Operations with elements of transferred density matrix via unitary   transformations at receiver side

A.I.Zenchuk

arXiv:1812.01408·quant-ph·May 6, 2020

Operations with elements of transferred density matrix via unitary transformations at receiver side

A.I.Zenchuk

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TL;DR

This paper introduces a method using unitary transformations at the receiver to manipulate transferred density matrix elements, enabling operations like zeroing, rearranging, and linear combination, demonstrated through numerical simulations on a spin chain.

Contribution

It presents a novel protocol for performing matrix element operations via unitary transformations at the receiver side in quantum state transfer.

Findings

01

Successfully zeroed out specific matrix elements

02

Rearranged matrix elements as desired

03

Simulated operations on a 42-node spin chain

Abstract

Using the tool of unitary transformations of the extended receiver we perform simple operations with the non-diagonal elements of the initial sender's density matrix after their transferring to the receiver. These operations are following: turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. We also propose a protocol for solving a system of linear algebraic equations with a system of two equations as an example. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubit sender and receiver.

Tables2

Table 1. Table 1: Numerical coefficients in Eqs.( 54 ), ( 66 ) and ( 78 ).

	$f_{01; 10}$	$a_{01; 01}$	$a_{10; 10}$	$c_{01; 01}$	$c_{10; 10}$	$d$
Eq.(54)	$0$	$0$	$0.9182 e^{- 1.5135 i}$	$0$	$0.4393 e^{1.1232 i}$	$0.4784 e^{- 0.3903 i}$
Eq.(66)	$0$	$0.6993 e^{- 1.4322 i}$	$0$	$0.4129 e^{0.0917 i}$	$0$	$0.5904 e^{- 1.3405 i}$
Eq.(78)	$0.3464 e^{2.9264 i}$	$0.3846 e^{- 2.9224 i}$	$0.9009 e^{0.0040 i}$	$0$	$0$	$0$

Table 2. Table 2: Families of the φ 𝜑 \varphi -parameters of unitary transformation ( 136 ) corresponding to the manipulations with the receiver’s density matrix elements discussed in Secs. V.2 - V.5 . The lines 1 – 4 correspond, respectively, to the transformation ( 49 ), rearrangements ( 84 ) and ( 86 ), linear combination of elements ( 100 ) - ( 104 ) and solving the system of two linear algebraic equations ( 121 ).

$#$	$φ_{2, 3}^{(1)}$	$φ_{2, 5}^{(1)}$	$φ_{2, 8}^{(1)}$	$φ_{3, 5}^{(1)}$	$φ_{3, 8}^{(1)}$	$φ_{4, 6}^{(1)}$	$φ_{4, 7}^{(1)}$	$φ_{4, 9}^{(1)}$	$φ_{4, 10}^{(1)}$	$φ_{4, 11}^{(1)}$	$φ_{5, 8}^{(1)}$	$φ_{6, 7}^{(1)}$	$φ_{6, 9}^{(1)}$
1.	0.2871	1.2703	5.3837	0.0157	4.4081	4.3842	1.4712	0.7807	5.7679	1.4273	5.7371	5.7107	5.0750
2.	6.2612	1.3199	4.6848	2.7895	3.1939	4.9799	1.0863	3.7417	0.3186	1.2688	0.1286	4.5929	0.1008
3.	1.4280	1.1782	3.5900	0.3906	0.8475	1.1926	5.9163	5.5803	4.6112	5.8579	4.0289	3.5539	1.9443
4.	3.7020	3.5217	3.9665	5.0063	2.3526	5.2017	1.0258	0.3117	1.0826	1.5443	3.9441	0.9333	2.0108

Equations153

ρ (0) = ρ^{(S)} (0) \otimes ρ^{(T L, E R)} (0),

ρ (0) = ρ^{(S)} (0) \otimes ρ^{(T L, E R)} (0),

ρ_{N^{3}; M^{3}}^{(R)} (t) = I^{1}, J^{1} \sum T_{N^{3} M^{3}; I^{1} J^{1}} (t) ρ_{I^{1} J^{1}}^{(S)} (0),

ρ_{N^{3}; M^{3}}^{(R)} (t) = I^{1}, J^{1} \sum T_{N^{3} M^{3}; I^{1} J^{1}} (t) ρ_{I^{1} J^{1}}^{(S)} (0),

T_{N^{3} M^{3}; I^{1} J^{1}} = N^{1} N^{2} \sum I^{2}, I^{3}, J^{2}, J^{3} \sum W_{{N}; {I}} ρ_{I^{2} I^{3}; J^{2} J^{3}}^{(T L, E R)} W_{{J}; N^{1} N^{2} M^{3}}^{+},

I^{k} = {i_{1}^{k} i_{2}^{k}}, k = 1, 3, I^{2} = {i_{1}^{2} \dots i_{N - 4}^{2}} .

I^{k} = {i_{1}^{k} i_{2}^{k}}, k = 1, 3, I^{2} = {i_{1}^{2} \dots i_{N - 4}^{2}} .

W = (E^{(S, T L)} \otimes U^{(E R)}) V (t) .

W = (E^{(S, T L)} \otimes U^{(E R)}) V (t) .

