Exploring postselection-induced quantum phenomena with time-bidirectional state formalism Evgeniy O. Kiktenko1 2 3 4

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Exploring postselection-induced quantum phenomena

with time-bidirectional state formalism

Evgeniy O. Kiktenko1, 2, 3, 4

1Department of Mathematical Methods for Quantum Technologies,

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia

2Russian Quantum Center, Skolkovo, Moscow 121205, Russia

3Geoelectromagnetic Research Center, Schmidt Institute of Physics of the Earth,

Russian Academy of Sciences, Troitsk 108840, Russia

4National University of Science and Technology “MISIS”, Moscow 119049, Russia∗

(Dated: March 24, 2023)

Here we present the time-bidirectional state formalism (TBSF) unifying in a general manner

the standard quantum mechanical formalism with no postselection and the time-symmetrized two-

state (density) vector formalism, which deals with postselected states. In the proposed approach,

a quantum particle’s state, called a time-bidirectional state, is equivalent to a joined state of two

particles propagating in opposite time directions. For a general time-bidirectional state, we derive

outcome probabilities of generalized measurements, as well as mean and weak values of Hermitian

observables. We also show how the obtained expressions reduce to known ones in the special cases of

no postselection and generalized two-state (density) vectors. Then we develop tomography protocols

based on mutually unbiased bases and a symmetric informationally complete positive operator-

valued measure, allowing experimental reconstruction of an unknown single qubit time-bidirectional

state. Finally, we employ the developed techniques for tracking of a qubit’s time-reversal journey in

a quantum teleportation protocol realized with a cloud-accessible noisy superconducting quantum

processor. The obtained results justify an existence of a postselection-induced qubit’s proper time-

arrow, which is diﬀerent from the time-arrow of a classical observer, and demonstrate capabilities

of the TBSF for exploring quantum phenomena brought forth by a postselection in the presence of

noise.

I. INTRODUCTION

The standard quantum formalism is commonly used

for calculating a probability distribution of measurement

outcomes, given a complete characterization of prepara-

tion and evolution of a measured quantum system. This

consideration with respect to a preselected initial state

contains an implicit time asymmetry related to the con-

cept of ‘collapsing’, or ‘reduction’, of system’s state in the

measurement process [1,2]. Combining the preselection

with a postselection, i.e. the consideration of a partic-

ular outcome of the measurement, removes this asym-

metry and gives rise to the two-state vector formalism

(TSVF) [2,3]. Within the TSVF, the quantum particle’s

state is described by a pair of vectors (|ψi,hφ|), where

|ψi, determined by the preselection, can be considered

evolving forward in time, while hφ|, determined by the

postselection, can be considered as evolving back from

the future to the past (see an example of an optical ex-

periment with pre- and postselection in Fig. 1).

From a practical point of view, one of the most impor-

tant concepts appearing in the framework of postselec-

tion and the TSVF is weak values of observables [5,6].

Despite some criticism (see e.g. [7–9]), weak values and

related techniques for a weak value ampliﬁcation [10–13]

appear to be extremely useful in the context of quan-

tum metrology [14–23] (for a review, see [4]). Moreover,

∗e.kiktenko@rqc.ru

|V⟩⟨D|

Birefringent crystal

Single photon

source

Polarizer

(vertical)

Polarizer

(diagonal)

Detection

system

click!

Figure 1. An example of an optical experimental setup

dealing with both pre- and postselected states. Here pho-

tons pass through two polarizers, the ﬁrst of which ﬁlters out

vertically polarized states |Viand the second one ﬁlters out

diagonally polarized states |Di(| hV|Hi| = 2−1/2). By in-

troducing a two-state vector (|Vi,hD|), the TSVF provides

a complete description of photons’ polarization, given their

detection after the second polarizer (the postselection condi-

tion). In particular, the two-state vector allows describing a

transformation of photons’ spatial degree of freedom due to

a coupling with polarization realized by a birefringent crys-

tal (see a comprehensive discussion in Ref. [4]).

taking postselection into account also plays an important

role in studying complexity theory [24], quantum contex-

tuality [25–29], fundamentals of quantum physics [30–40],

design of quantum computing algorithms [41], and quan-

tum communication protocols [42–46].

