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Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device
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Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device
Yu Guo ,1,2,3 ,Zixuan Liu,4,5,Hao Tang,1,2,3Xiao-Min Hu,1,2,3Bi-Heng Liu,1,2,3 ,?Yun-Feng Huang,1,2,3
Chuan-Feng Li,1,2,3 ,?Guang-Can Guo,1,2,3and Giulio Chiribella4,6,5,7 ,§
1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
2CAS Center For Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, 230026, China
3Hefei National Laboratory, University of Science and Technology of China, Hefei, 230088, China
4QICI Quantum Information and Computation Initiative, Department of Computer Science,
The University of Hong Kong, Pokfulam Road, Hong Kong
5HKU-Oxford Joint Laboratory for Quantum Information and Computation
6Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, United Kingdom
7Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada
(Received 11 October 2023; accepted 20 March 2024; published 16 April 2024)
Quantum theory allows information to flow through a single device in a coherent superposition of two
opposite directions, resulting into situations where the input-output direction is indefinite. Here we introduce
a theoretical method to witness input-output indefiniteness in a single quantum device, and we experimentally
demonstrate it by constructing a photonic setup that exhibits input-output indefiniteness with a statisticalsignificance exceeding 69 standard deviations. Our results provide a way to characterize input-output
indefiniteness as a resource for quantum information and photonic quantum technologies and enable tabletop
simulations of hypothetical scenarios exhibiting quantum indefiniteness in the direction of time.
DOI: 10.1103/PhysRevLett.132.160201
Introduction. —A cornerstone of quantum theory is the
CPT theorem [1,2], stating that the fundamental dynamics
of quantum fields is invariant under inversion of timedirection, charge, and parity. The theorem implies that, atthe fundamental level, the roles of past and future are
symmetric: while we normally treat systems at earlier times
as the inputs and systems at later times as the outputs, thedynamical laws of quantum mechanics are indifferent to the
direction of time. The time symmetry of the fundamental
quantum dynamics was later extended to scenarios involv-ing measurements by Aharonov and collaborators [3–5].
With the advent of quantum information, the role of time
symmetry in quantum theory has attracted renewed atten-
tion, due to its connection with the structure of quantumprotocols [6], multitime quantum states [7–9], simulation
of closed timelike curves [10–12], inversion of unknown
quantum evolutions [13–15], quantum retrodiction [16,17] ,
and the origin of irreversibility [18,19] . Time-symmetric
frameworks for quantum theory [20,21] and more general
physical theories [22,23] have been developed and analyzed.
Recently, Refs. [24,25] extended the notion of time
reversal to a broader notion of input-output inversion,which applies whenever the roles of the input and output
ports of a quantum device can be exchanged. Thisincludes, for example, the case of linear optical devices,which can be traversed in two opposite spatial directions.Notably, all kinds of input-output inversions turned out toshare the same mathematical structure. As a consequence,
hypothetical scenarios involving the reversal of the timedirection between two spacetime events can be simulatedby real-world setups that reverse the direction of a pathbetween two points in space. Building on the notion ofinput-output inversion, Ref. [24] then introduced a new
type of operations that utilize quantum devices in acoherent superposition of t wo alternative input-output
directions, giving rise to a feature called input-outputindefiniteness. This feature has been found to offeradvantages in information-theoretic [24,25] and thermo-
dynamical tasks [26,27] . Input-output indefiniteness is
also related to the notion of indefinite order [28–31],
whose applications to quantum information have been
extensively investigated in the past decade, both theo-retically [31–38]and experimentally [39–47]. An impor-
tant difference is that, while indefinite order requiresmultiple devices (or multiple uses of the same device),input-output indefiniteness can already arise at the single-device level, enabling quantum protocols that could notbe achieved with indefinite order (see SupplementalMaterial [48] for examples in the tasks of gate trans-
formation, estimation, and testing).