[H, I_{z}] = 0, [U^{(E R)}, I_{z}^{(E R)}] = 0,

[H, I_{z}] = 0, [U^{(E R)}, I_{z}^{(E R)}] = 0,

T_{N^{3} M^{3}; I^{1} J^{1}} = N^{1} N^{2} \sum I^{1}, I^{2}, I^{3}, J^{1} \sum W_{N^{1} N^{2} N^{3}; I^{1} I^{2} I^{3}} ρ_{I^{2} I^{3}; I^{2} I^{3}}^{(T L, E R)} (0) W_{J^{1} I^{2} I^{3}; N^{1} N^{2} M^{3}}^{+} .

T_{N^{3} M^{3}; I^{1} J^{1}} = N^{1} N^{2} \sum I^{1}, I^{2}, I^{3}, J^{1} \sum W_{N^{1} N^{2} N^{3}; I^{1} I^{2} I^{3}} ρ_{I^{2} I^{3}; I^{2} I^{3}}^{(T L, E R)} (0) W_{J^{1} I^{2} I^{3}; N^{1} N^{2} M^{3}}^{+} .

W_{N^{1} N^{2} N^{3}; I^{1} I^{2} I^{3}} = 0 \mbox i f i = 1 \sum 3 ∣ N^{i} ∣ \neq = i = 1 \sum 3 ∣ I^{i} ∣,

W_{N^{1} N^{2} N^{3}; I^{1} I^{2} I^{3}} = 0 \mbox i f i = 1 \sum 3 ∣ N^{i} ∣ \neq = i = 1 \sum 3 ∣ I^{i} ∣,

W_{0^{1} 0^{2} 0^{3}; 0^{1} 0^{2} 0^{3}} = 1,

W_{0^{1} 0^{2} 0^{3}; 0^{1} 0^{2} 0^{3}} = 1,

T_{N^{3} M^{3}; I^{1} J^{1}} = λ_{N^{3} M^{3}} δ_{N^{3} I^{1}} δ_{M^{3} J^{1}}, (N^{3}, M^{3}) \neq = (0^{3}, 0^{3}),

T_{N^{3} M^{3}; I^{1} J^{1}} = λ_{N^{3} M^{3}} δ_{N^{3} I^{1}} δ_{M^{3} J^{1}}, (N^{3}, M^{3}) \neq = (0^{3}, 0^{3}),

T_{0^{3} 0^{3}; I^{1} J^{1}} = δ_{I^{1} J^{1}} (1 - λ_{I^{1} I^{1}}), (I^{1}, J^{1}) \neq = (0^{1}, 0^{1}),

T_{0^{3} 0^{3}; 0^{1} 0^{1}} = 1.

ρ_{N^{3}; M^{3}}^{(R; n)} = λ_{N^{3} M^{3}} ρ_{N^{3}; M^{3}}^{(S; n)}, (N^{3}, M^{3}) \neq = (0^{3}, 0^{3}),

ρ_{N^{3}; M^{3}}^{(R; n)} = λ_{N^{3} M^{3}} ρ_{N^{3}; M^{3}}^{(S; n)}, (N^{3}, M^{3}) \neq = (0^{3}, 0^{3}),

ρ_{0^{3}; 0^{3}}^{(R; n)} = 1 - I^{1} \neq = 0^{1} \sum λ_{I^{1} I^{1}} ρ_{I^{1}; I^{1}}^{(S; n)} .

λ_{N_{0}^{3} M_{0}^{3}} = 0

λ_{N_{0}^{3} M_{0}^{3}} = 0

ρ_{N_{0}^{3} M_{0}^{3}}^{(R; n)} = 0.

ρ_{N_{0}^{3} M_{0}^{3}}^{(R; n)} = 0.

T_{N_{0}^{3} M_{0}^{3}; I_{0}^{3} J_{0}^{3}} = δ_{N_{0}^{3} J_{0}^{3}} δ_{M_{0}^{3} I_{0}^{3}} λ_{I_{0}^{3} J_{0}^{3}}, T_{I_{0}^{3} J_{0}^{3}; N_{0}^{3} M_{0}^{3}} = δ_{N_{0}^{3} J_{0}^{3}} δ_{M_{0}^{3} I_{0}^{3}} λ_{N_{0}^{3} M_{0}^{3}},

T_{N_{0}^{3} M_{0}^{3}; I_{0}^{3} J_{0}^{3}} = δ_{N_{0}^{3} J_{0}^{3}} δ_{M_{0}^{3} I_{0}^{3}} λ_{I_{0}^{3} J_{0}^{3}}, T_{I_{0}^{3} J_{0}^{3}; N_{0}^{3} M_{0}^{3}} = δ_{N_{0}^{3} J_{0}^{3}} δ_{M_{0}^{3} I_{0}^{3}} λ_{N_{0}^{3} M_{0}^{3}},

ρ_{N_{0}^{3}; M_{0}^{3}}^{(R; n)} = λ_{I_{0}^{3} J_{0}^{3}} ρ_{I_{0}^{3}; J_{0}^{3}}^{(S; n)}, ρ_{I_{0}^{3}; J_{0}^{3}}^{(R; n)} = λ_{N_{0}^{3} M_{0}^{3}} ρ_{N_{0}^{3}; M_{0}^{3}}^{(S; n)} .

ρ_{N_{0}^{3}; M_{0}^{3}}^{(R; n)} = λ_{I_{0}^{3} J_{0}^{3}} ρ_{I_{0}^{3}; J_{0}^{3}}^{(S; n)}, ρ_{I_{0}^{3}; J_{0}^{3}}^{(R; n)} = λ_{N_{0}^{3} M_{0}^{3}} ρ_{N_{0}^{3}; M_{0}^{3}}^{(S; n)} .