The postselection with respect to entangled states

gives rise to time-reversal phenomena, including an ap-

pearance of closed timelike curves (CTC), considered

both theoretically [47–56] and experimentally [48,53].

arXiv:2210.01583v2 [quant-ph] 23 Mar 2023

The basic idea behind these phenomena is that Bell state

preparation and Bell state measurement can be consid-

ered as a kind of “time mirrors” reﬂecting a quantum

state’s propagation in time. Note that this interpreta-

tion perfectly agrees with experimental results on delayed

entanglement swapping [57,58].

Originally, the TSVF was formulated with respect to a

pair of pure states [2,3]. An important extension comes

with introducing an ancillary particle and performing

postselection with respect to an entangled state. This

creates an entanglement between forward and backward

evolving states of a two-state vector, and yields the con-

cept of a generalized two-state vector [3,59]. Study-

ing statistical ensembles of generalized two-state vectors

bring forth a notion of two-state density vectors [60],

which can be considered as a manifestation of density ma-

trices in the framework of the TSVF. Another approach

of introducing mixedness into the TSVF is presented in

Ref. [37], where the case of forward and backward evolv-

ing states described with density matrices is considered.

The present work is devoted to a further development

of eﬀective description of quantum states in the presence

of postselection and pursues the following two main goals.

The ﬁrst goal is closing the gap between the previous

approaches for describing mixed, or randomized, quan-

tum states in the presence of the postselection and the

standard quantum formalism without any postselection

at all. This goal is achieved by extending the two-state

density vector formalism [60] with a more general time-

bidirectional state formalism (TBSF), where the posts-

election is performed with respect to an arbitrary posi-

tive operator-valued measure (POVM) eﬀect (see Fig. 2).

Within the the developed formalism, a state of a parti-

cle is described by a bipartite, generally mixed, state,

called a time-bidirectional state, which is equivalent to a

joint state of two particles propagating in opposite time-

directions. We show that in the absence of a postse-

lection, i.e. identity postselection eﬀect, the backward

evolving part appears to be in the maximally mixed state,

while the forward evolving one coincides with a density

matrix from the standard formalism. An important fea-

ture of the TBSF is its ability to account for any kind of

decoherence noise, aﬀecting both pre- and postselection.

The second goal is developing practical schemes for to-

mography of both pre- and postselected states. In the

current work, we focus on the case of a single qubit that

can be easily generalized to a multiqubit one. Compared

to a high level recipe for making a complete set of Kraus

operators, suﬃcient for reconstructing an unknown pre-

and postselected state, presented in Ref. [60], here we

obtain explicit circuits for running tomography proto-

cols on an arbitrary quantum processor. For this pur-

pose, we borrow two basic approaches for single-qubit

tomography: the one based on mutually unbiased bases

(MUBs) corresponding to measuring three components

of a Bloch vector, and the second based on a symmetric

informationally complete POVM (SIC-POVM) allowing

reconstruction of unknown state with a measurement of

ρpre A

ρprobe PUμ

pass|fail

record /

discard

AliceBob

postselection result

(pass / fail)

Time

POVM

{E(μ′ )}μ′

POVM with

Epost

Figure 2. General scheme of a postselection experiment giving

rise to a concept of a time-bidirectional state. First, Alice

prepares particles Q and A in some arbitrary state and sends

Q to Bob. On his side, Bob applies a unitary operation U

to Q and P, then performs an arbitrary measurement on P,

and ﬁnally returns Q back to Alice. Then Alice makes a joint

measurement on A and Q, described by a POVM containing

a particular eﬀect Epost . If the outcome given by Epost is

realized, then Alice tells Bob to keep his measurement result

µ, otherwise µis discarded.

a single type.

