\displaystyle|f_{\parallel}\rangle=(\sigma_{3}\otimes_{321}\mathbb{I})|f_{\parallel}\rangle,$
(168) $\displaystyle(P_{0}\otimes_{321}\mathbb{I})|f_{\perp}\rangle$
$\displaystyle=$ $\displaystyle 0,$ (169)
$\displaystyle(P_{1}\otimes_{321}\mathbb{I})|f_{\perp}\rangle$
$\displaystyle=$ $\displaystyle|f_{\perp}\rangle,$ (170)
$\displaystyle(P_{0}-P_{1})\otimes_{321}\mathbb{I})|f_{\perp}\rangle$
$\displaystyle=$
$\displaystyle-|f_{\perp}\rangle=(\sigma_{3}\otimes_{321}\mathbb{I})|f_{\perp}\rangle,$
(171)

which shows that

$\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|f_{\parallel}\rangle$
$\displaystyle=$ $\displaystyle|f_{\parallel}\rangle,$ (172)
$\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|f_{\perp}\rangle$
$\displaystyle=$ $\displaystyle-|f_{\perp}\rangle.$ (173)

The particular cases of (172)–(173) are

$\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|00_{321}\rangle$
$\displaystyle=$
$\displaystyle|00_{321}\rangle=|00\rangle=(\sigma_{3}\otimes_{321}\mathbb{I})|00\rangle=(\sigma_{3}\otimes\sigma_{3})|00\rangle,$
(174) $\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|01_{321}\rangle$
$\displaystyle=$
$\displaystyle|01_{321}\rangle=|11\rangle=(\sigma_{3}\otimes_{321}\mathbb{I})|11\rangle=(\sigma_{3}\otimes\sigma_{3})|11\rangle,$
(175) $\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|11_{321}\rangle$
$\displaystyle=$
$\displaystyle-|11_{321}\rangle=-|01\rangle=(\sigma_{3}\otimes_{321}\mathbb{I})|01\rangle=(\sigma_{3}\otimes\sigma_{3})|01\rangle,$
(176) $\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|10_{321}\rangle$
$\displaystyle=$
$\displaystyle-|10_{321}\rangle=-|10\rangle=(\sigma_{3}\otimes_{321}\mathbb{I})|10\rangle=(\sigma_{3}\otimes\sigma_{3})|10\rangle,$
(177)

which explicitly cross-checks (154). Analogously, one verifies that the two
forms of negation of the left bit are indeed equivalent:

$\displaystyle(\sigma_{1}\otimes_{321}\mathbb{I})(f_{00}|00_{321}\rangle+f_{11}|01_{321}\rangle)$
$\displaystyle=$ $\displaystyle
f_{00}|10_{321}\rangle+f_{11}|11_{321}\rangle,$ (178)
$\displaystyle(\sigma_{1}\otimes\mathbb{I})(f_{00}|00\rangle+f_{11}|11\rangle)$
$\displaystyle=$ $\displaystyle f_{00}|10\rangle+f_{11}|01\rangle$ (179)
$\displaystyle=$ $\displaystyle
f_{00}|10_{321}\rangle+f_{11}|11_{321}\rangle.$ (180)

Equations (162) and (163) show something peculiar as if an entangled state was
at the same time a product state. This apparent paradox originates from the
relativity of tensor product structures. The equation (163) stores hidden
global information about the source of the correlations and stores information
about temporal correlations (and causal relation between Alice and Bob) which
is not manifested at the level of purely spatial correlations between Alice
and Bob for the plain $\otimes$. This phenomenon exemplifies the duality of
quantum entanglement whose structure also varies based on the chosen tensor
algebra associated with observers.