Published by the American Physical Society under the terms of
theCreative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI.PHYSICAL REVIEW LETTERS 132, 160201 (2024)
0031-9007 =24=132(16) =160201(7) 160201-1 Published by the American Physical SocietyHere we develop a general method for witnessing input-
output indefiniteness in the laboratory, and we use it to
experimentally demonstrate a photonic setup that probes asingle quantum device in a coherent superposition of twoalternative directions. By optimizing the choice of witness,we demonstrate incompatibility of our setup with a definiteinput-output direction by more than 69 statistical deviations.Notably, our setup applies not only to reversible quantumdevices, such as polarization rotators, but also to a class ofirreversible devices including postselected polarization mea-surements. In addition to single-device indefiniteness, we
experimentally demonstrate the combination of two devices
in a quantum superposition of two opposite input-outputdirections, building a setup that achieves 99.6% winningprobability in a quantum game where every strategy usingboth devices in the same direction fails with at least 11%probability. Our techniques enable a rigorous characteriza-tion of input-output indefiniteness as a resource for quantuminformation and photonic quantum technologies, and, at thesame time, could be used to simulate exotic physical modelswhere the arrow of time is subject to quantum indefiniteness.
Witnesses of input-output indefiniteness. —For many
processes in nature, the role of the input and output portscan be exchanged. An example is the transmission of asingle photon through an optical crystal, schematicallyillustrated in Fig. 1. Quantum devices with exchangeable
input-output ports, called bidirectional , can be used in two
alternative ways, conventionally referred to as the “forward
mode ”(with the inputs entering at port Aand the outputs
exiting from port B)a n d “backward mode ”(with the inputsentering at port Band the outputs exiting from port A). In the
special case where ports AandBare associated with two
moments of time t
Athe standard use of the device in the forward time direction,while the backward mode corresponds to a hypothetical useof the device in the reverse time direction [24].
Reference [24] showed that a device is bidirectional if
and only if the corresponding transformation of density
matrices is a bistochastic quantum channel [77,78] , that is,
a linear map Cof the form C?ρ??P
iCiρC?
i, where ρis the
input density matrix, and ?Ci?are square matrices satisfy-
ing the conditionsP
iC?
iCi?P
iCiC?
i?I,Ibeing the
identity matrix. If a bistochastic channel Cdescribes the
state change in the forward mode, then the state change in
the backward mode is described by a (generally different)bistochastic channel Θ?C?given by Θ?C?∶ρ?P
iθ?Ci?ρθ?Ci??, where the square matrix θ?Ci?is either
unitarily equivalent to CT
i, the transpose of Ci, or unitarily
equivalent to C?
i, the adjoint of Ci[24]. The map Θis called
aninput-output inversion . Physically, it can represent a
time reversal (if the two ports of the device correspond
to two moments of time), an inversion of spatial directions(as in the example of the optical crystal), or any othersymmetry transformation obeying a set of general axioms
specified in Ref. [24]. In the following, we will focus on the
case where the input-output inversion is (unitarily equiv-alent to) the transpose. This case includes in particular thecanonical time reversal in quantum mechanics [79,80] and
quantum thermodynamics [81] (see[48] for more details.)
In principle, quantum mechanics allows for setups that
coherently control the input-output direction, such as the
setup shown in Fig. 1(d). We now develop a method for
witnessing input-output indefiniteness in the laboratory. Awitness for a given quantum resource, such as entangle-ment [82], indefinite causal order [83], and causal con-
nection [84], is an observable quantity that distinguishes
between resourceful and nonresourceful setups [85]. In our
case, the nonresourceful setups are those that use the devicein a well-defined direction. Setups that use it in the forward(backward) mode are described by a suitable set of positive
operators, denoted by S
fwd(Sbwd). The explicit characteri-
zation of these operators is provided in the SupplementalMaterial [48]. For the following discussion, it will suffice to
know that they act on the tensor product Hilbert space
H
AI?HAO?HBI?HBO, where HAI(HAO) is the Hilbert
space of the input (output) system of the device, while HBI
(HBO) is the Hilbert space of the input (output) system of
the overall process obtained by inserting the device into
the setup.
A setup that uses the device in a random mixture of the
forward and backward modes corresponds to an operator of
the form
S?pSfwd??1?p?Sbwd; ?1?
