# Purifying teleportation

K. Roszak  Institute of Physics (FZU), Czech Academy of Sciences, Na Slovance
2, 182 00 Prague, Czech Republic<EMAIL_ADDRESS>J. K. Korbicz  Center for
Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668
Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract

Coupling to the environment typically suppresses quantum properties of
physical systems via decoherence mechanisms. This is one of the main obstacles
in practical implementations of quantum protocols. In this work we show how
decoherence effects can be reversed/suppressed during quantum teleportation in
a network scenario. Treating the environment quantumly, we show that under a
general pure dephasing coupling, performing a second teleportation step can
probabilistically reverse the decoherence effects if certain commutativity
conditions hold. This effect is purely quantum and most pronounced for qubit
systems, where in 25$\%$ of instances the decoherence can be reversed
completely. As an example, we show the effect in a physical model of a qubit
register coupled to a bosonic bath. We also analyze general $d$-dimensional
systems, identifying all instances of decoherence suppression. Our results are
proof-of-concept but we believe will be relevant for the emerging field of
quantum networks as teleportation is the key building block of network
protocols.

##  1 Introduction

The role of the environment and decoherence in quantum systems has been a
subject of intensive studies due to its great importance for both fundamental
understanding of the quantum theory [1, 2, 3] and for practical
implementations of various landmark quantum effects [4, 5, 6, 7, 8], which we
believe will lead to novel technological developments. One of such landmark
effects is the widely-known quantum state teleportation. It relies on quantum
entanglement as a resource for transmitting an unknown quantum state from one
particle to another via a well defined protocol. Since its discovery [9],
quantum teleportation has been extensively studied theoretically, e.g. in [10,
11, 12, 13, 14], and demonstrated experimentally [15, 16, 17, 18, 19, 20]. It
has become especially important recently due to its fundamental role for the
emerging quantum networks [21, 22]. One of the real-life challenges in the
practical implementation of teleportation is the inevitable coupling to the
environment and the resulting decoherence. The entangled resource, necessary
for establishing the teleportation channel, is especially sensitive to
decoherence because entanglement requires some level of non-locality, i. e.
coherences between distinct states of two completely distinguishable
particles, and non-local phase relations are always more fragile than local
ones, since they are easier to distinguish for the environment. This motivated
more realistic studies of quantum teleportation with noisy, non-ideal
entanglement resources [10, 11, 12, 13]. The role of the environment has been
analyzed e.g. in [11, 12], showing, as one would expect, its detrimental
influence on the teleportation fidelity. However, this does not always have to
be so and we can take advantage of the quantum nature of the interactions to
suppress decoherence.

In this work we show how to reverse detrimental effects of coupling to the
environment during teleportation. In particular, we demonstrate that repeating
a noisy teleportation, i.e. performing a second non-ideal teleportation
process coupled to the environment, can, under certain general conditions,
probabilistically reverse the decoherence effects instead of accumulating
them. This counterintuitive effect is purely quantum and based on a fully
quantum treatment of the environment. In particular, we assume a realistic
coupling via a pure dephasing interaction [23, 24, 25, 26], which appears in
many physical systems, e.g. a spin coupled to a bosonic bath. The effect is
most pronounced for the lowest dimensional quantum systems, i.e. spin-$1/2$ or
qubits, where we show that in 25$\%$ of the instances, the second
teleportation completely purifies the teleported state, resulting in a perfect
copy of the input state. For higher dimensional systems, the effect is still
there but less pronounced.

This decoherence reversal effect has been unnoticed before since environments’
influence has been usually modeled with the help of quantum channels or other
methods operating only at the level of the reduced density matrix of the
system of interest [10, 11, 12, 13]. The problem with this treatment is that
it effectively reduces the environment to a source of noise, which as turns
out is not always sufficient for a description of its effects. A fully quantum
inclusion of the environment, on the other hand, has lead to such
experimentally important decoherence reversal techniques as the spin echo (see
e.g. [27, 28, 29, 30]) and dynamical decoupling [31, 32, 33, 34]. In the case
of teleportation, purely quantum treatment of the environment was analyzed in
[35], where the teleportation of correlations with the environment in a cyclic
protocol where studied, but the reversal effects were not noticed.

##  2 Purifying teleportation protocol.

Figure 1: The two-step teleportation protocol along a chain allowing for
purification. a) Initialization and the action of decoherence on the entangled
resource $BC$. The state to be teleported is $|\psi\rangle_{A}$. b) Bell
measurement on qubits $AB$, followed by the teleportation $A\to C$ and a feed-
forward of the Bell measurement outcome $\gamma$ to qubits
$A^{\prime}B^{\prime}$. c) Initialization of qubits $A^{\prime}B^{\prime}$ in
the Bell state $|\Psi_{\gamma}\rangle$, corresponding to the measurement
outcome in the previous step, followed by the action of decoherence on
$A^{\prime}B^{\prime}$. d) The second, final teleportation $C\to A^{\prime}$.

We first consider a spin-$1/2$ system, composed of three particles (qubits):
$A$ in an unknown state $|\psi\rangle$ to be teleported and $BC$ prepared
initially in a maximally entangled state (the entanglement resource). The
environment $E$ is included in the quantum description and we initialize the
whole system in the state:

$\sigma(0)=|\psi\rangle_{A}\langle\psi|\otimes|\Phi_{+}\rangle_{BC}\langle\Phi_{+}|\otimes
R(0),$ (1)

where

$|\psi\rangle_{A}=\alpha|0\rangle+\beta|1\rangle$ (2)

is the state to be teleported, $|\Phi_{+}\rangle\equiv
1/\sqrt{2}(|00\rangle+|11\rangle)$ is one of the Bell states in some chosen
basis $|0\rangle$, $|1\rangle$, and $R(0)$ is an arbitrary state of the
environment. The particular choice of the Bell state is irrelevant for the
idea and we have chosen $|\Phi_{+}\rangle$ only for definiteness’ sake (the
remaining choice of $|\Psi_{\pm}\rangle$ is analyzed in Appendix B). The
interaction with the environment is modeled via a general pure decoherence
(dephasing) dynamics:

$H_{dec}=\sum_{i,j=0,1}|ij\rangle\langle ij|\otimes V_{ij},$ (3)

where $V_{ij}=V_{ij}^{\dagger}$ are some observables on the environment $E$.
The resulting evolution is controlled by the two-qubit state:

$U(\tau)=\sum_{i,j=0,1}|ij\rangle\langle ij|\otimes w_{ij}(\tau),\quad
w_{ij}(\tau)\equiv e^{-i\tau V_{ij}}.$ (4)

This is a very general formalism that can describe any source of pure
dephasing: entangling or separable [36], with or without a self Hamiltonian of
the environment in $V_{ij}$, systems interacting with the same or separate
environments, Markovian and non-Markovian etc.

The two-step noisy teleportation procedure [35] is constructed as follows; see
Fig. 1. The decoherence process (4) acts for some time $\tau$ on qubits $BC$,
coupling them with the environment. As mentioned before, we assume that the
decoherence affects only the entangled resource as it is more sensitive than a
single system (see e.g. [37]). Afterwards, following the standard
teleportation procedure, a two qubit Bell measurement is performed on $AB$ and
depending on the measurement outcome, the appropriate corrective unitary
operation is applied to the qubit $C$. This concludes the first step. The
outcome, conditioned on the Bell measurement result, is:

$\sigma_{\gamma}(\tau)=|\Psi_{\gamma}\rangle_{AB}\langle\Psi_{\gamma}|\otimes\rho^{(\gamma)}_{CE}(\tau).$
(5)

Here $|\Psi_{\gamma}\rangle_{AB}$ denotes the Bell state that was measured.
The state of $CE$ depends on the measurement result $\gamma$ as the corrective
unitary operations are applied to $C$ only and leave the environment part
alone. This leads to an effective outcome-dependent permutation on the $E$
side:

$\rho^{(\gamma)}_{CE}(\tau)=\sum_{j,j^{\prime}=0,1}\psi_{j}\psi_{j^{\prime}}^{*}|j\rangle\langle
j^{\prime}|\otimes R_{j\oplus m,j^{\prime}\oplus m}$ (6)

where $\psi_{k}$ are from eq. (2),
$R_{ij}=w_{ii}(\tau)R(0)w_{jj}^{\dagger}(\tau)$ are matrices on the
environment (which in principle can be very large), $\oplus$ denotes the
addition modulo 2, and $m=0$ for $|\Phi_{\pm}\rangle$ outcomes and $m=1$ for
$|\Psi_{\pm}\rangle$. Obviously without decoherence $\rho_{CE}(\tau)$ would be
a product state $\rho_{CE}(\tau)=|\psi\rangle_{C}\langle\psi|\otimes R(0)$,
but in the presence of decoherence (4), the state (6) gets correlated,
possibly entangled [36, 38]. This, as expected, alters the state of the qubit
$C$, which for $\gamma=\Phi_{\pm}$ reads:

$\varrho_{C}(\tau)=\begin{pmatrix}|\alpha|^{2}&\alpha\beta^{*}c(\tau)\\\
\alpha^{*}\beta c^{*}(\tau)&|\beta|^{2}\end{pmatrix},$ (7)

where

$c(\tau)\equiv{\rm Tr}_{E}[w_{11}^{\dagger}(\tau)w_{00}(\tau)R(0)]$ (8)

is a decoherence factor between the states $|00\rangle$, $|11\rangle$ (for
$\gamma=\Psi_{\pm}$, factors $c(\tau)$ and $c^{*}(\tau)$ are interchanged; see
Appendix A for details). Decoherence clearly affects the fidelity
$F_{1}=\langle\psi|\varrho_{C}|\psi\rangle$ between the teleported state (7)
and the original state (2):

$F_{1}(\tau)=1-2|\alpha\beta|^{2}\left[1-\mathrm{Re}\left[c(\tau)\right]\right].$
(9)

We note that the ideal decoherence, $c(\tau)=0$, leads to
$F_{min}=1-2|\alpha\beta|^{2}$, which is in general greater than zero as we
are only destroying the coherences in (7) and that is not enough to make (7)
orthogonal to (2).