Projectors such as $P_{\alpha}\otimes_{321}\mathbb{I}$ represent observables
that can be measured only in case we know simultaneously the results of local
measurements performed in subsystems Alice123 and Bob123, a typical situation
in Bell-type experiments or tests for eavesdropping in entangled-state quantum
cryptography. Notice that states from the Bell basis,

$\displaystyle|\Psi_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle),$ (181)
$\displaystyle|\Phi_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle)$ (182)

satisfy

$\displaystyle(P_{0}\otimes_{321}\mathbb{I})|\Psi_{\pm}\rangle$
$\displaystyle=$ $\displaystyle 0,$ (183)
$\displaystyle(P_{1}\otimes_{321}\mathbb{I})|\Psi_{\pm}\rangle$
$\displaystyle=$ $\displaystyle|\Psi_{\pm}\rangle,$ (184)
$\displaystyle(P_{0}\otimes_{321}\mathbb{I})|\Phi_{\pm}\rangle$
$\displaystyle=$ $\displaystyle|\Phi_{\pm}\rangle,$ (185)
$\displaystyle(P_{1}\otimes_{321}\mathbb{I})|\Phi_{\pm}\rangle$
$\displaystyle=$ $\displaystyle 0.$ (186)

Hence, a definite truth value of the proposition
$P_{\alpha}\otimes_{321}\mathbb{I}$ can be determined either at the source of
the pairs of particles (preselection: those who arrange the experiment know
which state is prepared), hence in the past of the detection events, or at the
final stage when the results of local measurements performed by Alice123 and
Bob123 are collected and thus can be compared with one another — hence in the
future of the detection events (postselection). In this sense $\otimes_{123}$
describes space-like correlations, whereas $\otimes_{321}$ describes the time-
like ones.

Considering the singlet state $|\Psi_{-}\rangle$, we know that it is an
entangled state with respect to the “usual” tensor product $\otimes_{123}$.
However, with respect to $\otimes_{321}$ it must be a product state because
the truth value of the proposition (Alice123 XOR Bob123) is 1, while at the
same time the truth value of “bit of Bob321 is 0” is either 0 or 1. And
indeed,

$\displaystyle|01\rangle-|10\rangle$ $\displaystyle=$
$\displaystyle|0\rangle\otimes_{123}|1\rangle-|1\rangle\otimes_{123}|0\rangle$
(187) $\displaystyle=$ $\displaystyle|11_{321}\rangle-|10_{321}\rangle$ (188)
$\displaystyle=$
$\displaystyle|1\rangle\otimes_{321}|1\rangle-|1\rangle\otimes_{321}|0\rangle$
(189) $\displaystyle=$
$\displaystyle|1\rangle\otimes_{321}\big{(}|1\rangle-|0\rangle\big{)},$ (190)

is entangled with respect to $\otimes_{123}$ and product with respect to
$\otimes_{321}$. The singlet state is space-like entangled and time-like
untangled.

### Local change of basis

Assume $|\alpha\beta_{abc}\rangle$ is an orthonormal basis. A local change of
basis is defined by

$\displaystyle|\alpha_{V}\beta_{Wabc}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\gamma,\delta}V_{\alpha\gamma}W_{\beta\delta}|\gamma\delta_{abc}\rangle$
(191) $\displaystyle=$
$\displaystyle(V\otimes_{abc}W)|\alpha\beta_{abc}\rangle,$ (192)

where the matrices $V_{\alpha\gamma}$ and $W_{\beta\delta}$ are unitary.

In order to formulate the Einstein-Podolsky-Rosen-Bohm argument [21, 22] in
our generalized framework, we have to discuss consequences of a change of
basis in (7)–(14) into

$\displaystyle|0_{V}0_{W123}\rangle$ $\displaystyle=$
$\displaystyle|0_{V}\rangle\otimes_{123}|0_{W}\rangle,$ (193)
$\displaystyle|0_{V}1_{W123}\rangle$ $\displaystyle=$
$\displaystyle|0_{V}\rangle\otimes_{123}|1_{W}\rangle,$ (194)
$\displaystyle|1_{V}0_{W123}\rangle$ $\displaystyle=$
$\displaystyle|1_{V}\rangle\otimes_{123}|0_{W}\rangle,$ (195)
$\displaystyle|1_{V}1_{W123}\rangle$ $\displaystyle=$
$\displaystyle|1_{V}\rangle\otimes_{123}|1_{W}\rangle,$ (196)

and

$\displaystyle P_{0_{V}0_{W123}}$ $\displaystyle=$ $\displaystyle
P_{0_{V}}\otimes_{123}P_{0_{W}},$ (197) $\displaystyle P_{0_{V}1_{W123}}$
$\displaystyle=$ $\displaystyle P_{0_{V}}\otimes_{123}P_{1_{W}},$ (198)
$\displaystyle P_{1_{V}0_{W123}}$ $\displaystyle=$ $\displaystyle
P_{1_{V}}\otimes_{123}P_{0_{W}},$ (199) $\displaystyle P_{1_{V}1_{W123}}$
$\displaystyle=$ $\displaystyle P_{1_{V}}\otimes_{123}P_{1_{W}}.$ (200)