(a)
(b)
(c)
(d)
FIG. 1. Input-output indefiniteness in a bidirectional quantum
device. A bidirectional device with ports AandBcan be traversed
in two opposite directions: from AtoB(a) or from BtoA(b).
When these two configurations take place in a quantum super-position (c), the direction of the information flow between Aand
Bbecomes indefinite. To generate the superposition, we intro-
duce a control qubit that coherently controls the direction, withbasis states j0iandj1icorresponding to directions A→Band
B→A, respectively. Our setup (d) sets the control qubit in the
state j?i
c?? j0i?j1i?=???
2p
and witnesses input-output indefi-
niteness by performing local measurements on the target andcontrol system, with the target initialized in a quantum state ρ
t.PHYSICAL REVIEW LETTERS 132, 160201 (2024)
160201-2with Sfwd∈Sfwd,Sbwd∈Sbwd, and p∈?0;1/C138. We will
denote by Sdefinite the set of all operators of the form (1).
The setups outside Sdefinite are incompatible with the use
of the given device in a definite input-output direction: inthese setups, the device is not used in the forward mode, nor
in the backward mode, nor in any random mixture thereof.
For an operator Soutside S
definite , we define a witness of
input-output indefiniteness to be a self-adjoint operator W
such that
Tr?WS?<0; ?2?
and
Tr?WS0?≥0;?S0∈Sdefinite : ?3?
The condition (3)is characterized in the following
Theorem, which provides a systematic way to construct
witnesses of input-output indefiniteness.
Theorem 1 .—A Hermitian operator Wsatisfies Eq. (3)if
and only if there exist operators W0andW1such that
W≥W0,W≥W1,W0??BO/C138W0??AOBO/C138W0??AIAOBO/C138W0?
?AIAOBIBO/C138W0, and W1??BO/C138W1??AIBO/C138W1??AIAOBO/C138W1?
?AIAOBIBO/C138W1, having used the notation?X/C138S?TrX?S/C138?
?IX=dX?for a system Xof dimension dX.
The proof is provided in the Supplemental Material [48],
where we also show that the expectation value of any
witness can be decomposed into a linear combination ofoutcome probabilities arising from settings in which a
device is inserted in the setup and the resulting process is
probed on multiple input states.
Experimental demonstration of input-output indefiniteness
of a single quantum device. —Our experimental setup,
illustrated in Fig. 2, is inspired by a theoretical primitive
known as the quantum time flip (QTF) [24]. The QTF takes
in input an arbitrary bidirectional device and adds quantum
control to the direction in which the device is used. When
applied to a bidirectional device that acts as channel Cin
the forward direction, the QTF generates a new quantum
channel F?C?, acting jointly on the target system and
on a control qubit. Explicitly, the Kraus operators of thenew quantum channel F?C?, denoted by fF
ig, are related to
the Kraus operators of the original channel C, denoted
byfCig,a s
Fi?Ci?j0ih0j?CT
i?j1ih1j; ?4?
where fj0i;j1igare two orthogonal states of the control
qubit. When the control qubit is initialized in a coherentsuperposition of j0iand j1i, the new channel F?C?
implements a superposition of channel Cand its input-
output inversion C
T, in the sense of Refs. [86–93].
In our experiment, schematically illustrated in Fig. 1(d),
a heralded single photon is generated through spontaneous
parametric down-conversion [48]. The polarization qubit,
serving as the target system in the QTF, is initialized in an
HWP QWP PBS BS RM SPD Filter FCLaser
ppKTP
DL BS 1
LCZZBS2Port0
Port1
LC 1L C 2
FIG. 2. Experimental setup. A 2.5 mW continuous wave violet laser at 404 nm pumps a type-II cut ppKTP crystal, effectively working
as a heralded single photon source when the idler photons trigger a single photon detector (SPD). The single photon ’s polarization serves
as the target qubit and is initialized with a fiber polarizer controller, a half-wave plate (HWP), and a quarter-wave plate (QWP). Spatialmodes of the photon serve as the control qubit, and BS1 is used to coherently control the input-output direction. Measure-and-reprepareoperations on the polarization are implemented by two HWPs, two QWPs, and a PBS (dotted rectangle), while measurements on spatialmodes are implemented by two liquid crystal variable retardes (LCs) and BS2. A trombone-arm delay line (DL) and a piezoelectrictransducer are used to set the path length and the relative phases of the interferometer. [reflection mirror (RM), fiber coupler (FC)].PHYSICAL REVIEW LETTERS 132, 160201 (2024)
160201-3arbitrary fixed state, using a fiber polarizer controller, a
half-wave plate (HWP), and a quarter-wave plate (QWP).The photon is sent to a 50=50beamsplitter (BS1) to pre-
pare the spatial qubit in the superposition state j?i ?