We now perform the second noisy teleportation. We prepare the additional
qubits $A^{\prime}B^{\prime}$ in the Bell state corresponding to the first
Bell measurement outcome and this state will serve as the entanglement
resource for the teleportation. Thus the initial state for the second step is
of the form (5) but with the resource prepared on a new system to propagate
the state further rather than coming back to the original qubit $A$ like it
was in [35]:

$\sigma^{\prime}_{\gamma}(\tau)=|\Psi_{\gamma}\rangle_{A^{\prime}B^{\prime}}\langle\Psi_{\gamma}|\otimes\rho^{(\gamma)}_{CE}(\tau).$
(10)

The decoherence process (4) is assumed to couple to the same environment and
last for the same time $\tau$ as in the first teleportation and this is
crucial for our protocol. This is of course an idealized situation, but we are
concerned with a proof-of-principle here, investigating how much suppression
is theoretically possible. Coupling to the same environment can be a good
approximation to the situations when the nodes are close enough compared to
the effective range of the interaction with the environment. Robustness with
respect to small time mismatches will be discussed below. We assume that
decoherence again affects the entangled resource $A^{\prime}B^{\prime}$ as the
most sensitive. After measuring qubits $B^{\prime}C$ in the Bell basis, we
apply a correction unitary to $A^{\prime}$ but this time the unitaries depend
not only on the result of the second Bell measurement, but also on the first
one as the latter determines the entanglement resource in (5); see Appendix A.
The final state of $A^{\prime}$ depends now on the measurement outcomes in
both steps but is the same for either sign of $|\Phi_{\pm}\rangle_{AB}$ and of
$|\Psi_{\pm}\rangle_{AB}$ in (5), reducing the number of all possibilities to
four. For $|\Phi_{\pm}\rangle_{AB}$ in (5), the second decoherence process
will again be governed by the $w_{00}(\tau),w_{11}(\tau)$ and the state of
$A^{\prime}$ will be given by (7) but with $c(\tau)$ replaced by (derived in
the Appendix A):

$\displaystyle C_{\Phi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{11}^{\dagger 2}(\tau)w_{00}^{2}(\tau)R(0)],$ (11) $\displaystyle
C_{\Psi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{11}^{\dagger}(\tau)w_{00}^{\dagger}(\tau)w_{11}(\tau)w_{00}(\tau)R(0)],$
(12)

depending on the result of the second Bell measurement. $C_{\Phi},C_{\Psi}$
correspond, respectively, to $|\Phi_{\pm}\rangle$ and $|\Psi_{\pm}\rangle$. An
interesting effect happens for

$[w_{11}(\tau),w_{00}(\tau)]=0,$ (13)

when

$C_{\Psi}(\tau)={\rm Tr}_{E}R(0)=1$ (14)

strictly and irrespectively of any other parameters. Thus, for those instances
when $|\Psi_{\pm}\rangle$ is measured, the second noisy teleportation actually
purifies the state (7) back to the initial form instead of adding more noise
as it is the case for $|\Phi_{\pm}\rangle$ measurements.

This does not happen for outcomes
$|\Psi_{\gamma}\rangle_{AB}=|\Psi_{\pm}\rangle_{AB}$ of the first measurements
as then the decoherence factors read:

$\displaystyle C^{\prime}_{\Phi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{00}^{\dagger}(\tau)w_{01}^{\dagger}(\tau)w_{10}(\tau)w_{11}(\tau)R(0)],$
(15) $\displaystyle C^{\prime}_{\Psi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{00}^{\dagger}(\tau)w_{10}^{\dagger}(\tau)w_{01}(\tau)w_{11}(\tau)R(0)],$
(16)

and no commutativity condition can make it equal to unity for all $\tau$. It
is easy to check that the probabilities of each of the four possible outcomes
(11), (12), (15), (16) are all equal to $1/4$. Thus, the purifying
teleportation (14) happens in $25\%$ of the cases. The the statistical state
at the end of our protocol $\varrho_{A^{\prime}f}(\tau)$ is the mixture with
equal weights $1/4$ of the conditional states corresponding to the each set of
the measurement outcomes (see Appendix A). Its fidelity with respect to the
original state (2) is given by the average:

$F_{2}(\tau)=1-2|\alpha\beta|^{2}\left[1-\mathrm{Re}\left[C_{av}(\tau)\right]\right],$
(17)

where

$C_{av}(\tau)\equiv\frac{1}{4}\left[1+C_{\Phi}(\tau)+C^{\prime}_{\Phi}(\tau)+C^{\prime}_{\Psi}(\tau)\right].$
(18)

In many interesting situations, like the one described below, all the
$C(\tau)$-factors in (9) and (18) tend asymptotically to zero with $\tau$.
Then the presence of the constant factor $1/4$ in (18) causes $F_{2}>F_{min}$
asymptotically and as a result $F_{2}>F_{1}$. Thus the purifying effect
reveals itself not only in postselected events but also in the average state.

It is clear how the purifying effect propagates down a network: After a third
teleportation there will be instances when the decoherence is reduced to
single-step values, after a fourth teleportation there will again appear
complete reversals, and so on.

The commutativity condition (13) is not as restrictive as it may seem.
Obviously it is satisfied when the environmental observables in (3) commute:
$[V_{ij},V_{kl}]=0$, which includes such a fundamentally important case [2] as
$H\approx A\otimes V$, where $A$, $V$ are some observables on the system and
the environment respectively. Let us also briefly address the question of
robustness, leaving more detailed analysis to a future work. First of all,
decoherence factors $C_{\Psi}(\tau)$, $C_{\Phi}(\tau)$ are continuous
functions of $V_{ij}$, $R(0)$, and $\tau$ and thus small variations in these
parameters will lead to small variations of the coefficients. As an example
let us examine a small time mismatch. Assuming for simplicity
$[V_{00},V_{11}]=0$, which ensures (13) for all times, we obtain:

$\displaystyle|C_{\Psi}(\tau_{1},\tau_{2})|^{2}=\left|{\rm
Tr}_{E}\left[w_{11}(\Delta\tau)w_{00}(-\Delta\tau)R(0)\right]\right|^{2}$
$\displaystyle\approx
1-\Delta\tau^{2}\langle\Delta(V_{11}-V_{00})^{2}\rangle\approx
e^{-\Delta\tau^{2}\langle\Delta(V_{11}-V_{00})^{2}\rangle},$

so that to the lowest order, the departure from the ideal canceling (14)
decays quadratically with $\Delta\tau=\tau_{2}-\tau_{1}$ and with the rate
controlled by the variance of $(V_{11}-V_{00})$ in the initial state $R(0)$.

Figure 2: [Color online] Example of purifying teleportation in the spin-boson
model. Red solid line shows the single-step fidelity (9) and green, thick
solid line the two-step average fidelity (17) as functions of decoherence
duration $\tau$. Time $\tau$ is plotted in cutoff units $\Lambda^{-1}$,
temperature is set to $T=1/10\Lambda$, time-of-flight $\bar{t}=3\Lambda^{-1}$,
and Ohmicity parameter is $s=3$. Initial state (2) is chosen with
$\alpha=\beta=1/\sqrt{2}$. The plot of (9) shows a well known recoherence
impulse at $\tau=\bar{t}$ but asymptotically it decays to a value lower than
the assymptotic of (17). Thus, for long enough duration $\tau$ of the
decoherence processes, the second application of decoherence $+$ teleportation
actually increases the fidelity of the output state w.r.t. the one-step
process. To better visualize how it happens, we plot all the components of
(17), each entering with the equal weight of $1/4$: Black dotted line
corresponds to the partial fidelity associated with $C_{\Phi}(\tau)$, dashed-
dotted line to both factors (22), and the dashed line to the constant term
(14). Both $C_{\Phi}(\tau)$\- and $C^{\prime}_{\Phi,\Psi}$-dependent factors
decay quickly to $F_{min}=1/2$ as expected, but the presence of the constant
term makes final fidelity asymptoticaly higher than the first-step fidelity.
For short durations $\tau\ll\Lambda^{-1}$, the situation is reversed and
$F_{2}(\tau)<F_{1}(\tau)$.

##  3 Spin register example

We consider a well known model of a two-qubit register coupled to a bosonic
bath (see e.g. [37]), described by the Hamiltonian:

$H=\sum_{m=1,2}\frac{\sigma_{z}^{(m)}}{2}\otimes\sum_{k}(g_{mk}a_{k}^{\dagger}+g_{mk}^{*}a_{k})+\sum_{k}H_{k},$
(20)

where $\sigma_{z}^{(m)}$ are the $m$-th qubit $z$-axis Pauli matrices,
$a_{k},a_{k}^{\dagger}$ are the $k$-th mode annihilation and creation
operators, $g_{mk}$ are complex coupling constants, and
$H_{k}=\omega_{k}a_{k}^{\dagger}a_{k}$. Here
$V_{ij}=\sum_{k}(\epsilon_{i}g_{1k}a_{k}^{\dagger}+\epsilon_{j}g_{2k}a^{\dagger}_{k}+h.c.+H_{k})$,
where $\epsilon_{i}=\mp 1/2$ for $i=0,1$ respectively. Although
$[V_{ij},V_{lm}]\neq 0$, one checks that condition (13) is satisfied:
$[w_{11},w_{00}]=0$ (also $[w_{01},w_{10}]=0$ as any other pair with co-linear
$\bm{\epsilon}$, $\bm{\epsilon}^{\prime}$). Assuming the initial state of the
environment to be thermal, $R(0)=1/Ze^{-\beta\sum_{k}H_{k}}$, with the inverse
temperature $\beta$, the decoherence factor (8) has been found in e.g. [37,
39]:

$\displaystyle\ln c(\tau)=$ $\displaystyle-2\int_{0}^{\infty}d\omega
J(\omega)\frac{1-\cos\omega\tau}{\omega^{2}}\left(1+\cos\omega\bar{t}\right)\coth\frac{\omega\beta}{2},$
(21)

Here $\bar{t}\equiv k|r_{1}-r_{2}|/\omega$ is the time-of-flight between the
qubit positions, assuming that ${g}_{mk}=g_{k}e^{-kr_{m}}$, which is the case
e.g. for the electromagnetic bath in the dipole approximation. Further
assuming the spectral density of the form
$J(\omega)=\omega(\omega/\Lambda)^{s-1}e^{-\omega/\Lambda}$, where $\Lambda$
is the cut-off frequency, the above integral can be calculated analytically
[39], but we will not need it here. The factors
$C_{\Phi},C^{\prime}_{\Phi},C^{\prime}_{\Psi}$ can be also calculated (see
Appendix C). Like $c(\tau)$, they are real and read: $\ln C_{\Phi}(\tau)=4\ln
c(\tau)$,

$\displaystyle\ln C^{\prime}_{\Phi}(\tau)$ $\displaystyle=\ln
C^{\prime}_{\Psi}(\tau)$ (22) $\displaystyle=-4\int d\omega
J(\omega)\frac{1-\cos\omega\tau}{\omega^{2}}\coth\frac{\omega\beta}{2}.$

Note the absence of the time-of-flight $\bar{t}$ above, indicating collective
decoherence, when the register behaves like a single system [37]. Thus, after
the second step there are three types of decoherence present: No decoherence
(14), true $2$-qubit decoherence (21), and collective decoherence (22). Sample
plots of the fidelities, corresponding to those three processes, as well as
the average fidelity (17) and the fidelity after the first teleportation (9),
are presented in Fig. 2. We see that for short durations $\tau\ll\Lambda^{-1}$
of the decoherence processes, teleportation fidelity after one step is higher
than after two steps as one would expect. However, for a longer duration all
the contributions to $F_{2}(\tau)$ from
$C_{\Phi},C^{\prime}_{\Phi},C^{\prime}_{\Psi}$ decay quickly to $F_{min}$, but
the constant term due to (14) forces that $F_{2}>F_{1}$ after the initial
transients, including a propagation of an disturbance between the qubits.
Thus, in this example, the purifying effect of the second teleportation is
present in the statistical state too, if the decoherence duration is long
enough.