The formula

$\displaystyle P_{\alpha_{V}\beta_{W123}}$ $\displaystyle=$
$\displaystyle(V\otimes_{123}W)P_{\alpha\beta_{123}}(V\otimes_{abc}W)^{\dagger},$
(201)

trivially generalizes to any $abc$,

$\displaystyle P_{\alpha_{V}\beta_{Wabc}}$ $\displaystyle=$
$\displaystyle(V\otimes_{123}W)P_{\alpha\beta_{abc}}(V\otimes_{123}W)^{\dagger},$
(202)

because $P_{\alpha\beta_{abs}}$ is related to $P_{\gamma\delta_{123}}$ by a
permutation $\pi$ of pairs of the Greek indices,
$\alpha\beta\mapsto\gamma\delta=\pi(\alpha\beta)$, hence

$\displaystyle(\ref{163})$ $\displaystyle=$
$\displaystyle(V\otimes_{123}W)P_{\pi(\alpha\beta)_{123}}(V\otimes_{123}W)^{\dagger}$
(203) $\displaystyle=$ $\displaystyle
P_{\pi(\alpha_{V}\beta_{W})_{123}}=P_{\alpha_{V}\beta_{Wabc}}.$ (204)

Let us further note that (202) gives an example of a map

$\displaystyle P_{\alpha}\otimes_{abc}P_{\beta}\mapsto
U\big{(}P_{\alpha}\otimes_{abc}P_{\beta}\big{)}U^{\dagger}=P_{\alpha_{U}}\otimes_{abc}P_{\beta_{U}},$
(205)

that should not be regarded as a change $\otimes_{abc}\mapsto\otimes_{Uabc}$,

$\displaystyle P_{\alpha}\otimes_{abc}P_{\beta}\mapsto
P_{\alpha}\otimes_{Uabc}P_{\beta},$ (206)

of the tensor product itself. So, in case of $U=V\otimes W$ we obtain

$\displaystyle P_{\alpha_{U}}\otimes P_{\beta_{U}}$ (207)

and not

$\displaystyle P_{\alpha}\otimes_{Uabc}P_{\beta},$ (208)

Subtleties of this sort are essential for the EPR argument, as seen from the
perspective of generalized tensor products. The physical meaning of the result
is intuitively obvious: in case of $U=V\otimes_{123}W$ one does not influence
the splitting of the system into subsystems (Alice and Bob are intact), but
only changes bases in the subsystems. This is exactly what we need in order to
formulate the EPR argument. Interestingly, what was not obvious from the very
outset, although the product structure refers to one of the six tensor
products only, it automatically applies to all the remaining tensor products
as well.

If we additionally assume $V=W\in\mathrm{SU}(2)$, the singlet state remains
unchanged. Indeed, on the one hand,

$\displaystyle|01_{123}\rangle-|10_{123}\rangle$ $\displaystyle=$
$\displaystyle|11_{321}\rangle-|10_{321}\rangle$ (209) $\displaystyle=$
$\displaystyle|1\rangle\otimes_{321}\big{(}|1\rangle-|0\rangle\big{)},$ (210)

while on the other,

$\displaystyle|01_{123}\rangle-|10_{123}\rangle$ $\displaystyle=$
$\displaystyle|0_{V}1_{V123}\rangle-|1_{V}0_{V123}\rangle$ (211)
$\displaystyle=$ $\displaystyle|1_{V}1_{V321}\rangle-|1_{V}0_{V321}\rangle$
(212) $\displaystyle=$
$\displaystyle|1_{V}\rangle\otimes_{321}\big{(}|1_{V}\rangle-|0_{V}\rangle\big{)}$
(213)

At first glance, the equality of (210) and (213) is surprising for two
reasons. First of all, it shows that product states can be invariant under
changes of basis, a property typically attributed to entangled states.
Secondly, it shows that the correlatum at the sides of Alice321 of (210)–(213)
(i.e. the correlation with respect to $\otimes_{123}$) changes with the change
of basis. So, there are infinitely many correlations in the singlet state.