A x = b,

A x = b,

ρ_{N_{i}^{3}; M_{i}^{3}}^{(S; n)} = b_{i}, i = 1, \dots, M .

ρ_{N_{i}^{3}; M_{i}^{3}}^{(S; n)} = b_{i}, i = 1, \dots, M .

ρ_{N_{i}^{3}; M_{i}^{3}}^{(S; n)} = j = 1 \sum M A_{ij} x_{j}, i = 1, \dots, M .

ρ_{N_{i}^{3}; M_{i}^{3}}^{(S; n)} = j = 1 \sum M A_{ij} x_{j}, i = 1, \dots, M .

ρ_{N_{i}^{3}; M_{i}^{3}}^{(R; n)} = j = 1 \sum M α_{ij} x_{j}, i = 1, \dots, M,

ρ_{N_{i}^{3}; M_{i}^{3}}^{(R; n)} = j = 1 \sum M α_{ij} x_{j}, i = 1, \dots, M,

α_{ij} = α δ_{ij}, α = co n s t,

α_{ij} = α δ_{ij}, α = co n s t,

ρ_{N_{i}^{3}; M_{i}^{3}}^{(R; n)} = α x_{i}, i = 1, \dots, M,

ρ_{N_{i}^{3}; M_{i}^{3}}^{(R; n)} = α x_{i}, i = 1, \dots, M,

H = j > i \sum D_{ij} (I_{i x} I_{j x} + I_{i y} I_{j y}),

H = j > i \sum D_{ij} (I_{i x} I_{j x} + I_{i y} I_{j y}),

[H, I_{z}] = 0,

ρ^{(T L, E R)} (0)

ρ^{(T L, E R)} (0)

N^{3} = {n_{1} n_{2}}, M^{3} = {m_{1} m_{2}}, I^{1} = {i_{1} i_{2}}, J^{1} = {j_{1} j_{2}},

N^{3} = {n_{1} n_{2}}, M^{3} = {m_{1} m_{2}}, I^{1} = {i_{1} i_{2}}, J^{1} = {j_{1} j_{2}},

{N^{1} N^{2}} = J_{N - 2} = {j_{1} \dots j_{N - 2}}, ρ_{I^{2} I^{3}; I^{2} I^{3}}^{(T L, E R)} = δ_{I^{2} 0^{2}} δ_{I^{3} 0^{3}},

T_{n_{1} n_{2} m_{1} m_{2}; i_{1} i_{2} j_{1} j_{2}} = J_{N - 2} \sum W_{J_{N - 2} n_{1} n_{2}; i_{1} i_{2} 0_{N - 2}} W_{j_{1} j_{2} 0_{N - 2}; J_{N - 2} m_{1} m_{2}}^{+},

T_{n_{1} n_{2} m_{1} m_{2}; i_{1} i_{2} j_{1} j_{2}} = J_{N - 2} \sum W_{J_{N - 2} n_{1} n_{2}; i_{1} i_{2} 0_{N - 2}} W_{j_{1} j_{2} 0_{N - 2}; J_{N - 2} m_{1} m_{2}}^{+},

m_{1}, m_{2} \sum J_{N - 2} \sum W_{j_{1} j_{2} 0_{N - 2}; J_{N - 2} m_{1} m_{2}}^{+} W_{J_{N - 2} m_{1} m_{2}; i_{1} i_{2} 0_{N - 2}} = δ_{i_{1} j_{1}} δ_{i_{2} j_{2}},

m_{1}, m_{2} \sum J_{N - 2} \sum W_{j_{1} j_{2} 0_{N - 2}; J_{N - 2} m_{1} m_{2}}^{+} W_{J_{N - 2} m_{1} m_{2}; i_{1} i_{2} 0_{N - 2}} = δ_{i_{1} j_{1}} δ_{i_{2} j_{2}},

ρ^{(S)} = k = - 2 \sum 2 ρ^{(S; k)}, ρ^{(R)} = k = - 2 \sum 2 ρ^{(R; k)},

ρ^{(S)} = k = - 2 \sum 2 ρ^{(S; k)}, ρ^{(R)} = k = - 2 \sum 2 ρ^{(R; k)},

D_{i (i + 1)} \equiv δ_{i}, i = 1, \dots, N - 1,

D_{i (i + 1)} \equiv δ_{i}, i = 1, \dots, N - 1,

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TopicsQuantum Information and Cryptography · Quantum Computing Algorithms and Architecture · Matrix Theory and Algorithms

Full text

**Operations with elements of transferred density matrix via unitary transformations on extended receiver **

A.I.Zenchuk

Institute of Problems of Chemical Physics RAS, Chernogolovka, Moscow reg., 142432, Russia

Abstract

We combine the long-distance quantum state transfer and simple operations with the elements of the transferred (nor perfectly) density matrix. These operations are turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. The basic tool performing these operations is the unitary transformation on the extended receiver. A system of linear algebraic equations can be solved in this way as well. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubit sender and receiver.