##  4 Generalization to dimension $d$

The generalized teleportation protocol is defined by the following $d^{2}$
measurement vectors and correction unitaries [9]:

$\displaystyle|\Psi_{nm}\rangle$
$\displaystyle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}e^{\frac{2\pi
ijn}{d}}|j,j\oplus m\rangle,$ (23) $\displaystyle U_{nm}$
$\displaystyle=\sum_{j=0}^{d-1}e^{\frac{2\pi ijn}{d}}|j\rangle\langle j\oplus
m|,$ (24)

where $n,m\in[0,d-1]$, same as the basis indicies
$j,j^{\prime},k,k^{\prime},\cdots$, and $\oplus$ is the addition modulo $d$.
We assume the following initial state:

$\sigma(0)=|\psi\rangle_{A}\langle\psi|\otimes|\Psi_{00}\rangle_{BC}\langle\Psi_{00}|\otimes
R(0),$ (25)

where $|\psi\rangle=\sum_{j=0}^{d-1}\psi_{j}|j\rangle$ and
$|\Psi_{00}\rangle=1/\sqrt{d}\sum_{j}|jj\rangle$. The final state after the
two-step teleportation reads (see Appendix D):

$\varrho^{A^{\prime}}_{mm^{\prime}}=\sum_{jj^{\prime}}\psi_{j}\psi^{*}_{j^{\prime}}C^{mm^{\prime}}_{jj^{\prime}}|j\rangle\langle
j^{\prime}|,$ (26)

where:

$\displaystyle C^{mm^{\prime}}_{jj^{\prime}}(\tau)=$ $\displaystyle{\rm
Tr}_{E}\left[w_{j\oplus M,j\oplus M\oplus m}w_{j\oplus m,j\oplus
m}R(0)\right.$ $\displaystyle\times\left.w^{\dagger}_{j^{\prime}\oplus
m,j^{\prime}\oplus m}w^{\dagger}_{j^{\prime}\oplus M,j^{\prime}\oplus M\oplus
m}\right]$ (27)

are the decoherence factors between the levels $jj^{\prime}$, with
$m,m^{\prime}$ labeling the results of the first and the second Bell
measurements (the states are independent of the $n$-indicies), and
$M=\left[d-(m\oplus m^{\prime})\right]\text{mod}\,d$. A detailed analysis
found in the Appendix D shows that under the commutation condition
$[w_{ii},w_{jj}]=0$, the decoherence suppression/reduction happens only when
$m=0$ and has the following pattern: i) For every $m^{\prime}$, there will be
$d$ levels $jj^{\prime}$ in the corresponding state $\varrho_{0m^{\prime}}$
(26), such that the decoherence factors (27) between them reduce to the one-
step form, i.e. like if only one decoherence process has happened; ii)
Additionally, when $d$ is even, the state corresponding to $m^{\prime}=d/2$
will have $d/2$ of its coherences fully restored to the original form like if
no decoherence has happened at all. These findings have two implications: a)
using higher-$d$ systems gives no advantage in terms of noise
cancellation/suppression in the network scenario considered here; to the
contrary, in relatively less instances coherence losses can be
suppressed/reversed than for qubits; b) if higher-$d$ systems must be used for
some reason, we provide the list of instances when the noise suppression
happens.

##  5 Concluding remarks

We have presented a decoherence reversal/suppression mechanism in the quantum
teleportation protocol, which allows to use quantum nature of decoherence to
our advantage. Working in a quantum open systems framework, we have studied a
fully quantum teleportation scheme, where the environment is modeled as a
quantum system, and shown that a repetition of the teleportation protocol can,
under suitable conditions, reverse the detrimental decoherence effects, or at
least suppress them, instead of accumulating. This rather counterintuitive
effect is due to the quantumness of the environment and is probabilistic. That
is, it is guaranteed to occur when a certain measurement result is obtained,
in the same manner as it happens e.g. in the famous Elitzur-Vaidman
interaction-free measurement [40]. It is also most visible for the lowest
dimensional quantum systems, which reflects its highly non-classical nature:
For qubits there is a complete decoherence reversal with probability $1/4$
while for quitrits in $1/3$ of the instances the decohrence reduces to the
single-step value.

Let us briefly discuss a connection to other decoherence reversal and
purification techniques. Spin-echo (see e.g. [27] or [30] for a more relevant
exposition), is an experimental technique, widely used to prolong coherence
times. While there are certain similarities, e.g. both protocols rely on a
two-step procedure and commutativity of the conditional environmental
evolutions, there are fundamental differences. Firstly, decoherence here is
acting on physically distinct systems in each step, while in the spin-echo it
acts on the same system. Secondly, and most importantly, unlike the spin-echo,
which is basically a single-system effect, purifying teleportation is a
multipartite effect, which fully relies on entanglement. The latter allows to
shift decoherence effects from one system to another [35] and opens the chance
for their cancellation. Superbroadcasting effect [41], is in turn a
purification effect, where $N$ input copies of a sufficiently mixed qubit
state can be broadcasted in form of correlated copies to $M>N$ receivers, such
that each of the local states has greater purity, ${\rm Tr}\varrho^{2}$, than
the initial state. The difference is that superboradcasting assumes an
inherent noise, which must be present at the input, and transfers it to
correlations among the copies, purifying along a Bloch vector, while purifying
teleportation physically reduces the noise dynamically acquired from the
environment.

The significance of our effect is two-fold. First, it shows how to improve the
fidelity of a single noisy teleportation – quite surprisingly by performing a
second one (this is most notable for qubits, but also for other even-
dimensional systems). But more interestingly it shows, at least in theory, a
way to mitigate teleportation noise in quantum networks, where teleportation
is the key primitive for quantum information distribution. By a proper design
of the teleportation processes, the noise can be reduced instead of
accumulated during multiple teleportations between the nodes. Of course the
practical issues must be first addressed. We have already briefly commented on
the time mismatch. Another important point is the common environment. For the
effect to happen there must be at least some overlap between the environments
in both steps and we expect the magnitude of the effect will approach the
theoretical bounds presented here with the growing overlap. The common
environment may be also viewed as a condition on the physical node separation,
i.e. that it is not too big compared to the effective interaction range, as we
have already mentioned. Other practical issues include possible difference in
the coupling observables and the neglected decoherence of the individual
particles. We leave those important questions for a future work.

Acknowledgements—JKK acknowledges the financial support of the Polish National
Science Center (NCN) through the grant no. 2019/35/B/ST2/01896 and the
hospitality of Institute of Physics (FZU) of the Czech Academy of Sciences. KR
acknowledges support from project 20-16577S of the Czech Science Foundation.

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##  Appendix A The two-step teleportation procedure

We present the details of the two-step teleportation; see Fig. 1. We
initialize the whole system in the state:

$\sigma(0)=|\psi\rangle_{A}\langle\psi|\otimes|\Phi_{+}\rangle_{BC}\langle\Phi_{+}|\otimes
R(0),$ (28)

and then apply the decoherence process (4) to $BCE$.

Then the standard teleportation procedure is performed: A joint measurement in
the Bell basis $|\Phi_{\pm}\rangle,|\Psi_{\pm}\rangle$ on the qubits $AB$
followed by a corrective unitary operation of qubit $C$, conditioned on the
measurement outcome. For completeness we recall this by now famous set of
unitaries:

$U_{\Phi_{+}}=\bm{1},\ U_{\Phi_{-}}=\sigma_{z},\ U_{\Psi_{+}}=\sigma_{x},\
U_{\Psi_{-}}=i\sigma_{y}.$ (29)

Consequently, the post-teleportation states of the whole system (each obtained
with probability $1/4$) are given by:

$\displaystyle\sigma_{\Phi}(\tau)$
$\displaystyle=|\Phi_{\pm}\rangle\langle\Phi_{\pm}|\otimes\left(\begin{array}[]{cc}|\alpha|^{2}R_{00,00}&\alpha\beta^{*}R_{00,11}\\\
\alpha^{*}\beta R_{11,00}&|\beta|^{2}R_{11,11}\end{array}\right)_{CE},$ (32)
$\displaystyle\sigma_{\Psi}(\tau)$
$\displaystyle=|\Psi_{\pm}\rangle\langle\Psi_{\pm}|\otimes\left(\begin{array}[]{cc}|\alpha|^{2}R_{11,11}&\alpha\beta^{*}R_{11,00}\\\
\alpha^{*}\beta R_{00,11}&|\beta|^{2}R_{00,00}\end{array}\right)_{CE}.$ (35)

Here

$R_{ii,jj}(\tau)=w_{ii}(\tau)R(0)w_{jj}^{\dagger}(\tau)$ (36)

(in this Section we use a full index notation $R_{ii,jj}$ for clarity rather
than the abbreviated one $R_{ij}$ used elsewhere). In the main text these
states are written in a compact way in eqs (5) and (6). Note that the
difference between the two states in the $CE$ part lies in the environmental
operators $R_{ii,jj}$ having their indices interchanged. This leads to the
similar states of $C$ of the form (7) with eventually $c(\tau)\leftrightarrow
c^{*}(\tau)$. Thus, the degree of coherence of qubit $C$ is the same, but
there may be a difference in the phases.

We are now ready for the second teleportation process. Its aim is to propagate
the state of $C$ further on the same time increasing its fidelity w.r.t. the
original state. To this aim, we prepare additional qubits
$A^{\prime}B^{\prime}$ in the Bell state corresponding to the first Bell
measurement result. This will serve as the entanglement resource. Thus, the
second teleportation step starts with the state (32) or (35) as the initial
state, but with the Bell states prepared on additional qubits
$A^{\prime}B^{\prime}$. In both cases, the first the decoherence process acts
on the Bell states. We assume it acts for the same duration $\tau$ as in the
first step. Then a Bell measurement is performed on qubits $B^{\prime}C$,
followed by the appropriate unitary operation on qubit $A^{\prime}$. Since now
the initial resource for the teleportation is an arbitrary Bell state, the
corrective unitary operations will depend on the results of both measurements
and will be given by some permutation of the set (29). We will explicitly
state the correct unitaries in what follows.