After a little thought, one realizes what happens. There are indeed infinitely
many correlations — projections of spin on the same direction $\vec{a}$ are in
a one-to-one relation for any $\vec{a}$ if the two-particle state is the
singlet. The whole analysis we have performed so far referred to the eigen-
basis of $\sigma_{3}$ (or, more precisely, we took an arbitrary basis and then
defined $\sigma_{3}=|0\rangle\langle 0|-|1\rangle\langle 1|$). This is why the
left bit corresponding to the subsystem defined by means of $\otimes_{321}$
(i.e. the correlation for $\otimes_{123}$) might seem unique. Our analysis
shows that the correlation with respect to $\otimes_{123}$ behaves as a
typical qubit, if we regard it as a correlatum for $\otimes_{321}$.

For a nontrivial $V$, equality (211) is possible only in the singlet case.
Accordingly, the formula

$\displaystyle|1\rangle\otimes_{321}\big{(}|1\rangle-|0\rangle\big{)}$
$\displaystyle=$
$\displaystyle|1_{V}\rangle\otimes_{321}\big{(}|1_{V}\rangle-|0_{V}\rangle\big{)},$
(214)

provides another unique characterization of the singlet state. Moreover, if
(214) is true for a single nontrivial $V$, it remains true for any
$V\in\mathrm{SU}(2)$. Accordingly, for any unitary $V$, the singlet state is
the eigenstate of

$\displaystyle V\sigma_{3}V^{\dagger}\otimes_{321}V\sigma_{1}V^{\dagger},$
(215)

corresponding to the eigenvalue $+1$, a fact that generalizes (241).

### Implications for the EPR argument

Einstein-Podolsky-Rosen-Bohm argumentation is implicit in the following
collection of equivalent formulas,

$\displaystyle|\Psi_{-}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|-_{3}\rangle\otimes_{123}|+_{3}\rangle-|+_{3}\rangle\otimes_{123}|-_{3}\rangle\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|-_{3V}\rangle\otimes_{123}|+_{3V}\rangle-|+_{3V}\rangle\otimes_{123}|-_{3V}\rangle\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|-_{3}\rangle\otimes_{321}\big{(}|-_{3}\rangle-|+_{3}\rangle\big{)}$
(218) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|-_{3V}\rangle\otimes_{321}\big{(}|-_{3V}\rangle-|+_{3V}\rangle\big{)}$
(219) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|-_{3}\rangle-|+_{3}\rangle\big{)}\otimes_{231}|-_{3V}\rangle$
(220) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|-_{3V}\rangle-|+_{3V}\rangle\big{)}\otimes_{231}|-_{3V}\rangle.$
(221)

Here,

$\displaystyle\sigma_{3}$ $\displaystyle=$
$\displaystyle|+_{3}\rangle\langle+_{3}|-|-_{3}\rangle\langle-_{3}|,$ (222)
$\displaystyle\sigma_{1}$ $\displaystyle=$
$\displaystyle|+_{3}\rangle\langle-_{3}|+|-_{3}\rangle\langle+_{3}|,$ (223)
$\displaystyle\sigma_{3V}$ $\displaystyle=$ $\displaystyle
V\sigma_{3}V^{\dagger},$ (224) $\displaystyle\sigma_{1V}$ $\displaystyle=$
$\displaystyle V\sigma_{1}V^{\dagger},$ (225)

and $V\in\mathrm{SU}(2)$ is arbitrary.

The argumentation based on (Implications for the EPR argument)–(LABEL:178!!)
is well known. Formulas (218)–(219) and (220)–(221) involve either Alice231 or
Bob321, but never both Alice and Bob at the same time. The other qubit is the
correlation between Alice123 and Bob123. Notice that in the latter two cases
the qubits of Alice231 or Bob321 are always described by pure states, because
the singlet is a product state with respect to $\otimes_{321}$ and
$\otimes_{231}$.