I Introduction

The development of quantum counterparts of classical algorithms is a beneficial direction in quantum information processing. Since unitary transformations play a fundamental role in quantum mechanics, the problem of their implementation and control is of general interest. A principal role plays a simplest two-qubit operator called controlled-NOT (CNOT) which entangles states of two qubits. This operation was generalized to controlled- $U$ operator ( $C^{n}(U)$ ) with an arbitrary one-qubit unitary transformation $U$ controlled by $n$ ( $n\geq 1$ ) qubits BBCVMSSSW ; NCh ; LLS . Selecting a proper combination of unitary operators (perhaps, combined with certain measurements) allows one to realize various algorithms useful in a wide range of applications. To such algorithms we refer the quantum Fourier transform (QFT) NCh , Hamiltonian simulation BACS ; Ch and phase estimation NCh ; LP ; CEMM . These algorithms are used as subroutines in other algorithms such as the HHL-algorithm for solving linear systems HHL ; WZRL , the algorithms for solving systems of nonlinear equations QH and for quntum realization of elementary matrix operations using Trotter product formula ZZRF . The HHL-algorithm was first performed in photon systems CWSCGZLLLP ; BKRLDAW and recently it was implemented on the basis of superconducting qubit system to solve the system of two linear equations ZSCXLG . The HHL-algorithm is also effectively used in the quantum algorithm for data fitting over the exponentially large data set WBL . Review on quantum machine learning is represented in BWPRWL . The problem of quantum speedup, which is crucial for justifying the advantages of quantum computations, is discussed in the above references as well. Basic quantum algorithms and their implementations in quantum information theory can be found in book NCh . The research on realization of quantum gates goes further to work out the effective remote control of unitary operations (operator teleportation) HPXLG ; QZACB with multi-qubit controlled-phase gates WY among them.

In our paper we consider the remote preparation of initial state at one side of the communication line (sender) which after been (non-perfectly) transferred to the (extended) receiver is subjected to the unitary transformation with the purpose of obtaining the desired state. In other words, we combine the long-distance non-perfect mixed state transfer and the operations with the elements of the transferred density matrix. Thus, the communication line includes the sender (where the initial state is to be prepared), the receiver (where the final state is to be detected) the extended receiver (which includes receiver and several additional nodes (ancilla) and serves as a platform for applying the unitary transformation aimed to perform required operations) and the transmission line (connecting the sender with the extended receiver). During the evolution due to the spin-spin interactions, each element of the receiver’s density matrix becomes, in general, a linear combination of all the elements of the initial sender’s density matrix. This phenomenon is an obstacle for the state transfer but not for the remote manipulations with the matrix elements as is shown further in this paper. To (partially) remove this mixture of elements we chose the evolution preserving the number of excitations in the system FZ_2017 . This constraint provides the independent evolution of the multiple-quantum (MQ) coherence matrices, so that the matrix elements mix only inside of each MQ-coherence matrix of a particular order. We recall that the $n$ -order MQ-coherence matrix collects those elements of the density matrix which correspond to the state-transitions changing the $z$ -projection of the total spin by $n$ . The importance of such independent evolution of MQ-coherence matrices is associated with different meaning of the matrix elements assembled inside of each coherence matrix. For instance, the diagonal elements (a part of the zero-order coherence matrix) represent the probabilities of state realization and therefore must be nonnegative. In addition, their sum equals one.

We deal with the spin chain considered in Ref.Z_2018 where the unitary transformation of the extended receiver was used for the structural restoring of the nondiagonal part of the transferred matrix. We show that such transformation can be an effective tool for performing simple operations with a mixed state of a quantum system. Of course, the problem of practical realization of required unitary transformations is directly related to such approach. In this regards we, first of all, relay on the so-called Solovay-Kitaev theorem NCh . According to this theorem, any unitary transformation can be constructed using one-qubit rotations and CNOTs (simplest two-qubit operations). In addition, the operator formation is a central point of the duality computing Long ; LY ; LYCh ; G . In this method, the quantum system consists of two parts. The first one is the internal subsystem which serves for computing. The second part is the (multi-qubit) ancila, on which two new quantum gates are realized: quantum wave divider and quantum wave combiner. Using controlled unitary operations and controlled measurement acting on the internal subsystem one can construct various non-unitary (in general) operators acting on the internal subsystem (allowable generalized quantum gate). Realization of such gates is discussed in ZCL .

As operations with the elements of the transferred density matrix we, consider the following manipulations.

•

Structural restoring of the nondiagonal part of the density matrix supplemented with turning some nondiagonal matrix elements to zero. In this case, each nondiagonal element multiplied by its scale factor maps into the appropriate nondiagonal element of the receiver’s density matrix thus establishing the structural restoring Z_2018 . However, there is a group of elements having zero scale factors.

•

Rearranging matrix elements. Unlike the usual structural restoring, some nondiagonal elements of the sender’s density matrix multiplied by the scale factor exchange their positions at the receiver’s state.

•

Constructing certain linear combinations of matrix elements. This operation results in the linear combination (with needed coefficients) of several nondiagonal elements of the sender’s density matrix and assigns this combination to a proper element of the receiver’s density matrix.

•

Solving systems of linear algebraic equations. Unlike the HHL-algorithm, our protocol is based on the mixed states. Certain elements of the sender’s density matrix are used as the input data (the right-hand side of the linear system). Then the evolution following by the unitary transformation on the extended receiver makes such linear combinations of these elements that equals to the solutions of the considered linear system and place the result in the appropriate elements of the receiver’s density matrix.

In all these manipulations, we use the sender’s state as the input information, the receiver’s state as the output information and the unitary transformation of the extended receiver as a tool allowing to manipulate the elements of the output state. The transmission line is an optional part, it connects the sender with the extended receiver and serves to perform the remote manipulation with the input data. It can be absent if we are interested in the local manipulations. We emphasise that our protocols do not include a classical communication channel and therefore can be referred to completely quantum processes.