###  A.1 Second step: first measurement outcome $|\Phi_{\pm}\rangle$

We take (32) as the starting point. In this case, the decoherence is described
by the same conditional environmental operators $w_{ii}(\tau)$ as in the first
teleportation. The state of the whole system after the action of the
decoherence (4) on $A^{\prime}B^{\prime}E$ is given by:

$\sigma_{\Phi_{\pm}}^{dec}(\tau)=\frac{1}{2}\left(\begin{array}[]{cccc}|\alpha|^{2}R_{00,00}^{00,00}&\alpha\beta^{*}R_{00,11}^{00,00}&\pm|\alpha|^{2}R_{00,00}^{00,11}&\pm\alpha\beta^{*}R_{00,11}^{00,11}\\\
\alpha^{*}\beta
R_{11,00}^{00,00}&|\beta|^{2}R_{11,11}^{00,00}&\pm\alpha^{*}\beta
R_{11,00}^{00,11}&\pm|\beta|^{2}R_{11,11}^{00,11}\\\
\pm|\alpha|^{2}R_{00,00}^{11,00}&\pm\alpha\beta^{*}R_{00,11}^{11,00}&|\alpha|^{2}R_{00,00}^{11,11}&\alpha\beta^{*}R_{00,11}^{11,11}\\\
\pm\alpha^{*}\beta
R_{11,00}^{11,00}&\pm|\beta|^{2}R_{11,11}^{11,00}&\alpha^{*}\beta
R_{11,00}^{11}&|\beta|^{2}R_{11,11}^{11,11}\end{array}\right).$ (37)

The rows and columns are ordered as:
$|000\rangle,|001\rangle,|110\rangle,|111\rangle$ and the trivial rows and
columns have been omitted. The environmental operators are given by:

$R_{ii,jj}^{kk,qq}(\tau)=w_{kk}(\tau)w_{ii}(\tau)R(0)w_{jj}^{\dagger}(\tau)w_{qq}^{\dagger}(\tau),$
(38)

the indices $i,j$ describe the decoherence process before the first
teleportation, while the indices $k,q$ correspond to the second one.

Now the Bell measurement is performed on the qubits $B^{\prime}C$ and suitable
correction unitaries are applied to qubit $A^{\prime}$. Like in the first
step, there are four cases here, arranged in two families.

The second outcome $|\Phi_{\pm}\rangle$. It is easy to check that the
following operations will perform the suitable corrections on $A^{\prime}$ in
this case:

$U_{\Phi_{+}\Phi_{+}}=U_{\Phi_{-}\Phi_{-}}=\bm{1},\
U_{\Phi_{+}\Phi_{-}}=U_{\Phi_{-}\Phi_{+}}=\sigma_{z},$ (39)

where $U_{\Phi\Phi^{\prime}}$ corresponds to the case when the first Bell
measurement resulted in $\Phi$, while the second in $\Phi^{\prime}$. The final
state of $AE$ is in this case given by:

$\rho_{A^{\prime}E}^{\Phi\Phi}(\tau)=\left(\begin{array}[]{cc}|\alpha|^{2}R_{00,00}^{00,00}&\alpha\beta^{*}R_{00,11}^{00,11}\\\
\alpha^{*}\beta
R_{11,00}^{11,00}&|\beta|^{2}R_{11,11}^{11,11}\end{array}\right).$ (40)

The coherence of $A^{\prime}$ is obtained by tracing out the environment from
(40) and is given by eq. (11).

The second outcome $|\Psi_{\pm}\rangle$. The corrective unitaries are now
given by:

$U_{\Phi_{+}\Psi_{+}}=U_{\Phi_{-}\Psi_{-}}=\sigma_{x},\
U_{\Phi_{+}\Psi_{-}}=U_{\Phi_{-}\Psi_{+}}=i\sigma_{y},$ (41)

and lead to the following state:

$\rho_{A^{\prime}E}^{\Phi\Psi}(\tau)=\left(\begin{array}[]{cc}|\alpha|^{2}R_{00,00}^{11,11}&\alpha\beta^{*}R_{00,11}^{11,00}\\\
\alpha^{*}\beta
R_{11,00}^{00,11}&|\beta|^{2}R_{11,11}^{00,00}\end{array}\right).$ (42)

The coherence of $A^{\prime}$ obtained from the state (42) is given by (12).
For commuting conditional evolution operators $w_{ii}$, it is in this case
that we observe the purification.

###  A.2 Second step: first measurement outcome $|\Psi_{\pm}\rangle$

The starting state is now (35). There is no qualitative difference between
this case and the teleportation procedure described in the previous section,
with the exception of the decoherence process, which is now governed by
different environmental evolution operators: $w_{01}(\tau)$ and
$w_{10}(\tau)$. This is because the Bell states are now superpositions of
states from a different subspace of the two qubit Hilbert space.

After the decoherence process, the state (35) takes the form

$\sigma_{\Psi_{\pm}}^{dec}(\tau)=\frac{1}{2}\left(\begin{array}[]{cccc}|\alpha|^{2}R_{11,11}^{01,01}&\alpha\beta^{*}R_{11,00}^{01,01}&\pm|\alpha|^{2}R_{11,11}^{01,10}&\pm\alpha\beta^{*}R_{11,00}^{01,10}\\\
\alpha^{*}\beta
R_{00,11}^{01,01}&|\beta|^{2}R_{00,00}^{01,01}&\pm\alpha^{*}\beta
R_{00,11}^{01,10}&\pm|\beta|^{2}R_{00,00}^{01,10}\\\
\pm|\alpha|^{2}R_{11,11}^{10,01}&\pm\alpha\beta^{*}R_{11,00}^{10,01}&|\alpha|^{2}R_{11,11}^{10,10}&\alpha\beta^{*}R_{11,00}^{10,10}\\\
\pm\alpha^{*}\beta
R_{00,11}^{10,01}&\pm|\beta|^{2}R_{00,00}^{10,01}&\alpha^{*}\beta
R_{00,11}^{10,10}&|\beta|^{2}R_{00,00}^{10,10}\end{array}\right).$ (43)

The second outcome $|\Phi_{\pm}\rangle$. The correct unitaries are given by:

$U_{\Psi_{+}\Phi_{+}}=U_{\Psi_{-}\Phi_{-}}=\sigma_{x},\
U_{\Psi_{+}\Phi_{-}}=U_{\Psi_{-}\Phi_{+}}=i\sigma_{y},$ (44)

which leads to the following state:

$\rho_{A^{\prime}E}^{\Psi\Phi}(\tau)=\left(\begin{array}[]{cc}|\alpha|^{2}R_{11,11}^{10,10}(\tau)&\alpha\beta^{*}R_{11,00}^{10,01}(\tau)\\\\[4.0pt]
\alpha^{*}\beta
R_{00,11}^{01,10}(\tau)&|\beta|^{2}R_{00,00}^{01,01}(\tau)\end{array}\right),$
(45)

and the coherence of the qubit $A^{\prime}$ is:

$C^{\prime}_{\Phi}(\tau)={\rm
Tr}_{E}[w_{00}^{\dagger}(\tau)w_{01}^{\dagger}(\tau)w_{10}(\tau)w_{11}(\tau)R(0)].$
(46)

The second outcome $|\Psi_{\pm}\rangle$. The corrective unitaries are now
given by:

$U_{\Psi_{+}\Psi_{+}}=U_{\Psi_{-}\Psi_{-}}=\bm{1},\
U_{\Psi_{+}\Psi_{-}}=U_{\Psi_{-}\Psi_{+}}=\sigma_{z},$ (47)

and lead to the following state:

$\rho_{A^{\prime}E}^{\Psi\Psi}(\tau)=\left(\begin{array}[]{cc}|\alpha|^{2}R_{11,11}^{01,01}&\alpha\beta^{*}R_{11,00}^{01,10}\\\
\alpha^{*}\beta
R_{00,11}^{10,01}&|\beta|^{2}R_{00,00}^{10,10}\end{array}\right),$ (48)

so the coherence is given by:

$C^{\prime}_{\Psi}(\tau)={\rm
Tr}_{E}[w_{00}^{\dagger}(\tau)w_{10}^{\dagger}(\tau)w_{01}(\tau)w_{11}(\tau)R(0)].$
(49)

##  Appendix B $\Psi_{\pm}$ initial state

In case the first teleportation process is performed using the Bell state
$|\Psi_{\pm}\rangle_{BC}$ (instead of $|\Phi_{+}\rangle_{BC}$ in eq. (1)), the
difference in the whole scenario boils down to exchanging the conditional
evolution operators of the environment according to:

$\hat{w}_{00}(\tau)\leftrightarrow\hat{w}_{01}(\tau),\
\hat{w}_{11}(\tau)\leftrightarrow\hat{w}_{10}(\tau).$ (50)

This means that the operators governing the initial decoherence process are
now $\hat{w}_{01}(\tau)$ and $\hat{w}_{10}(\tau)$, influencing the state in
eq. (6) accordingly, so that the coherence of the qubit $C$ is now given by

$c(\tau)\equiv{\rm Tr}_{E}[w_{10}^{\dagger}(\tau)w_{01}(\tau)R(0)]$ (51)

instead of (8).

As before, the amount of coherence present in the state of qubit $A$ at the
end of the protocol depends on the measurement outcome in the first
teleportation, which determines the decoherence which happens before the
second teleportation, and on the outcome of the last measurement. If the first
measurement outcome is $|\Psi_{\pm}\rangle_{AB}$ then instead of (11) and (12)
we obtain:

$\displaystyle C_{\Psi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{01}^{\dagger 2}(\tau)w_{10}^{2}(\tau)R(0)],$ (52) $\displaystyle
C_{\Phi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{01}^{\dagger}(\tau)w_{01}^{\dagger}(\tau)w_{10}(\tau)w_{01}(\tau)R(0)],$
(53)

where the indexes $\Phi$ and $\Psi$ indicate two possible second measurement
outcomes. The purifing condition (13) now reads:

$[w_{01},w_{10}]=0,$ (54)

leading to

$C_{\Phi}(\tau)=1.$ (55)

If the first measurement outcome does not match the initial Bell state then
the coherence of qubit $A^{\prime}$ at the end is given by

$\displaystyle C^{\prime}_{\Psi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{10}^{\dagger}(\tau)w_{00}^{\dagger}(\tau)w_{11}(\tau)w_{01}(\tau)R(0)],$
(56) $\displaystyle C^{\prime}_{\Phi}(\tau)$ $\displaystyle\equiv{\rm
Tr}_{E}[w_{10}^{\dagger}(\tau)w_{11}^{\dagger}(\tau)w_{00}(\tau)w_{01}(\tau)R(0)].$
(57)

As before, each of the cases (52), (53), (56), (57) happen with the same
probability of $1/4$ so that the purifying teleportation (55) happens in
$25\%$ cases.