To demonstrate capabilities of the TBSF and devel-

oped tomography techniques, we consider a well-known

time-reversal phenomenon appearing in a quantum tele-

portation protocol [48,50,53]. Namely, we track propa-

gation of a qubit state, initially prepared by Alice, (i) for-

ward in time to the moment of a Bell measurement on her

qubit and a qubit from a pre-shared Bell pair, then (ii)

back in time on Alice’ qubit from the Bell pair to the mo-

ment of the Bell pair birth, and then (iii) forward in time

on Bob’s particle from the Bell pair. For this purpose we

use a seven-qubit cloud-accessible noisy superconducting

quantum processor provided by IBM. Although, experi-

ments on the observation of a postselection-induced time-

travel in quantum teleportation were considered previ-

ously [48,53], to the best of our knowledge, this is the

ﬁrst time where it is demonstrated, using the developed

formalism, how the state, prepared by Alice, propagates

back in time on Alice’s physical qubit taken from the

pre-shared Bell pair. As already mentioned, an impor-

tant advantage of the developed TBSF, compared, e.g.,

to the time-reversal formalism suggested in Ref. [48], is

that this formalism allows considering decoherence ef-

fects. In particular, we observe evidences of irreversible

corruption of the Alice’s quantum state during propaga-

tion along its proper postselection-induced time-arrow.

The rest of the paper is organized as follows. In Sec. II,

we introduce the concept of time-bidirectional states,

provide some illustrative examples, and derive their main

mathematical properties. In Sec. III, we discuss descrip-

tion of measurements made on a time-bidirectional state

with a focus on von Neuman measurement of Hermitian

observables and measurements of weak values. In Sec. IV,

we develop tomography protocols for experimental recon-

structing of an unknown single-qubit time-bidirectional

state. In Sec. V, we apply the developed formalism

and tomography techniques for observing a time-reversal

journey of a qubit’s state in a quantum teleportation pro-

tocol. We conclude and provide an outlook in Sec. VI.

II. INTRODUCING TIME-BIDIRECTIONAL

STATES

A. General postselection experiment

Let us consider a postselection experiment shown in

Fig. 2. The experiment is realized by two parties, named

Alice and Bob, that are able to communicate with quan-

tum particles and classical messages. At the start of the

experiment, Alice prepares two particles, Q and A, in an

arbitrary joint mixed state

ρpre =ρpre

ii0;mm0|iiQhi0|⊗|miAhm0|.(1)

Here |niXwith integer labels ndenote computational

basis states of particle X and ρpre

ii0;mm0are density ma-

trix elements providing standard conditions ρpre ≥0,

Trρpre = 1. Note that here and hereafter we apply the

Einstein summation convention and omit explicit sum-

mation with respect to repeated indices. After its prepa-

ration, Q is given to Bob, while A remains with Alice.

On his side, Bob takes an additional particle P, pre-

pared in some mixed state

ρprobe =ρprobe

kk0|kiPhk0|(2)

(ρprobe ≥0, Trρprobe = 1), and lets P and Q evolve

according to a unitary evolution operator

U=Uk;i

l;j|liPhk|⊗|jiQhi|.(3)

Then Bob makes a measurement on P described by

a POVM {E(µ0)}µ0, satisfying standard conditions

E(µ0)≥0, Pµ0E(µ0) = 1, where 1is the identity ma-

trix. Here we consider outcome labels µ0belonging to an

arbitrary ﬁnite set. Bob keeps the obtained measurement

outcome µ, corresponding to the realized eﬀect E(µ), and

returns Q back to Alice.