The paper is organized as follows. In Sec.II, we discuss the general protocol of operations with the transferred density matrix, including structure of the communication line, the initial state and the general form of the evolution operator. The set of manipulations with the receiver’s density matrix elements is introduced in Sec.III. In Sec.IV, we briefly describe a particular model of communication line used in Sec.V for numerical simulations of operations with density matrix elements. The paper concludes with Sec.VI. The general structure of the unitary transformation and examples of such transformation realizing the operations with matrix element are represented in the Appendix, Sec.VII.

II Initial state and evolution operator

We consider the spin-1/2 communication line Z_2018 which includes the sender ( $S$ ), transmission line ( $TL$ ), receiver ( $R$ ) and ancilla ( $A$ ) in the strong external magnetic field directed along the chain ( $z$ -direction), Fig.1. The receiver and ancilla form the extended receiver ( $ER$ ).

The general protocol performing certain manipulations with nondiagonal elements of $\rho^{(R)}$ is schematically represented in Fig.2. First of all, we start with the tensor-product initial state

[TABLE]

where $\rho^{(S)}$ is the density matrix of the sender $S$ , and $\rho^{(TL,ER)}$ is the density matrix of the subsystem $TL\cup ER$ . Let both the sender and receiver have the same dimensionality. If $W(t)$ is the evolution operator describing dynamics of the given quantum system, then the state of the receiver at some time instant $t$ reads

[TABLE]

where we use the short notation $\{N\}$ for $(N^{1}N^{2}N^{3})$ and similar notation $\{I\}$ , $\{J\}$ . The operator $T$ in this formula maps the sender’s initial state into the receiver’s state. Here the capital latin multi-indexes with superscripts 1, 2 and 3 are associated, respectively, with the sender, the transmission line joined with the ancilla, and the receiver. These indexes include set of elements each associated with the particular spin of the appropriate subsystem and can take either 0 or 1. Thus, for the $N$ -qubit communication line with the two-qubit sender and receiver we write

[TABLE]

Now we discuss the structure of the evolution operator $W$ including two blocks as shown in Fig.2.

First of all, it includes the spin dynamics described by some Hamiltonian $H$ : $V(t)=e^{-iHt}$ . In addition, the evolution operator $W$ includes the unitary transformation $U^{(ER)}$ of the extended receiver. This transformation involves the set of additional parameters $\varphi_{i}$ (hereafter we call them $\varphi$ -parameters) needed to control the state of the receiver. Then

[TABLE]

Here $E^{(S,TL)}$ is the identity operator in the state space of the subsystem $S\cup TL$ .

Hereafter we deal with such Hamiltonian $H$ and unitary transformation $U^{(ER)}$ that conserve the excitation number during the spin evolution, i.e.,

[TABLE]

where $I_{z}$ and $I_{z}^{(ER)}$ are the $z$ -projections of the total spin momentum of, respectively, the whole communication line and the subsystem $ER$ . Then, according to Ref. FZ_2017 , the MQ-coherence matrices of the whole system do not interact during such evolution. In addition, to transfer the MQ-coherence matrices from the sender $S$ to the receiver $R$ without mixing, the initial density matrix $\rho^{(TL,ER)}(0)$ must include only the zero-order coherence matrix FZ_2017 . In particular, it can be the diagonal matrix $\rho^{(TL,ER)}(0)$ (for instance, the thermodynamic equilibrium state). Then eq.(3) becomes

[TABLE]

Commutativity (6) implies the following constraint for the evolution operator $W$ :

[TABLE]

where $|N^{i}|$ and $|I^{i}|$ mean the sum of entries of the multi-indexes $N^{i}$ and $I^{i}$ . We also set zero the energy of the ground state (the state without excitations), which means

[TABLE]

where $0^{i}$ , $i=1,2,3$ , are the multi-indexes with all zero entries.

III Manipulations with receiver’s density matrix elements

Before proceed to the numerical simulation of the particular operations with matrix elements in Sec.V, we describe general structures of these operations in terms of elements (7) of the $T$ -operator. This operator includes the $\varphi$ -parameters of the unitary transformation $U^{(ER)}$ , which allow to control the structure of the receiver’s density matrix elements. In particular, the following operations with elements inside of $n$ -order coherence matrix are of our interest, where we assume that the time instant is properly fixed (for instance, as in Sec.IV).

Structural restoring of initial sender’s state Z_2018 . In this case, by choosing the $\varphi$ -parameters, we have to satisfy the following constraints:

[TABLE]

As the result, the structurally restored state reads

[TABLE]

Hereafter, we refer to the parameters $\lambda_{I^{i}J^{i}}$ as scale factors.

Turning some matrix elements to zero. For some fixed multi-indexes $N^{3}=N^{3}_{0}$ and $M^{3}=M^{3}_{0}$ in (13) we can find the $\varphi$ -parameters such that, along with system (10)-(12), the additional constraint

[TABLE]

is satisfied. This constraint makes the appropriate matrix element vanish:

[TABLE]

Rearranging matrix elements. For the fixed pair of multi-indexes $N^{3}=N^{3}_{0}$ , $M^{3}=M^{3}_{0}$ and $I^{3}=I^{3}_{0}$ , $J^{3}=J^{3}_{0}$ in (2) we change formulas (10) as follows:

[TABLE]

and solve system (11), (12), (17) for the $\varphi$ -parameters. Then two equations in (13) which correspond to this fixed pair must be replaced with

[TABLE]

Thus, we exchange the elements $\rho^{(S;n)}_{I^{3}_{0};J^{3}_{0}}$ and $\rho^{(S;n)}_{N^{3}_{0};M^{3}_{0}}$ .