##  Appendix C $C$-factors in the spin-boson model

Decoehernce factors (11), (12) and (15), (16) correspond to a “stroboscopic
decoherence” – a decoherence process, interrupted by measurements. They
contain four conditional evolution operators $w_{ij}$ unlike the usually
studied decoherence factors which contain two. We will find them for the
$\Phi_{\pm}$ initial state as studied in the main text. The calculations for
$\Psi_{\pm}$ are analogous. Let us first recall the well known form of
$w_{ij}$. We will first slightly change the notation for the later
convenience, orginizing the double index $ij$ into a vector
$\bm{\epsilon}=(\epsilon_{1},\epsilon_{2})$, where $\epsilon_{m}=\mp 1/2$
corresponds to $i=0,1$. In the interaction picture, $w_{\bm{\epsilon}}$ are
then given by [37] (we are dropping the environmental mode index $k$ for
clarity):

$w_{\bm{\epsilon}}^{I}=D\left[\alpha(\tau)\bm{\epsilon}\cdot\bm{g}\right]e^{i\xi(\tau)|\bm{\epsilon}\cdot\bm{g}|^{2}},$
(58)

where $D(\cdot)$ is the displacement operator,
$\bm{g}=g(e^{-ikr_{1}},e^{-ikr_{2}})$ is the vector of coupling constants, and
the time-dependent functions read:

$\displaystyle\alpha(\tau)$ $\displaystyle=\frac{1-e^{i\omega\tau}}{\omega},$
(59) $\displaystyle\xi(\tau)$
$\displaystyle=\frac{\omega\tau-\sin\omega\tau}{\omega^{2}}.$ (60)

We calculate

${\rm Tr}[w_{\bm{a}}^{\dagger}w_{\bm{b}}^{\dagger}w_{\bm{c}}w_{\bm{d}}R(0)].$
(61)

Since $R(0)$ is a thermal state, it commutes with the free Hamiltonian of the
environment and we can use the interaction picture. Using multiple times the
composition law for the displacement operators
$D(a)D(b)=e^{(ab^{*}-a^{*}b)/2}D(a+b)$ and the thermal average formula:

${\rm
Tr}[D(a)\varrho_{T}]=\exp\left[-\frac{|a|^{2}}{2}\coth(\frac{\beta\omega}{2})\right]$
(62)

we find:

$\displaystyle{\rm
Tr}[w_{\bm{a}}^{\dagger}w_{\bm{b}}^{\dagger}w_{\bm{c}}w_{\bm{d}}R(0)]$
$\displaystyle=e^{i(\phi_{0}+\phi+\phi^{\prime}+\phi_{+})}\times$
$\displaystyle\exp\left[-\frac{|\alpha(\tau)|^{2}}{2}\left|(\bm{c}+\bm{d}-(\bm{a}+\bm{b}))\cdot\bm{g}\right|^{2}\coth(\frac{\beta\omega}{2})\right],$
(63)

where

$\displaystyle\phi_{0}$
$\displaystyle=\xi(\tau)\left(|\bm{c}\cdot\bm{g}|^{2}+|\bm{d}\cdot\bm{g}|^{2}-|\bm{a}\cdot\bm{g}|^{2}-|\bm{b}\cdot\bm{g}|^{2}\right)$
(64) $\displaystyle i\phi$
$\displaystyle=\frac{|\alpha(\tau)|^{2}}{2}\left[(\bm{a}\cdot\bm{g})(\bm{b}\cdot\bm{g^{*}})-(\bm{a}\cdot\bm{g^{*}})(\bm{b}\cdot\bm{g})\right]$
(65) $\displaystyle i\phi^{\prime}$
$\displaystyle=\frac{|\alpha(\tau)|^{2}}{2}\left[(\bm{c}\cdot\bm{g})(\bm{d}\cdot\bm{g^{*}})-(\bm{c}\cdot\bm{g^{*}})(\bm{d}\cdot\bm{g})\right]$
(66) $\displaystyle i\phi_{+}$
$\displaystyle=\frac{|\alpha(\tau)|^{2}}{2}\big{[}((\bm{a}+\bm{b})\cdot\bm{g^{*}})((\bm{c}+\bm{d})\cdot\bm{g})-((\bm{a}+\bm{b})\cdot\bm{g})((\bm{c}+\bm{d})\cdot\bm{g^{*}})\big{]}.$
(67)

Calculation is now straightforward. For example, $C_{\Phi}$ corresponds to
$\bm{a}=\bm{b}=(\frac{1}{2},\frac{1}{2})$,
$\bm{c}=\bm{d}=(-\frac{1}{2},-\frac{1}{2})$. The phase factors in (63) all
vanish and

$\displaystyle|\alpha(\tau)|^{2}$
$\displaystyle=\frac{2(1-\cos\omega\tau)}{\omega^{2}},$ (68)
$\displaystyle\left|(\bm{c}+\bm{d}-(\bm{a}+\bm{b}))\cdot\bm{g}\right|^{2}$
$\displaystyle=8|g|^{2}[1+\cos(k(r_{1}-r_{2}))].$ (69)

Restoring the multimode character of the environment, labeled by the index
$k$, introducing spectral density
$J(\omega)=\sum_{k}|g_{k}|^{2}\delta(\omega-\omega_{k})$, and the time-of-
flight parameter $\bar{t}$ via $k|r_{1}-r_{2}|=\omega\bar{t}$, we obtain from
(63):

$\ln C_{\Phi}=-8\int d\omega
J(\omega)\frac{1-\cos\omega\tau}{\omega^{2}}\coth(\frac{\omega\beta}{2})\left(1+\cos\omega\bar{t}\right).$
(70)

The other factors are calculated analogously. For all of them the phase
factors in (63) vanish and the resulting quantities are real. In particular,
$C_{\Psi}(\tau)\equiv{\rm
Tr}_{E}[w_{11}^{\dagger}(\tau)w_{00}^{\dagger}(\tau)w_{11}(\tau)w_{00}(\tau)R(0)]$
corresponds to $\bm{a}=\bm{c}=(\frac{1}{2},\frac{1}{2})$,
$\bm{b}=\bm{d}=(-\frac{1}{2},-\frac{1}{2})$ for which

$\left|(\bm{c}+\bm{d}-(\bm{a}+\bm{b}))\cdot\bm{g}\right|^{2}=0$ (71)

so that $C_{\Psi}(\tau)=1$. This looks formally like a Decoherence Free
Subspace (DFS) condition but it is defined in a two-step decoherence process.

##  Appendix D The general $d$-dit case

Here we study a generalization to arbitrary $d$-dimensional systems, using the
original teleportation protocol from [9]. It is defined by the following
measurements and correction unitaries:

$\displaystyle|\Psi_{nm}\rangle$
$\displaystyle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}e^{\frac{2\pi
ijn}{d}}|j,j\oplus m\rangle,$ (72) $\displaystyle U_{nm}$
$\displaystyle=\sum_{j=0}^{d-1}e^{\frac{2\pi ijn}{d}}|j\rangle\langle j\oplus
m|.$ (73)

where $n,m\in[0,d-1]$ just like the basis indicies
$j,j^{\prime},k,k^{\prime},\cdots$ and $\oplus$ is the addition modulo $d$
here. We start with the following initial state:

$\sigma(0)=|\psi\rangle_{A}\langle\psi|\otimes|\Psi_{00}\rangle_{BC}\langle\Psi_{00}|\otimes
R(0),$ (74)

where

$|\psi\rangle=\sum_{j=0}^{d-1}\psi_{j}|j\rangle$ (75)

and $|\Psi_{00}\rangle$ denotes now the maximally entangled state
$1/\sqrt{d}\sum_{j}|jj\rangle$, according to the notation of (72), which we
will use throughout this Section. From (4), (72), and (73)) it is easy to see
that the total state after the first teleportaion step and conditioned on the
measurement result $nm$ is given by:

$\tilde{\sigma}_{nm}(\tau)=|\Psi_{nm}\rangle_{AB}\langle\Psi_{nm}|\otimes\frac{1}{d^{2}}\sum_{j,j^{\prime}}\psi_{j}\psi_{j^{\prime}}^{*}|j\rangle_{C}\langle
j^{\prime}|\otimes R_{j\oplus m,j^{\prime}\oplus m},$ (76)

where

$R_{ij}=w_{ii}(\tau)R(0)w^{\dagger}_{jj}(\tau).$ (77)

The first step thus introduces the following decoherence factors into the
teleported state:

$c^{m}_{jj^{\prime}}(\tau)={\rm Tr}_{E}\left[w^{\dagger}_{j^{\prime}\oplus
m,j^{\prime}\oplus m}w_{j\oplus m,j\oplus m}R(0)\right],$ (78)

(they do not depend on the index $n$ which is responsible for differenciating
the entangled states by the phase factors).

In the second step, we first prepare the additional qudits
$A^{\prime}B^{\prime}$ in the state corresponding to the first measurement,
i.e. if the measurement result was $nm$, the initial state is given by
$|\Psi_{nm}\rangle_{A^{\prime}B^{\prime}}\langle\Psi_{nm}|\otimes R_{k\oplus
m,k^{\prime}\oplus m}$. The decoherence process (4) then affects the entangled
state. It is important that it couples to the same environment $E$ as in the
first step. The resulting state reads:

$\displaystyle
U_{A^{\prime}B^{\prime}E}\tilde{\sigma}_{nm}(\tau)U^{\dagger}_{A^{\prime}B^{\prime}E}$
$\displaystyle=\frac{1}{d^{3}}\sum_{jj^{\prime}kk^{\prime}}e^{\frac{2\pi
in}{d}(k-k^{\prime})}\psi_{j}\psi^{*}_{j^{\prime}}|k\rangle_{A^{\prime}}\langle
k^{\prime}|$ $\displaystyle\bm{\otimes}|k\oplus
m,j\rangle_{B^{\prime}C}\langle k^{\prime}\oplus
m,j^{\prime}|\bm{\otimes}w_{k,k\oplus m}R_{j\oplus m,j^{\prime}\oplus
m}w^{\dagger}_{k^{\prime},k^{\prime}\oplus m}.$ (79)

Now we perfom the second teleportation: i) We measure the system $B^{\prime}C$
in the basis $|\Psi_{nm}\rangle$; let us assume the result is
$n^{\prime}m^{\prime}$; ii) We apply a correction unitary from the set (73).
We note that unlike in the first step, the entangled resource is now a general
state $|\Psi_{nm}\rangle$ (cf. (76)) rather then $\Psi_{00}$ to which the sets
(72), (73) were tailored. Because of this, the corresponding correction
unitary is in general not given by $U_{n^{\prime}m^{\prime}}$ but is some
permuted one $U_{NM}$, where the indicies $N,M$ will be determined below to
reproduce the initial state $|\psi\rangle$. To avoid excessive algebraic
complications in one step, we first apply the measurement and the corrective
operation to the appropriate systems:

$\langle\Psi_{n^{\prime}m^{\prime}}|k\oplus m,j\rangle\cdot
U_{NM}|k\rangle=\frac{1}{\sqrt{d}}\sum_{lr}e^{\frac{2\pi
i}{d}(-n^{\prime}l+Nr)}\delta_{l,k\oplus m}\delta_{j,l\oplus
m^{\prime}}\delta_{k,r\oplus M}|r\rangle$ (80)

The delta symbols force that $k=r\oplus M$ and $j=r\oplus M\oplus m\oplus
m^{\prime}$. Applying (80) to (79), we obtain the following (unnormalized)
state of $A^{\prime}E$:

$\tilde{\varrho}_{A^{\prime}E}^{(2)}(\tau)=\sum_{jj^{\prime}rr^{\prime}}\frac{\psi_{j}\psi^{*}_{j^{\prime}}}{d^{4}}e^{i\varphi}\delta_{j,r\oplus
M\oplus m\oplus m^{\prime}}\delta_{j^{\prime},r^{\prime}\oplus M\oplus m\oplus
m^{\prime}}|r\rangle\langle r^{\prime}|\bm{\otimes}w_{r\oplus M,r\oplus
M\oplus m}R_{j\oplus m,j^{\prime}\oplus m}w^{\dagger}_{r^{\prime}\oplus
M,r^{\prime}\oplus M\oplus m},$ (81)

where for brevity’s sake we neglected the indices $nmn^{\prime}m^{\prime}$
labeling the state $\tilde{\varrho}_{A^{\prime}E}$ just substituting them with
the superscript $\phantom{\rho}{}^{(2)}$ indicating the sate after the second
step of the protocol. The phase $\varphi$ is equal to:

$\varphi=\frac{2\pi}{d}\bigg{[}n\left(r\oplus M-r^{\prime}\oplus
M\right)-n^{\prime}\left(r\oplus M\oplus m-r^{\prime}\oplus M\oplus
m\right)+N(r-r^{\prime})\bigg{]}\equiv\frac{2\pi}{d}(r-r^{\prime})\Big{[}n-n^{\prime}+N\Big{]},$
(82)

since the differences of modular sums generate at most an extra $\pm d$ term,
which in turn shifts the phase by an irrelevant factor $\pm
2\pi(n-n^{\prime})$. To complete the teleportation, we have to find the proper
values of the indices $N,M$, properly correcting the state of $A^{\prime}$.
From (82) it is clear that in order to get rid of the phase factor, $N$ must
be chosen as:

$N=n^{\prime}\ominus n=(n^{\prime}-n)\text{mod}d.$ (83)

Looking at the arguments of delta symbols in (81), we see that in order to
reproduce the correct state on $A^{\prime}$, we must have $r=j$ and
$r^{\prime}=j^{\prime}$. Let us choose

$M=\left[d-(m\oplus m^{\prime})\right]\text{mod}\,d.$ (84)

Then it is easy to check that

$M\oplus m\oplus m^{\prime}=0,$ (85)

because by definition $a=m\oplus m^{\prime}\in[0,d-1]$ and $M\oplus
a=d-a+a-d=0$ from the definition of the modular sum (the other case is trivial
as for $a=0$, $M=0$ from its definition). Hence

$r\oplus M\oplus m\oplus m^{\prime}=r$ (86)

in (81), forcing through the delta symbols that $r=j$ and
$r^{\prime}=j^{\prime}$ as needed. We thus finally obtain:

$\tilde{\varrho}_{A^{\prime}E}^{(2)}(\tau)=\frac{1}{d^{4}}\sum_{jj^{\prime}}\psi_{j}\psi^{*}_{j^{\prime}}|j\rangle\langle
j^{\prime}|\bm{\otimes}w_{j\oplus M,j\oplus M\oplus m}R_{j\oplus
m,j^{\prime}\oplus m}w^{\dagger}_{j^{\prime}\oplus M,j^{\prime}\oplus M\oplus
m},$ (87)

so that the protocol outputs the following family of teleported states:

$\varrho^{A^{\prime}}_{mm^{\prime}}=\sum_{jj^{\prime}}\psi_{j}\psi^{*}_{j^{\prime}}C^{mm^{\prime}}_{jj^{\prime}}|j\rangle\langle
j^{\prime}|,$ (88)

where:

$C^{mm^{\prime}}_{jj^{\prime}}(\tau)={\rm Tr}_{E}\left[w_{j\oplus M,j\oplus
M\oplus m}w_{j\oplus m,j\oplus m}R(0)w^{\dagger}_{j^{\prime}\oplus
m,j^{\prime}\oplus m}w^{\dagger}_{j^{\prime}\oplus M,j^{\prime}\oplus M\oplus
m}\right].$ (89)

are decoherence factors between the levels $jj^{\prime}$ and $M$ is defined by
(84). We note that the teleported states do not depend on the indicies
$n,n^{\prime}$ so there is a $d^{2}$-fold degeneracy, leading to
$d^{2}/d^{4}=d^{2}$ distinct states with equal probabilities $1/d^{2}$

Let us study the possibility of reversing the decoherence. Analogously to the
qubit case, we assume the commutativity condition

$[w_{ii}(\tau),w_{jj}(\tau)]=0,\text{ for every }i,j$ (90)

motivated by the fact that it is satisfied in many important models. Then, the
only possibility to reverse the decoherence completely, i.e. on all the off
diagonal terms of (88), is to find combinations of $m,m^{\prime}$, such that
the corresponding states have all of the $d(d-1)/2$ decoherence factors of the
form :

${\rm Tr}_{E}\left[w_{ll}R_{kl}w_{kk}^{\dagger}\right]={\rm
Tr}_{E}\left[w_{kk}^{\dagger}w_{ll}^{\dagger}w_{kk}w_{ll}R(0)\right].$ (91)

Looking at (89), the necessary condition for this is

$m=0$ (92)

as $m$ is shifting the second index of $w_{j\oplus M,j\oplus M\oplus m}$ and
we need both indicies to be the same. For $m=0$ the decoherence factors take
the form:

${\rm
Tr}_{E}\left[w^{\dagger}_{j^{\prime}j^{\prime}}w^{\dagger}_{j^{\prime}\oplus
M,j^{\prime}\oplus M}w_{j\oplus M,j\oplus M}w_{jj}R(0)\right].$ (93)

In order for all of them to be of the form (91), the following condition would
have to be satisfied:

$\forall jj^{\prime},j\neq j^{\prime}\ \exists M\neq 0:j=j^{\prime}\oplus
M\text{ and }j^{\prime}=j\oplus M$ (94)

($M=0$ leads to the factors containing
$[w^{\dagger}_{kk}]^{2}w_{ll}^{2}\neq\bm{1}$, cf. (11)). In particular, it
would imply that:

$M=j\ominus j^{\prime}$ (95)

and on the same time $M$ would be independent of $jj^{\prime}$ for all $j\neq
j^{\prime}$. But this can happen only in $d=2$, where there are only two
levels and one nonzero $M=1$ switching between them.

One may ask if a partial purification is possible, when only some of the
coherences are restored. This is indeed the case as one of the two conditions
(94) can always be satisfied for any $M$. It happens for the levels
$jj^{\prime}$ of the form:

$(j,j\oplus M),0\leq j\leq d-1.$ (96)

For every such a pair $jj^{\prime}$, the decoherence factor (93) does not in
general reduce to unity like it was in the qubit case but instead reduces
under (90) to the one-step form (11):

${\rm Tr}_{E}\left[w^{\dagger}_{j\oplus 2M,j\oplus 2M}w_{jj}R(0)\right].$ (97)

There are in general $d$ such reduced decoherence factors in each of the $d$
states (88) with $m=0$, $m^{\prime}$ arbitrary. In particular, for a qutrit
$d=3$ all decoherence factors are reduced in each of the $3$ states
$(m=0,m^{\prime})$ as there are $d(d-1)/2=3$ off-diagonal elements in each
state.

For an even dimension $d$, a further, full reduction can happen for a special
state with $M=d/2$. Indeed, since $M=d/2$, $2M=d$ and $j\oplus M=j$ in (97).
Due to the symmetry around $d/2$, the distinct levels for which this happens
are given by:

$(j,j+d/2),0\leq j<d/2,$ (98)

and there are $d/2$ of them in the state $m=0,M=d/2=m^{\prime}$ (cf. (84)).
The rest of the $d(d-2)/2$ decoherence factors are not reduced. We recall that
we were considering above only cases with $m=0$ as otherwise there is no
chance for even a partial purification under (90).

Summarizing, for an arbitrary dimension $d>2$ a partial supression/reversal of
decoherence is possible. It can happen only for those instances when the first
Bell measurement gives $m=0$ result. We then have:

  * •

For every $m^{\prime}$, there will be levels $jj^{\prime}$ in the
corresponding state $\varrho_{0m^{\prime}}$ (88), such that the decoherence
factors (89) between them reduce to the one-step form (97), i.e. like if only
one decoherence process has happened. There are $d$ such levels, described by
(96).

  * •

Additionally, when $d$ is even, the state corresponding to $m^{\prime}=d/2$
will have $d/2$ of its coherences (98) fully restored to the original form
like if no decoherence has happened at all.


#
Performance Analysis Of Binaural Signal Matching (BSM) in the Time-Frequency
Domain

Ami Berger(1) Vladimir Tourbabin(2) Jacob Donley(2) Zamir Ben-Hur(2) and Boaz
Rafaely(1)

### ABSTRACT

The capture and reproduction of spatial audio is becoming increasingly
popular, with the mushrooming of applications in teleconferencing,
entertainment and virtual reality. Many binaural reproduction methods have
been developed and studied extensively for spherical and other specially
designed arrays. However, the recent increased popularity of wearable and
mobile arrays requires the development of binaural reproduction methods for
these arrays. One such method is binaural signal matching (BSM). However, to
date this method has only been investigated with fixed matched filters
designed for long audio recordings. With the aim of making the BSM method more
adaptive to dynamic environments, this paper analyzes BSM with a parameterized
sound-field in the time-frequency domain. The paper presents results of
implementing the BSM method on a sound-field that was decomposed into its
direct and reverberant components, and compares this implementation with the
BSM computed for the entire sound-field, , to compare performance for binaural
reproduction of reverberant speech in a simulated environment.