On her side, Alice makes a joint measurement on Q

and A, described by another POVM, whose collection of

eﬀects includes a particular eﬀect

Epost =Ejj0;mm0

post |j0iQhj|⊗|m0iAhm|(4)

(0 ≤Epost ≤1). If an outcome of Alice’s measurement

corresponds to Epost, then we say that the postselection

passed and set a special ﬂag variable ps := 1; postselec-

tion failed and ps := 0 otherwise. Alice transmits the

result of the postselection, i.e. single bit ps, to Bob, who

keeps his measurement outcome µif postselection has

passed, or discards µotherwise. We note that the only

constraints on particular time moments, when the de-

scribed operations take place, correspond to the general

ordering: Preparations of ρpre and ρprobe are in the past

light cone of U, while Alice’s and Bob’s measurements

are in the future light cone of Uand in the past light

cone of the ﬁnal decision on µ.

The main object of our study is a conditional proba-

bility distribution of Bob’s measurement outcomes, given

the passing postselection on Alice’s side (here Mdenotes

a random variable of Bob’s outcome). According to ax-

iomatics of quantum mechanics, the probability of the

joint event of M=µand the postselection passing is

given by

Pr[M=µ∪ps = 1]

=ρpre

ii0;kk0ρprobe

mm0Ui;m

l;jUi0;m0

l0;j0E(µ)ll0Ejj0;kk0

post ,(5)

where the overbar denotes the complex conjugate. The

probability of the postselection passing takes the form

Pr[ps = 1] = ρpre

i˜

i0;˜

k˜

k0ρprobe

˜mm0U˜

i; ˜m

l;˜

jU˜

i0; ˜m0

l;˜

j0E˜

j˜

j0;˜

k˜

post ≡Ppost.

(6)

Putting these expressions in Bayes’ rule leads to

Pr[M=µ|ps = 1] = Pr[M=µ∪ps = 1]

Ppost

=ρpre

ii0;kk0ρprobe

mm0Ui;m

l;jUi0;m0

l0;j0E(µ)ll0Ejj0;kk0

post

ρpre

i˜

i0;˜

k˜

k0ρprobe

˜mm0U˜

i; ˜m

l;˜

jU˜

i0; ˜m0

l;˜

j0E˜

j˜

j0;˜

k˜

post ≡P(µ),(7)

given Ppost >0, and P(µ) = 0 otherwise. It appears

that Eq. (7) is much easier to follow in the form of a

tensor network, shown in Fig. 3(a). We note that if Zis

a tensor, then by Zwe denote a tensor obtained from Z

by taking complex conjugation of all element. It diﬀers

from the Hermitian conjugate of Z, which we denote as

Z†.

Within Eq. (7) we can separate two mathematical

structures, which are related to Alice’s and Bob’s actions

correspondingly. The pre- and postselection, which are

performed by Alice, are described by a tensor

ηjj0

ii0:= ρpre

ii0;mm0Ejj0;mm0

post ,(8)

while the indirect Bob’s measurement is described by a

collection of tensors

K(µ)ii0

jj0:= ρprobe

kk0Uk;i

l;jUk0;i0

l0;j0E(µ)ll0.(9)

We then can rewrite Eqs. (7) and (6) in a compact form

P(µ) = K(µ)ii0

mm0ηmm0

ii0

K˜

i˜

˜m˜m0η˜m˜m0

i˜

i0≡K(µ)•η

K•η,

Ppost =K•η,

(10)

where K:= PµK(µ) and •, in line with Ref. [60], de-

notes contraction with respect to proper indices [see also

Fig. 3(a)].

In what follows, we refer to η,K(µ), and Kas a time-

bidirectional state, operation outcome µtensor, and op-

eration tensor correspondingly. The time-bidirectional

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Exploringpostselection-inducedquantumphenomenawithtime-bidirectionalstateformalismEvgeniyO.Kiktenko1,2,3,41DepartmentofMathematicalMethodsforQuantumTechnologies,SteklovMathematicalInstituteofRussianAcademyofSciences,Moscow119991,Russia2RussianQuantumCenter,Skolkovo,Moscow121205,Russia3Geoelectromagn...

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Exploring postselection-induced quantum phenomena with time-bidirectional state formalism Evgeniy O. Kiktenko1 2 3 4

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