Linear combination of matrix elements with required coefficients. Since the elements of $n$ -order receiver’s coherence matrix $\rho^{(R;n)}$ are linear combinations of the elements of $\rho^{(S;n)}$ , this operation means providing the required values to the corresponding elements of the $T$ -operator (2) using the proper choice of the $\varphi$ -parameters. In particular, these combinations can yield the polynomial expansion of certain degree for some functions, see Sec.V.4.
Solving systems of linear algebraic equations

[TABLE]

where $A$ is a nondegenerate square matrix, $\vec{x}$ is a vector of unknowns and $\vec{b}$ is a constant vector. We consider the extended receiver (consisting of the two-qubit ancilla and receiver, see Fig.1) as a single block for applying the unitary transformation establishing the desired operation. Our protocol is based on mixed states and operates as follows. If there are $M$ elements $\rho^{(S;n)}_{N^{3}_{i};M^{3}_{i}}$ , $i=1,\dots,M$ in an $n$ -order coherence matrix which map into $\tilde{M}\leq M$ linearly independent combinations at the receiver side (elements of $\rho^{(R;n)}$ ), then we can solve a system of $\tilde{M}$ equations for $\tilde{M}$ unknowns as follows. Let $\tilde{M}=M$ for simplicity. We first identify elements $\rho^{(S;n)}_{N^{3}_{i};M^{3}_{i}}$ with the appropriate entries of $\vec{b}$ :

[TABLE]

From the other hand, we can write

[TABLE]

Accordingly, we can represent the elements of the receiver’s $n$ -order coherence matrix as

[TABLE]

where $\alpha_{ij}$ , in turn, are some linear combinations of $A_{ij}$ with coefficients depending on $\varphi$ -parameters and $t$ . Now we impose the constraints on the coefficients $\alpha_{ij}$ :

[TABLE]

which can be satisfied by the proper choice of the $\varphi$ -parameters for certain value of the constant $\alpha$ at some time instant. Then eq.(22) yields

[TABLE]

i.e., the scaled values of $x_{i}$ appear in the entries of the receiver’s $n$ -order coherence matrix. We note that solving system (23) is equivalent to inverting the matrix $A$ with the classical tool. However, been calculated, the $\varphi$ -parameters can be used for solving a family of equations with a given matrix $A$ and different vectors $\vec{b}$ .

Similarly, we can consider the case $\tilde{M}<M$ , see Sec.V.5. We notice, that the maximal absolute value of the arbitrary parameter $\alpha$ is restricted because of the dispersion during evolution.

IV Model

We consider the communication line of $N=42$ nodes used in Ref.Z_2018 . It consists of the two-qubit sender $S$ (the first and second nodes), the two-qubit receiver $R$ (the 41st and 42nd nodes) and the two-qubit ancilla (the 39th and 40th nodes). The ancilla and receiver form the four-node extended receiver $ER$ , which is connected to the sender through the transmission line $TL$ . The spin dynamics is governed by the XX-Hamiltonian with the dipole-dipole interaction

[TABLE]

where $D_{ij}=\frac{\gamma^{2}\hbar}{r_{ij}^{3}}$ is the coupling constant between the $i$ th and $j$ th nodes, $\gamma$ is the gyromagnetic ratio, $\hbar$ is the Planck constant, $r_{ij}$ is the distance between the $i$ th and $j$ th nodes, $I_{i\alpha}$ ( $\alpha=x,y,z$ ) is the projection operator of the $i$ th spin on the $\alpha$ axis and $I_{z}=\sum_{i}I_{iz}$ . We also consider the tensor-product initial state (1) with an arbitrary $\rho^{(S)}(0)$ , while the subsystem $TL\cup ER$ is in the state without excitations, i.e.,

[TABLE]

Such initial state restricts the evolution of the spin dynamics to the two-excitation subspace which significantly simplifies calculations. In virtue of initial condition (27), eq.(7) with setting

[TABLE]

gets the following form:

[TABLE]

where $0_{N-2}$ is the set of $N-2$ zeros. In this case $W_{J_{N-2}n_{1}n_{2};i_{1}i_{2}0_{N-2}}$ can be considered as the elements of the Kraus operators Kraus , because they satisfy the constraint

[TABLE]

which follows from the definition (5) of the operator $W$ .

In our protocol, we use the expansion of the density matrices in the sums of the multiple-quntum (MQ) coherence matrices FL which read in the two-qubit case as

[TABLE]

where $\rho^{(S;k)}$ and $\rho^{(R;k)}$ are the $k$ -order coherence matrices of, respectively, the sender and receiver.

Following Ref.Z_2018 , we introduce notation

[TABLE]

and consider the chain with the following coupling constants:

[TABLE]

Here $\delta_{1}$ , $\delta_{2}$ and $t$ are chosen to maximize the intensity of the second-order coherence. For $N=42$ , their optimal values read Z_2018 :

[TABLE]

V Operations with nondiagonal elements of density matrix

In this section, we simulate the operations with density matrix elements discussed in Sec.III. According to eqs. (2) and (29), the general structure of the upper nondiagonal elements of the receiver’s density matrix is following:

[TABLE]

where

[TABLE]

V.1 Partial structural restoring of initial sender’s state

In view of the structure of matrix elements (37)-(38), system (10)-(12) for the structural restoring of the upper nondiagonal elements Z_2018 now reads

[TABLE]

which must be solved for the $\varphi$ -parameters. This system is satisfied if

[TABLE]

Then the restored system reads

[TABLE]

The structural restoring was considered in Z_2018 , therefore we do not discuss it further.