Keywords: Binaural reproduction, BSM, Dynamic-arrays, Head tracking

## 1 INTRODUCTION

Binaural reproduction of acoustic scenes that are captured by microphone-
arrays is becoming increasingly popular [1],[2],[3], with many applications in
teleconferencing and virtual and augmented reality. A popular method for
binaural reproduction involves convolving high order Ambisonics (HOA) signals
with the head related transfer function (HRTF) [4]. This method is fairly
accurate for sufficiently high spherical harmonics (SH) orders, and the
incorporation of head tracking enhances the immersion experience of the
listener. However, the main drawback of this method lies in the large number
of microphones required when the audio signal is recorded by a spherical
microphone array, and in the need for a spherical array geometry, which
restrict the practical use of this method [12].
In order to reproduce binaural signals using more flexible array geometries,
the beamforming-based binaural reproduction (BFBR) [5],[6],[7] method was
proposed. In this method the microphone signals are filtered using a set of
beamformers, and the output signals are later filtered using the HRTFs and
then summed to reproduce the binaural signals. A theoretical framework to set
the design parameters of BFBR, such as the look direction and beam number for
spherical and planar arrays, was presented in [8]. However, for more general
array geometries only limited guidelines were suggested, with no guarantee of
accurate binaural signal reproduction, and so a comprehensive design
methodology is still unavailable.
With the aim of overcoming the limitations of current beamforming-based
methods, and in order to accurately reproduce binaural signals recorded by
arrays of arbitrary geometry, the binaural signal-matching (BSM) [9],[10],[11]
method was developed. BSM refers to the estimation of the binaural signals
directly from the array measurements using optimal filters calculated for each
ear separately. Recently, the design of a BSM system was described and studied
for a varying number of microphones in a semi-circular array [13]
incorporating head tracking [20],[22]; it was shown that the accuracy of BSM
is sensitive to the position of the microphones rather than to their number.
In particular, it was shown that the closer a microphone in the array is to an
ear, the better the binaural reproduction in that ear. Therefore, the main
drawback of the BSM method is that it performs poorly for high frequencies,
especially in cases where one of the ears is relatively far from all the
microphones of the array. In order to improve the perceptual performance it
was proposed to use Magnitude Least-Squares (MagLS) [28] instead of Least-
Squares (LS) for high frequencies. However, while the use of MagLS improved
the overall perceptual experience of a reproduced binaural signal, BSM still
performs perceptually badly when one of the ears is relatively far from all of
the array microphones.
Parametric spatial audio and binaural reproduction have also been studied as
an alternative to the BSM and beamforming methods described above. In this
approach the sound-field is decomposed into components, typically direct
sources and reverberant parts, and each is estimated and reproduced separately
[21],[23],[24],[25],[26]. While these approaches show promising performance,
the latter may depend on estimation accuracy, which becomes challenging in
complex environments. Nevertheless, the parametric approach also motivated the
study in this paper.
With the limitations of previous method in mind, this paper aims to study and
analyze the performance of BSM with a parameterized sound-field in the time-
frequency domain. Specifically, an acoustic scene that is separated into
direct and reverberant components is investigated. The paper investigates the
potential of improving BSM by incorporating sound-field parameterization,
especially in cases where BSM currently fails.

## 2 MATHEMATICAL BACKGROUND

This section provides the mathematical model of BSM. Throughout the paper, the
standard spherical coordinate system is used, denoted by $(r,\theta,\phi)$,
where $r$ is the distance to the origin, $\theta$ is the elevation angle
measured from the Cartesian z axis downwards to the Cartesian xy plane, and
$\phi$ is the azimuth angle measured from the positive x axis towards the
positive y axis. A sound-field can be described as a linear combination of
plane-waves $a(k,\theta,\phi)$, where $k=\frac{2\pi}{\lambda}$ is the wave-
number, $\lambda$ is the wave-length and $(\theta,\phi)$ is the wave arrival
direction. Further, assume that the sound-field is composed of $L$ far-field
sound sources arriving from directions ${(\theta_{l},\phi_{l})}^{L}_{l=1}$,
with source signals ${s_{l}(k)}^{L}_{l=1}$ [14]. The sound-field is captured
by an array of $M$ microphones, which are located at
${(r_{m},\theta_{m},\phi_{m})}^{M}_{m=1}$, centered at the origin. The noisy
array measurements can be described by the following narrow-band model [14]:

$\mathbf{x}(k)=\mathbf{V}(k)\mathbf{s}(k)+\mathbf{n}(k)$ (1)

where $\mathbf{x}(k)=[x_{1}(k),x_{2}(k),...,x_{M}(k)]^{T}$ is the microphone-
signal vector (measurements),
$\mathbf{V}(k)=[\mathbf{v}(k,\theta_{1},\phi_{1}),\mathbf{v}(k,\theta_{2},\phi_{2}),...,\mathbf{v}(k,\theta_{L},\phi_{L})]$
is an $M\times L$ complex matrix with columns
$\mathbf{v}(k,\theta_{l},\phi_{l})$ representing the array steering vector
from the $l$-th source to the microphone positions for all $l=1,2,...,L$
sources,
$\mathbf{s}(k)=[s_{1}(k),s_{2}(k),...,s_{L}(k)]^{T}$ is the source-signal
vector, and $\mathbf{n}(k)$ is an additive-noise vector. In the BSM model we
further assume that a listener is positioned with the center of the head
coinciding with the origin, where $h^{l,r}(k,\theta,\phi)$ denotes the HRTF of
the left and right ears of the listener using the superscripts $(\cdot)^{l}$
for the left ear and $(\cdot)^{r}$ for the right ear. The signal at the left
and rights ears can now be written as [13]:

$p^{l,r}(k)=[\mathbf{h}^{l,r}(k)]^{T}\mathbf{s}(k)$ (2)

where
$\mathbf{h}^{l,r}=[h^{l,r}(\theta_{1},\phi_{1}),h^{l,r}(\theta_{2},\phi_{2}),...,h^{l,r}(\theta_{L},\phi_{L})]^{T}$
contains the HRTFs corresponding to the directions of the sources.
Next, assume that the configuration of the microphone array is known, such
that the steering matrix $\mathbf{V}(k)$ can be calculated analytically or
numerically. In the first step, the array measurements are filtered and
combined, in a similar manner to beamforming:

$z^{l,r}(k)=[\mathbf{c}^{l,r}(k)]^{H}\mathbf{x}(k)$ (3)

where $\mathbf{c}$ is an $M\times 1$ complex vector holding the filter
coefficients. Next, $\mathbf{c}$ is chosen to minimize the following mean-
squared error between $z^{l,r}(k)$ and $p^{l,r}(k)$, the binaural signals in
(2), for each ear separately:

$err^{l,r}_{bin}(k)=\mathbb{E}[|p^{l,r}(k)-z^{l,r}(k)|^{2}]$ (4)

where $\mathbb{E}[\cdot]$ is the expectation operator. Next, assume that the
noise is uncorrelated to the sources. This leads to the following error
formulation:

$err^{l,r}_{bin}(k)=\|[\mathbf{s}(k)]^{H}([\mathbf{V}(k)]^{H}\mathbf{c}^{l,r}(k)-[\mathbf{h}^{l,r}(k)]^{*})\|^{2}_{2}+\|[\mathbf{n}(k)]^{H}\mathbf{c}^{l,r}(k)\|^{2}_{2}$
(5)

Minimizing the error in (5) leads to the following solution:

$\mathbf{c}^{l,r}_{opt}(k)=(\mathbf{V}\mathbf{R_{s}}\mathbf{V}^{H}+\mathbf{R_{n}})^{-1}\mathbf{V}\mathbf{R_{s}}[\mathbf{h}^{l,r}]^{*}$
(6)

where $\mathbf{R_{s}}=E[\mathbf{s}\mathbf{s}^{H}]$,
$\mathbf{R_{n}}=E[\mathbf{n}\mathbf{n}^{H}]$. Next, assume that the sources
are uncorrelated as in [20], leading to
$\mathbf{R_{s}}=\sigma_{s}^{2}\mathbf{I}_{L}$, and also that the noise in
uncorrelated between microphones, such that
$\mathbf{R_{n}}=\sigma_{n}^{2}\mathbf{I}_{M}$. These assumptions lead to the
following simplification of (6):

$\mathbf{c}^{l,r}_{opt}(k)=(\mathbf{V}\mathbf{V}^{H}+\frac{1}{SNR}\mathbf{I}_{M})^{-1}\mathbf{V}[\mathbf{h}^{l,r}]^{*}$
(7)

where $SNR=\frac{\sigma_{s}^{2}}{\sigma_{n}^{2}}$. Equation (7) has been used
in [20]. The advantage of (7) is that it does not require the estimation of
$\mathbf{R_{s}}$, but this in turn may lead to reduced performance. The next
section presents an approach to enhance performance by incorporating limited
information on $\mathbf{R_{s}}$.

## 3 BSM WITH PARAMETERIZED SOUND-FIELD

This section investigates the BSM method described in [20], but extended to
incorporate a parameterization of the sound-field. The parameterization is
based on the assumption that the measured sound-field can be decomposed into
two components as follows:

$\mathbf{x}(n,k)=\mathbf{x}_{d}(n,k)+\mathbf{x}_{r}(n,k)+\mathbf{n}(n,k)$ (8)

where $\mathbf{n}(k)$ is an additive-noise vector, and $\mathbf{x}_{d}(n,k)$
represents the direct signal from the sound source in the time-frequency
domain, modeled as a single far-field plane wave, written as:

$\mathbf{x}_{d}(n,k)=\mathbf{v}(k,\theta_{d},\phi_{d}){s}_{d}(n,k)$ (9)

where $(\theta_{d},\phi_{d})$ represents the DOA of the direct signal, and
${s}_{d}(n,k)$ represents the source signal. $\mathbf{x}_{r}(n,k)$ represents
the reverberant part of the measured signal in the time-frequency domain and
is typically composed of a large, unknown number of sources that arrive from
unknown directions. This model can represent a single source in a room where
$\mathbf{x}_{d}$ are the measurements of the direct sound from the source and
$\mathbf{x}_{r}$ are the measurements of the source reflections from room
boundaries. There are methods available that estimate $\mathbf{x}_{d}(n,k)$
and its direction-of-arrival (DOA) for each time-frequency bin [21],[26],[27],
but in this paper $\mathbf{x}_{d}(n,k)$ and its DOA are assumed to be known.
Knowing $\mathbf{x}_{d}(n,k)$, there are methods that can reproduce the direct
component of the binaural signal, $p^{l,r}_{d}(n,k)$ [20],[21],[8],[1].
Nevertheless, in this paper the BSM method was chosen as this method is the
focus of this paper. Reproducing the binaural signal of the reverberant
component, $p^{l,r}_{r}(n,k)$, from the reverberant component of the
measurement, $\mathbf{x}_{r}(k)$, is a more challenging problem since it
usually requires an estimation of $\mathbf{R_{s}}$. The simplification of
$\mathbf{R_{s}}$ applied in (7) assumes that the sources are uncorrelated,
e.g. a diffuse sound-field, and so it could be a reasonable approximation for
the reverberant part.
In summary, computing the BSM filters separately for the direct and
reverberant components is expected to produce better results compared to
computing the BSM filters for the entire sound-field, because in the former
the direct sound components are expected to be reproduced more accurately.
These components are formulated as:

$\hat{p}^{l,r}(n,k)=\hat{p}^{l,r}_{d}(n,k)+\hat{p}^{l,r}_{r}(n,k)$ (10)

where $\hat{p}^{l,r}_{d}(n,k)$ and $\hat{p}^{l,r}_{r}(n,k)$ are the reproduced
binaural signals of the direct and reverberant components using the BSM
method, as described in (7). The performance of BSM with the proposed
decomposition is evaluated in this paper numerically using binaural signal
error and perceptually, using a listening test.