?j0i?j1i?=???
2p
, where j0iandj1icorrespond to the two
alternative paths shown green and carmine in Fig. 2. The input
device for the QTF is a bistochastic measure-and-reprepare
operation [21], implemented by an assemblage of two HWPs,
two QWPs, and a polarizing beam splitter (PBS), showninside the dotted rectangle in Fig. 2. The input-output
inversion is realized by routing the photon through the
same assemblage along a backward path sandwichedbetween two fixed Pauli gates Z?j0ih0j?j1ih1j.Ac o h e r -
ent superposition of the forward and backward measure-
and-reprepare operations is created by using the spatial qubitas a control qubit. Finally, two paths are coherently recom-
bined on BS2, followed by a measurement on the polariza-
tion qubit.
To certify input-output indefiniteness, we derived the
witness W
optwith maximum robustness to noise (see
Supplemental Material [48].) This witness can be estimated
by probing the setup on a set of bistochastic measure-and-
reprepare processes that measure the polarization qubit in
the eigenbasis of a Pauli gate and reprepare the output in astate in the eigenbasis of another Pauli gate. The overall
evolution induced by the setup is probed by initializing
the path qubit in the maximally coherent state j?iand the
polarization qubit in one of the states j0i,j1i,j?iand
?1=???
2p
??j0i?ij1i?. Finally, the target qubit and control
qubit are measured in the eigenbases of the three Pauligates. The measured probabilities, shown in Fig. 3, are
used to calculate the experimental value of the witness
Tr?W
optSQTF/C138, which we find to be ??0.345/C60.005?,corresponding to a violation of the condition of definite
input-output direction by more than 69 standard deviations.
To implement the optimal witness Wopt, we performed
local operations on 5 qubits, using a total of 794 settings[48]. The complexity of the experiment is less than that of a
full process tomography, which would require at least 1023
settings. To further reduce the complexity, we designed a
simplified witness, where the target qubit is initialized in afixed state j0iand is eventually discarded. This witness
involves only 3 qubits and 48 settings, which we show tobe the optimal values [48]. In the experiment, we find the
value ??0.140/C60.004?, which certifies incompatibility
with a definite input-output direction by more than
35 standard deviations.
Experimental demonstration of advantage in a quantum
game. —Input-output indefiniteness offers an advantage in a
quantum game where a referee challenges a player to
find out a hidden relation between two unknown quantumgates [24]. In this game, the referee provides the player with
two devices implementing unitary gates UandV, respec-
tively, promising that the two gates satisfy either the
relation UV
T?UTVor the relation UVT??UTV. The
player ’s task is to determine which of these two alternatives
holds. Reference [24]showed that a player that uses the two
gates in the QTF can win the game with certainty, while
every strategy that uses the two devices in the same input-
output direction will fail at least 11% of the times.
In our experiment, discussed in the Supplemental
Material [48], we observe an average success probability
of99.60/C60.18% over a set of 21 gate pairs. The worst-
case error probability is approximately 0.68/C60.19%,
which is 16 times smaller than 11%, the lower bound
on the error probability for all possible strategies with
FIG. 3. Experimental data for the optimal witness. The figure shows the outcome probabilities of different measurements on the
control and target qubits, with the control initialized in the state j?iand the target in one of the states j0i(red), j1i(cyan), j?i(yellow),
and?j0i?ij1i?=???