V.2 Turning matrix elements to zero

It is interesting to note that not any single element of the restored nondiagonal part of $\rho^{(R)}$ can be turned to zero in our model, because vanishing of one of the elements in (44) - (47) implies vanishing of some others. We consider three variants of this operation.

The coefficients $a_{01;01}$ , $c_{01;01}$ and $f_{01;10}$ have the common factor $W^{+}_{010_{N-2};0_{N-2}01}$ or its conjugate $W_{0_{N-2}01;010_{N-2}}$ . Therefore, if we choose the $\varphi$ -parameters in Eqs.(44)-(47) such that

[TABLE]

then

[TABLE]

The explicit general form of the described transformation reads:

[TABLE]

Similarly, the coefficients $a_{10;10}$ , $c_{10;10}$ and $f_{01;10}$ have the common factor $W^{+}_{100_{N-2};0_{N-2}10}$ or its conjugate $W_{0_{N-2}10;100_{N-2}}$ . Therefore, if the $\varphi$ -parameters in (44)-(47) are such that

[TABLE]

then

[TABLE]

The explicit general form of the described transformation reads:

[TABLE]

The coefficients $c_{01;01}$ and $c_{10;10}$ are proportional to $d=W^{+}_{110_{N-2};0_{N-2}11}$ . Therefore, if the $\varphi$ -parameters in (44)-(47) are such that

[TABLE]

then

[TABLE]

The explicit general form of the described transformation reads:

[TABLE]

Thus, in the numerical simulations of transformations (54), (66) and (78), we have to solve eqs. (41) - (43) and one of equations, respectively, (48), (60) or (72) for the $\varphi$ -parameters. The solution of this system is not unique. Therefore, to reveal the optimal one, we find 1000 different solutions and choose one corresponding to the maximal sum of the absolute values of the non-zero scale factors $\lambda_{I^{i}J^{i}}$ in the nondiagonal elements of the receiver’s density matrix. Results of numerical calculation of the factors $a_{ij;nm}$ , $c_{ij;nm}$ , $f_{01;10}$ and $d$ in transformations (54), (66) and (78) are collected in Table 1. The list of the $\varphi$ -parameters of the unitary transformation corresponding to the transformation (54) is given in Appendix, the first line in Table 2.

V.3 Rearrangement of matrix elements

We consider the rearrangement of the elements $\rho^{(S;1)}_{00;01}$ and $\rho^{(S;1)}_{00;10}$ , which implies equations

[TABLE]

instead of (45). Therefore we must replace $a_{n_{1}n_{2};n_{2}n_{1}}$ with $a_{n_{1}n_{2};n_{1}n_{2}}$ in system (40), which implies

[TABLE]

instead of (41). Consequently, according to (37) and (39), the elements $\rho^{(S;1)}_{01;11}$ and $\rho^{(S;1)}_{10;11}$ interchange as well, i.e., we have

[TABLE]

instead of (46). The explicit general form of the described transformation reads:

[TABLE]

The interchange between the pairs $\rho^{(S;1)}_{00;01}$ , $\rho^{(S;1)}_{00;10}$ and $\rho^{(S;1)}_{01;11}$ , $\rho^{(S;1)}_{10;11}$ is impossible because of the structure of the appropriate elements of $\rho^{(R)}$ (see eqs.(37) and (37), there are no elements $\rho^{(S;1)}_{00;01}$ and $\rho^{(S;1)}_{00;10}$ in the linear combination (37)).

Thus, we have to solve system (42), (43) and (85) for the $\varphi$ -parameters. Again, the solution is not unique. Therefore, we find 1000 different solutions and choose one corresponding to the maximal sum of the absolute values of the scale factors $\lambda_{I^{i}J^{i}}$ in the nondiagonal elements of the density matrix. The result of numerical calculations yields the following values for the factors $a_{ij;nm}$ , $c_{ij;nm}$ , $f_{01;10}$ and $d$ in transformation (91):

[TABLE]

The list of the $\varphi$ -parameters of the unitary transformation corresponding to rearrangement (91) is given in Appendix, the second line in Table 2.

V.4 Linear combinations of matrix elements

The structure of matrix elements (37) - (38) shows that, among the elements of the first order coherence matrix, only $\rho^{(R;1)}_{00;01}$ and $\rho^{(R;1)}_{00;10}$ can be two indepdendent linear combinations of all the elements of $\rho^{(S;1)}$ with arbitrary coefficients. Also the coefficient in $\rho^{(R;2)}_{00;11}$ (second-order coherence matrix) is independent on the coefficients in $\rho^{(R;1)}_{00;01}$ and $\rho^{(R;1)}_{00;10}$ . But the element $\rho^{(R;0)}_{01;10}$ has a different feature. Only the coefficient $f_{11;11}$ is independent on the other coefficients in $\rho^{(R;1)}_{00;01}$ , $\rho^{(R;1)}_{00;10}$ and $\rho^{(R;2)}_{00;11}$ . Thus, we can arrange the following independent linear combinations:

[TABLE]

where $\alpha_{i}$ , $\beta_{i}$ and $\gamma$ are arbitrary scalar coefficients (but such that $\rho^{(R)}$ remains the nonnegatively definite matrix with unit trace).