## 4 SIMULATION STUDY

This section presents a simulation-based analysis of the performance of the
proposed BSM method with sound-field decomposition, compared to the BSM method
computed from the entire measured signal without applying decomposition.

### 4.1 SET-UP

A point source was simulated inside a room of dimensions $8\times 5\times
3\,$m, having a reverberation time of ${T_{60}=0.68\,}$s, using the image
method [18]. The source location in the room was $(2.47,2.27,1.7)\,$m. The
source signal was a 5s long recording of female speech, taken from the TIMIT
database [19], sampled at $48\,$kHz. A semi-circular microphone array was
centered at $(2,2,1.7)\,$m, with a DRR value of $4.5\,$dB, compromising $M=6$
omni-directional microphones arranged on the horizontal plane. Microphone
positions were denoted using spherical coordinates by
$(r_{m},\theta_{m},\phi_{m})$ for $m=1,...,M$ relative to the array center,
where $r_{m}=10\,$cm, $\theta_{m}=\frac{\pi}{2}\,$rad and
$\phi_{m}=\pi-\frac{\pi(m-1)}{M-1}\,$rad. Microphone measurement signals are
denoted by $\mathbf{x}(t)=[x_{1}(t),x_{2}(t),...,x_{M}(t)]^{T}$. The component
of $\mathbf{x}(t)$ representing the direct contribution from the source,
$\mathbf{x}_{d}(t)$ as in (8), was calculated by assuming a free-field
environment. The HRTF in the simulation was taken from the Cologne database
[17] measurements of the Neumann KU100 manikin, sampled at a frequency of
$48\,$kHz. The HRTFs for the DOAs of the assumed sources were interpolated in
the SH domain using a SH order of 30. The head was centered at $(2,2,1.7)\,$m
and was aligned with the positive $x$ axis. An illustration of the array
position relative to the head position and alignment is presented in Figure 1.
Assuming the semi-circular array represents an array on AR glasses, for
example, the selected orientation of the array relative to the head was chosen
as this was found to be the most challenging for the BSM algorithm [20].

Figure 1: An illustration of a head and the semi-circular array where the blue
dots represent the array microphones, and the $x$ and $y$ grey arrows
represent the positive $x$ and $y$ axes, respectively.

### 4.2 METHODOLOGY

Having computed $\mathbf{x}(t)$ and $\mathbf{x}_{d}(t)$ as described above,
$\mathbf{x}(n,k)$ and $\mathbf{x}_{d}(n,k)$ were computed using the Short-
Time-Fourier-Transform (STFT), with a Hamming window of $32\,$ms and a hop
length of $16\,$ms. $\mathbf{x}_{r}(n,k)$ was then calculated using (8). Using
(7), two BSM filters were computed, one for the direct component and one for
the reverberant component of the signal, denoted $\mathbf{c}^{l,r}_{d}(k)$ and
$\mathbf{c}^{l,r}_{r}(k)$, respectively. As described in section 2, the BSM
method was developed assuming a sound-field composed of $L$ far-field sources.
When computing the filter for the reverberant component, it was assumed,
similarly, that the sound-field is composed of $L=240$ sources with DOAs that
correspond to a spirally-nearly-uniform distribution [15]. A single source
with $L=1$ and a DOA of $(\theta_{d}=\frac{\pi}{2},\phi_{d}=\frac{\pi}{6})$,
relative to the array’s center, was assumed when computing the filter of the
direct component. The steering vectors for the reverberant and direct
components were generated analytically in the SH domain according to section
4.2 in [16] using the corresponding number of $L$ sources, the array geometry
and the DOAs of the assumed sources. Next, $SNR=20\,$dB was assumed for the
reverberant components, while $SNR=\infty$ was assumed for the direct
component, and both values were substituted in (7). The BSM weights of the
reverberant component were calculated using MagLS [28] for frequencies in the
range of $[1500,24000]\,$Hz.
Next, $\hat{p}^{l,r}_{r}(n,k)$ and $\hat{p}^{l,r}_{d}(n,k)$, representing the
output of the BSM method for the reverberant and direct components, were
calculated according to (3) using
$\mathbf{c}^{l,r}_{r}(k)$,$\mathbf{x}_{r}(n,k)$ and
$\mathbf{c}^{l,r}_{d}(k)$,$\mathbf{x}_{d}(n,k)$, respectively.
$\hat{p}^{l,r}_{BSM}(n,k)$, representing the solution of implementing the BSM
method directly from the array measurements without sound-field decomposition,
were calculated according to (3) using $\mathbf{c}^{l,r}_{r}(k)$ and
$\mathbf{x}(n,k)$.
${p}^{l,r}_{ref}(n,k)$, the reference signals at the ears, were calculated by
convolving the HRTFs of the left and right ears with the HOA signals of order
14 that were calculated using the image method as described in section 4.1.
${p}^{l,r}_{d-ref}(n,k)$, the direct component of the reference signals at the
ears, were calculated by convolving the HRTFs of the left and right ears with
the HOA signals of order 14 that were calculated by assuming a free-field
environment as described in section 4.1.
${p}^{l,r}_{r-ref}(n,k)$, the reverberant component of the reference signals
at the ears, were calculated using (10).
The normalized mean-squared error (NMSE) of the binaural signals was
calculated as:

$NMSE(k)=\frac{\mathbb{E}_{n}[|\hat{p}^{l,r}(n,k)-{p}^{l,r}_{ref}(n,k)|^{2}]}{\mathbb{E}_{n}[|{p}^{l,r}_{ref}(n,k)|^{2}]}$
(11)

where ${p}^{l,r}_{ref}(n,k)$ is the reference binaural signal of a certain
time-frequency bin, and $\hat{p}^{l,r}(n,k)$ is the reproduced binaural signal
of a certain time-frequency bin.

### 4.3 SIMULATION RESULTS

In order to study the performance of the BSM method when applied to a
decomposed sound-field, the NMSE of the direct and reverberant components were
calculated and are presented in Figures 2 and 3. Figure 2 shows that the NMSE
of the reproduced direct component of the binaural signal is relatively low,
for both the left and the right ears, in particular at the low frequencies.
This result indicates a fairly accurate reproduction of the direct component,
which agrees with the expected performance of the BSM method designed for a
sound-field compromised of $L<M$ sources as shown in [13].
The NMSE of the reverberant component is presented in Figure 3. As can be
observed, the NMSE is higher at higher frequencies and lower for the ear
closest to the microphone positions (in this simulation the left ear), see
Figure 1. These results agree with the expected performance of the BSM method
as described in [20].
The NMSE of the reproduced binaural signal using the BSM method without sound-
field decomposition, $\hat{p}^{l,r}_{BSM}(n,k)$, and the reproduced binaural
signal using the BSM method with sound-field decomposition,
$\hat{p}^{l,r}_{r}+\hat{p}^{l,r}_{d}$, were also calculated and are presented
in Figure 4. As can be observed there is a clear overall improvement in the
NMSE of the reproduced binaural signals for the BSM method with sound-field
decomposition, compared to the standard BSM method, especially for the left
ear (which is closer to the array). This demonstrates that a more accurate
reproduction of the binaural signal using the BSM method can be achieved by
sound-field decomposition.
By comparing the NMSE of the direct and reverberant components, we can deduce
that the NMSE is dominated by the reverberant component. This simulation study
shows the potential for sound field-decomposition - if such decomposition can
be implemented in practice, it can significantly improve the performance of
the BSM algorithm.

Figure 2: The NMSE of the direct component of the binaural signal for the left
and right ears ($p_{d}^{l},p_{d}^{r}$), calculated using ${p}^{l,r}_{d-ref}$
as the reference binaural signals. Figure 3: The NMSE of the reverberant
component of the binaural signal for the left and right ears
($p_{r}^{l},p_{r}^{r}$) calculated using ${p}^{l,r}_{r-ref}$ as the reference
binaural signals. Figure 4: The NMSE of the left (a), and right (b) ears
calculated using $\hat{p}^{l,r}_{ref}$ as the reference binaural signals.

## 5 LISTENING EXPERIMENT

A preliminary listening test was reproduced with the aim of evaluating the
similarity between the reference binaural signal ${p}^{l,r}_{ref}$, and the
reproduce signals $\hat{p}^{l,r}_{BSM}$ and
$\hat{p}^{l,r}_{r}+\hat{p}^{l,r}_{d}$. The experiment was based on the
MUltiple Stimuli with Hidden Reference and Anchor (MUSHRA) test [29] and
incorporated 9 normal hearing participants. The preliminary listening test
scores are presented using a box-plot in Figure 5. As can be observed in
Figure 5, the scores of the reproduced binaural signal for the BSM with sound-
field decomposition are higher than the scores of the standard BSM method.
These results agree with the results presented in Figure 4; in particular, the
score of the BSM with sound-field decomposition was very high and close to the
reference, suggesting that this approach could also provide improvement that
is perceptually notable.

Figure 5: Box-plot of the listening experiment score. The median is marked by
a red line, the 25th and 75th percentiles are marked as the bottom and top
blue edges respectively, and the maximal and minimal grades are marked by
black lines. The scores are based on overall quality.

## 6 CONCLUSIONS

In this paper, binaural reproduction with the BSM method was studied with a
semi-circular array and a decomposed sound-field. It was shown that the
perceptual and overall accuracy of the reproduced binaural signal using a
decomposed sound-field is higher than in the case of using the standard BSM
method, especially where the distance between the listener ear position and
the microphones is large. A listening test demonstrated that the accurate
reproduction of the direct component was indeed important for perception.
Future work could include the incorporation of a spatial coding method to
implement sound-field decomposition, the development of a design framework
that better reproduces the reverberant component, and an extension of the
listening test performed in this work. Future work could also include the
study of the BSM method with other array configurations, and the development
of a design framework that improves binaural reproduction.

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