The EPR argument, when reformulated on the basis of (218)–(219) is somewhat
surprising, but reflects a tacit element of the standard argumentation. For
let us begin with (218). Bob321 performs his measurement of $\sigma_{3}$ and
finds $+1$ or $-1$. The fact that the result of Alice321 is always $-1$ means
that the result of a measurement performed by Alice123 (not Alice321!) will be
exactly opposite to the one found by Bob321. So, knowing the result obtained
by Bob321, and taking into account that the result obtained by Alice321 is
always the same (namely $-1$; hence the product structure of
$|\Psi_{-}\rangle$) we automatically know what will be the result of the
measurement of $\sigma_{3}$ if performed by Alice123. This is why the vertices
of the triangles in Fig. 4 correspond to simultaneously measurable correlata
and correlations (recall that the right-hand-sides of (42)–(53) commute with
one another).

If we repeat the same reasoning with (219), the only difference is that
$\sigma_{3}$ is now replaced by $\sigma_{3V}$. The result obtained by Alice321
is again always the same, namely $-1$. But now it means that the result found
by Alice123 will be exactly opposite to the one found by Bob321, if they both
decide to measure $\sigma_{3V}$.

## Conclusions

The duality of correlations and correlata seems to be an overlooked aspect of
the well known relativity of tensor structures. Quantum relativity of time and
space appears here in a completely different light, suggesting that
correlations may constitute the fabric of space-time itself.

The discussion is restricted to two qubits which allowed us to separate the
tensor ambiguities from those that occur in Hilbert spaces of other non-prime
dimensions [5, 7]. Moreover, the fact that we restrict the ambiguity to
permutations of the four basis vectors, restricts the automorphism freedom in
the sense of [6, 10] to the permutation group only, making it a kind of
fundamental gauge symmetry of any quantum theory, and simultaneously
guaranteeing that the alternative tensor structures are mutually coexisting
($P_{\alpha}\otimes_{abc}\mathbb{I}$ commute with one another for any
$a,b,c$). One should mention here that there exists yet another form of tensor
ambiguity, related to modular tensor products [14], which is not directly
related to the above problems and applies to Hilbert spaces of any dimension.

The old problem of the borderland between an observer and that which is
observed should be, perhaps, discussed anew from the perspective of
alternative tensor structures. It is in principle possible that correlations
within a system should be formulated by means of, say, $\otimes_{123}$,
whereas those between the system and an observer should involve a tensor
product analogous to $\otimes_{321}$ which turns entangled states into product
ones, thus allowing “system–universe” product states
$|\psi_{s}\rangle\otimes_{321}|\psi_{s}\rangle$ maintain system–observer
correlations.

Maybe this is the formal reason why predictions of Schrödinger equations can
lead to observable consequences?

## Acknowledgments

We are indebted to Maciej Cichosz, Marek Kuś, Kamil Nalikowski, Marcin
Pawłowski, Michał Szczepanik, and Karol Życzkowski for their remarks at
various stages of this project. Some calculations were carried out at the
Academic Computer Center in Gdańsk.

## Appendix

###  .1 Subsystems defined by $\otimes_{213}$

Tensor product $\otimes_{213}$ leads to bits defined by

$\displaystyle P_{0}\otimes_{213}\mathbb{I}$
$\displaystyle=P_{00}+P_{10},\textrm{(0 of Bob)},$ (226) $\displaystyle
P_{1}\otimes_{213}\mathbb{I}$ $\displaystyle=P_{01}+P_{11},\textrm{(1 of
Bob)}$ (227) $\displaystyle\mathbb{I}\otimes_{213}P_{0}$
$\displaystyle=P_{00}+P_{01},\textrm{(0 of Alice)}$ (228)
$\displaystyle\mathbb{I}\otimes_{213}P_{1}$
$\displaystyle=P_{10}+P_{11}.\textrm{(1 of Alice)}$ (229)

Hence, Alice213$~{}=~{}$Bob123, Bob213 = Alice123, i.e.