In particular, if

[TABLE]

the above combinations represent the power series in $x$ . For instance, let $\varphi$ -parameters satisfy the system

[TABLE]

then we have the 4-degree polynomial expansions of $\alpha(e^{x}-1)$ and $\alpha(e^{-x}-1)$ in the entries $\rho^{(R;1)}_{00;01}$ and $\rho^{(R;1)}_{00;10}$ respectively, while the entry $\rho^{(R;1)}_{00;10}$ yields a fifth-power term in the expansion of $\alpha(e^{x}-1)$ . The general form of the described transformation for real $x$ and $\alpha$ reads:

[TABLE]

where $\rho^{(R;1)}_{ij;nm}(x)$ are some polynomial in $x$ which we do not write explicitly.

In the numerical simulations, we find 1000 different solutions of system (104) and choose one that maximizes $|\alpha|$ , which yields $|\alpha|_{max}=0.5399$ . The list of $\varphi$ -parameters of the unitary transformation corresponding to this manipulation is given in Appendix, the third line in Table 2.

V.5 Solving system of linear algebraic equations

We show that the 2 $\times$ 2 linear algebraic system (19) with real coefficients

[TABLE]

can be solved using the tool of unitary transformations on the extended receiver. Let

[TABLE]

We also assume that

[TABLE]

After evolution under the operator $W$ we have

[TABLE]

where $\alpha_{ij}$ depend on the $\varphi$ -parameters and on the parameters $a_{ij}$ of linear system (121). We choose the $\varphi$ -parameters to satisfy the conditions

[TABLE]

Then system (124) gets the following form

[TABLE]

Thus, the elements $\rho^{(R;1)}_{00;01}$ and $\rho^{(R;1)}_{00;10}$ of the first-order coherence matrix represent solution of original system (121) up to the factor $c$ . The general form of the described transformation reads:

[TABLE]

where $\rho^{(R;1)}_{ij;nm}(x)$ are some linear functions of $x$ which we do not write explicitly. As an example, we take $a_{11}=0.4$ , $a_{12}=0.3$ , $a_{21}=0.6$ , $a_{22}=0.2$ and solve system (125) for the $\varphi$ -parameters. Again, the solution is not unique. We find the solution corresponding to the maximal possible value of the real positive $c$ in (126), $c_{max}=0.1094$ . The list of $\varphi$ -parameters of the unitary transformation corresponding to this operation is given in Appendix, the 4th line in Table 2.

VI Conclusions

The unitary transformations on the extended receiver can be an effective tool not only for structural restoring of the transferred state (as shown in Ref.Z_2018 ) but also for realizing a set of other manipulations with elements of the receiver’s density matrix, such as turning some matrix elements to zero (Sec.V.2), exchanging the positions of some elements (Sec.V.3), constructing certain linear combinations of the elements (Sec.V.4) and solving the systems of linear algebraic equations (Sec.V.5). Being larger than the receiver, the extended receiver admits larger number of free parameters ( $\varphi$ -parameters in this paper) in the applied unitary transformation, and these parameters can be effectively involved in the above structural manipulations with the mixed receiver’s state. This is another benefit of the extended receiver which first was used to improve the characteristics of the communication line in the protocol of remote state creation BZ_2016 . It is important that our protocols do not use a classical channel.

We shall notice that certain restrictions in manipulation with matrix elements in Sec.V in comparison with general formulas proposed in Sec.III (for instance, only triplets of matrix elements can be turned to zero rather than any particular element, restricted possibilities for rearranging the matrix elements) are associated with the particularly chosen initial state of the transmission line and receiver (27), which is the state without excitations. Using a different initial state (for instance, the thermal equilibrium one) we might extend possibilities of such manipulations.

The considered operations with matrix elements via the unitary transformations on the extended receiver represent simplest examples of quantum realization of algebraic operations. The further development of this approach with the purpose to simulate quantum counterparts for more complicated algebraic algorithms as well as methods of realizations of appropriate unitary transformations is of interest.

This work is performed in accordance with the state task of the Ministry of Education and Science of the Russian Federation, state registration No. 0089-2019-0002, and by the Russian Foundation for Basic Research (Grants No.20-03-00147).

VII Appendix: Unitary transformation as a tool for manipulations with matrix elements

For the matrix representation of the unitary transformation $U^{(ER)}$ and density matrix $\rho^{(ER)}$ we use the basis of eigenstates of $I^{(ER)}_{z}$ ,

[TABLE]

The unitary operator $U^{(ER)}$ can be written in the form

[TABLE]

where $\varphi^{(k)}_{n,m}$ are scalar parameters, the product is ordered in such a way that $n$ and $m$ increase from the right to the left and the Hermitian matrices $\gamma^{(1;ij)}$ , $\gamma^{(2;ij)}$ , $j>i$ have the following nonzero elements Z_2018 :

[TABLE]

In Table 2, we represent the families of $\varphi$ -parameters for the unitary transformations corresponding to the manipulations with matrix elements discussed in Secs.V.2-V.5. The first, second and third lines in this table correspond, respectively, to transformation (49), to rearrangements (84),(86) and to linear combination (100) - (104). The parameters of the unitary transformation solving the system of two linear algebraic equations (121) are collected in the fourth line of this table.

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