2p
(green). The measurement outcomes are labeled by numbers 0, 1, 2, 3, corresponding to projections on the states
j0i;j1i;j?i;?j0i?ij1i?=???
2p
, respectively. For example, “03”labels the outcome that projects the control qubit onto j0iand the target
qubit onto ?j0i?ij1i?=???
2p
. The bars show the theoretical predictions, while the blue diamonds show the experimental data. We omit the
experimental data for outcomes that are irrelevant to the evaluation of the optimal witness. All the data in this figure refer to the settingwhere the device inside our setup implements a measure-and-reprepare process. Specifically, they refer to the event where the target ismeasured on the Xeigenstate j?iand reprepared in the Zeigenstate j0i. The experimental data for the remaining settings are shown in
the Supplemental Material [48].PHYSICAL REVIEW LETTERS 132, 160201 (2024)
160201-4definite-input-output direction. We also show that the
advantage of input-output indefiniteness persists even ifthe player has coherent control on each of the gates Uand
V: every strategy using the controlled gates ctrl?U?I?
j0ih0j?U?j1ih1jand ctrl?V?I?j0ih0j?V?
j1ih1jin the same input-output direction will necessarily
have an error probability of at least 5.6%. Overall, this
game can be regarded as a bipartite witness of global input-output indefiniteness. In the Supplemental Material [48],
we provide a general theory of such witnesses.
Conclusions. —In this Letter we introduced the notion of
witness of input-output indefiniteness and used it to
experimentally demonstrate input-output indefiniteness in
a single photonic device. Our results provide a way torigorously characterize input-output indefiniteness in the
laboratory, and represent a counterpart to recent experi-
ments on indefinite order of quantum gates [39–47].
Overall, input-output indefiniteness provides a new re-
source for quantum information protocols, and could
potentially lead to advantages in photonic quantumtechnologies.Our setup and its generalizations could also
be used to simulate exotic physics in which the arrow of
time is a quantum variable. These hypothetical phenomenafit into a broad framework developed by Hardy [28],w h o
suggested that a full-fledged theory of quantum gravity
would require spacetime structures to be subject to quan-tum indefiniteness. While an explicit physical model for
scenarios with indefinite time direction has yet to be
proposed, the availability of a mathematical frameworkfor their study and an experimental platform for their
simulation represent valuable tools for understanding their
operational implications.
Note added. —After the arXiv submission of our paper,
another research group [94] independently reported an
experimental demonstration of the advantage of indefinite
input-output direction in the game introduced in Ref. [24].
The main difference between Ref. [94] and this paper is
that, in addition to the experimental demonstration of the
game, here we provide a framework for witnessing indefi-
nite input-output direction and use it to experimentallydemonstrate input-output indefiniteness within a single
quantum device.
We acknowledge helpful discussions with Jonathan
Barrett, Hl? er Kristjánsson, Kavan Modi, Andreas Winter,
Ge Bai, and Fei Meng. This work was supported by theNational Key Research and Development Program of
China (No. 2021YFE0113100), NSFC (No. 12374338,
No. 11904357, No. 12174367, No. 12204458, andNo. 17326616), the Hong Kong Research Grant Council
(No. 17300920, SRFS2021-7S02, and T45-406/23-R), the
Innovation Program for Quantum Science and Technology(No. 2021ZD0301200), the Fundamental Research Funds
for the Central Universities, USTC Tang Scholarship,
Science and Technological Fund of Anhui Province forOutstanding Youth (2008085J02), China Postdoctoral
Science Foundation (2021M700138), China Postdoctoral
for Innovative Talents (BX2021289), and the JohnTempleton Foundation through the ID No. 62312 grant,
as part of the “The Quantum Information Structure of
Spacetime ”Project (QISS). This work was partially carried
out at the USTC Center for Micro and Nanoscale Researchand Fabrication. The opinions expressed in this publication
are those of the authors and do not necessarily reflect the
views of the John Templeton Foundation.
These authors contributed equally to this letter.
?bhliu@ustc.edu.cn
?cfli@ustc.edu.cn
§giulio@cs.hku.hk
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