$\displaystyle P_{\alpha}\otimes_{213}\mathbb{I}$ $\displaystyle=$
$\displaystyle\mathbb{I}\otimes_{123}P_{\alpha},$ (230)
$\displaystyle\mathbb{I}\otimes_{213}P_{\alpha}$ $\displaystyle=$
$\displaystyle P_{\alpha}\otimes_{123}\mathbb{I},$ (231)
$\displaystyle\sigma_{\mu}\otimes_{213}\sigma_{\nu}$ $\displaystyle=$
$\displaystyle\sigma_{\nu}\otimes_{123}\sigma_{\mu}.$ (232)

One may wonder if the resulting decomposition into subsystems should be
treated as different from the one implied by $\otimes_{123}$. If not, we
should identify systems that differ only by permutations of Alice and Bob,
which would further reduce the number of different tensor products to three.

###  .2 Subsystems defined by $\otimes_{231}$

Tensor product $\otimes_{231}$ is similar to $\otimes_{321}$, but now Bob231 =
Alice123 = Bob213,

$\displaystyle P_{0}\otimes_{231}\mathbb{I}$
$\displaystyle=P_{00}+P_{11},\textrm{(Alice {\tt IFF} Bob)}$ (233)
$\displaystyle P_{1}\otimes_{231}\mathbb{I}$
$\displaystyle=P_{01}+P_{10},\textrm{(Alice {\tt XOR} Bob)}$ (234)
$\displaystyle\mathbb{I}\otimes_{231}P_{0}$
$\displaystyle=P_{00}+P_{01},\textrm{(0 of Alice)}$ (235)
$\displaystyle\mathbb{I}\otimes_{231}P_{1}$
$\displaystyle=P_{10}+P_{11},\textrm{(1 of Alice)}$ (236)

and Alice231 = Alice321.

###  .3 Singlet state once again

The singlet state satisfies simultaneously

$\displaystyle(\sigma_{3}\otimes_{321}\mathbb{I})|\Psi_{-}\rangle$
$\displaystyle=$ $\displaystyle-|\Psi_{-}\rangle,$ (237)
$\displaystyle(\mathbb{I}\otimes_{321}\sigma_{1})|\Psi_{-}\rangle$
$\displaystyle=$ $\displaystyle-|\Psi_{-}\rangle,$ (238)
$\displaystyle(\mathbb{I}\otimes_{231}\sigma_{3})|\Psi_{-}\rangle$
$\displaystyle=$ $\displaystyle-|\Psi_{-}\rangle,$ (239)
$\displaystyle(\sigma_{1}\otimes_{231}\mathbb{I})|\Psi_{-}\rangle$
$\displaystyle=$ $\displaystyle-|\Psi_{-}\rangle,$ (240)

and

$\displaystyle(\sigma_{3}\otimes_{321}\sigma_{1})|\Psi_{-}\rangle$
$\displaystyle=$ $\displaystyle|\Psi_{-}\rangle,$ (241)
$\displaystyle(\sigma_{1}\otimes_{231}\sigma_{3})|\Psi_{-}\rangle$
$\displaystyle=$ $\displaystyle|\Psi_{-}\rangle,$ (242)

implying (here $\sigma_{k}|\pm_{k}\rangle=\pm|\pm_{k}\rangle$, $k=1,3$)

$\displaystyle|\Psi_{-}\rangle$ $\displaystyle=$
$\displaystyle|-_{3}\rangle\otimes_{321}|-_{1}\rangle$ (243) $\displaystyle=$
$\displaystyle|-_{1}\rangle\otimes_{231}|-_{3}\rangle,$ (244)

which agrees with (190).

Bearing in mind that single qubits are defined as $|0\rangle=|+_{3}\rangle$,
$|1\rangle=|-_{3}\rangle$, we observe that (243) agrees with the structure of
the observable $\sigma_{3}\otimes_{321}\sigma_{1}$ if we interpret the left
qubit satisfying $\sigma_{3}|-_{3}\rangle=-|-_{3}\rangle$ as the positive
answer to the question “Are the bits of Alice and Bob different?”

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