# Virtual Correlations in Single Qutrit

Authors: Alexey A. Strakhov; Vladimir I. Man’ko
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

We construct the positive invertible map of the mixed states of a single qutrit onto the antisymmetrized bipartite qutrit states (quasifermions). It is shown that using this one-to-one correspondence between qutrit states and states of two three-dimensional quasifermions one may attribute hidden entanglement to a single mixed state of qutrit.


---

## Body

## 1. Introduction

Quantum entanglement [1] is one of the most valuable nonclassical resources in quantum information science. By virtue of entanglement the protocols of superdense coding [2] and quantum teleportation [3] were discovered. Entangled states are essential in quantum computing [4, 5] and are used in various cryptography protocols [6, 7]. Entangled states are also used in more applied areas of physics such as quantum metrology [8, 9].Naturally it is hard to prepare and maintain maximally entangled states. In order to control the amount of entanglement in quantum systems several measures were proposed. In case of bipartite pure states an entropy of entanglement [10] was proved to be an asymptotic measurement of distillating maximally entangled states by means of local operations and classical communication (LOCC) from the supply of initial pure states and vice versa. The case of mixed states is more complicated. In particular for mixed states the following measures were introduced: the asymptotic entanglement of formation and distillation [11] and the relative entropy of entanglement [12]. However, all of them include variational step and thus are generally hardly calculable (notable exceptions are mixed qubit states fully characterized byconcurrence [13]). From the Peres-Horodecki criterion [14, 15] the notions ofnegativity andlogarithmical negativity [16–18] emerged. They are relatively easy calculable and the latter provides an upper bound of distillable entanglement. Negativity will be used further in the paper.Recently several attempts to study quantum correlations of noncomposite qudit systems were conducted [19–23]. The physical realization of such systems is convenient with nonlinear quantum circuits on Josephson junctions [24, 25]. This paper is an attempt to study correlations between different degrees of freedom in qutrit via the invertible map to a bipartite system of antisymmetrized states of qutrits (as the dimension of them is odd we shall call them quasifermions). The source of correlations in such case would be antisymmetrization, that is, restriction to the subspace of the whole tensor product space of two qutrits. Actually using the same method one can construct an isomorphism between six-dimensional noncomposite qudit and two indistinguishable four-dimensional fermions. Correlations between particles and modes in the systems of indistinguishable particles were recently studied [26–30]. For mixed states of indistinguishable four-dimensional fermions obtained by the isomorphism discussed in the present paper both the dualization concurrence and Slater number will show that the states are separable in the sense of antisymmetrized fermionic subspace. However, we would like to emphasize that due to the virtuality of the division of the noncomposite system into bipartite quasifermion system it is not necessary to treat these quasifermions as indistinguishable particles. One might view correlations resulting from antisymmetrization as internal correlations between different degrees of freedom of the initial qudit. Thus one might treat the initial six-dimensional mixed state as a fully separable state of indistinguishable Dim=4 fermions or as an entangled state of distinguishable Dim=4 particles. The present paper will be about analyzing negativity of qutrit viewed as an antisymmetrized biqutrit state. An open question in such approach is operating with the corresponding virtual subsystems.


## 2. Partial Trace in Matrix Representation

The state of a subsystem of a quantum system is defined by an operation called partial trace, taking mean value over states of another subsystem:(1)ρI=TrJ∖IρJ⟺i1ρIi2=∑j∈J∖Ii1jρJji2,where i1 and |i2〉 are basis vectors of the subsystem I and |j〉 is basis vectors of the subsystem J∖I. Despite the explicit choice of basis vectors of space J, partial trace is invariant with respect to unitary transformations of I.Let us consider a spaceJ=H3⊗H2 and choose a lexicographically ordered basis |e1〉=|a1〉⊗|b1〉, |e2〉=|a1〉⊗|b2〉,…,|e6〉=|a3〉⊗|b2〉. Then an arbitrary state ρ will be presented as matrix:(2)ρ=ρ11ρ12ρ13ρ14ρ15ρ16ρ21ρ22ρ23ρ24ρ25ρ26ρ31ρ32ρ33ρ34ρ35ρ36ρ41ρ42ρ43ρ44ρ45ρ46ρ51ρ52ρ53ρ54ρ55ρ56ρ61ρ62ρ63ρ64ρ65ρ66.Formula (1) then might be represented by square block matrices whose sizes are determined by the numbers of dimensions of spaces H3 and H2:(3)
ρ
=
ρ
11
ρ
12
ρ
13
ρ
14
ρ
15
ρ
16
ρ
21
ρ
22
ρ
23
ρ
24
ρ
25
ρ
26
ρ
31
ρ
32
ρ
33
ρ
34
ρ
35
ρ
36
ρ
41
ρ
42
ρ
43
ρ
44
ρ
45
ρ
46
ρ
51
ρ
52
ρ
53
ρ
54
ρ
55
ρ
56
ρ
61
ρ
62
ρ
63
ρ
64
ρ
65
ρ
66
=
R
11
R
12
R
13
R
21
R
22
R
23
R
31
R
32
R
33
;
ρ
1
=
Tr
⁡
R
11
Tr
⁡
R
12
Tr
⁡
R
13
Tr
⁡
R
21
Tr
⁡
R
22
Tr
⁡
R
23
Tr
⁡
R
31
Tr
⁡
R
32
Tr
⁡
R
33
,
ρ
2
=
R
11
+
R
22
+
R
33
,where ρ1 is a state of the first subsystem H3 and ρ2 is a state of the second subsystem H2. Partial traces in matrix form will be constantly used further in this paper.


## 3. Qutrit asH=H3∧H3 Quasifermionic System

Let us pick a basis{|e1〉,|e2〉,|e3〉} in an arbitrary three-dimensional state space Q. The general mixed state of qutrit is described by a density operator ρI∈S(Q): ρI=∑i,j=13‍(ρI)ij|ei〉〈ej|, in matrix form: (4)ρI=ρ11ρ12ρ13ρ21ρ22ρ23ρ31ρ32ρ33.Now we will consider a system of two three-dimensional quasifermions, antisymmetrized qutrits. Mathematically it is an exterior product, that is, the quotient of the tensor product by the subspace generated byx⊗x elements, of two three-dimensional spaces: H=H3∧H3. Let us pick orthonormal basis vectors in H3 as |1〉,|2〉,|3〉. Then in the normalized basis {|g1〉,|g2〉,|g3〉} of the exterior product an arbitrary pure state |Ψ〉∈H looks as follows:(5)Ψ=a1g1+a2g2+a3g3,g1=121∧2=121⊗2-2⊗1,g2=121∧3=121⊗3-3⊗1,g3=122∧3=122⊗3-3⊗2.We will further omit the⊗ symbol of tensor product. Thus we see that the space H is isomorphic to qutrit space Q. The general form of density operator ρO∈S(H) on space H is ρO=∑i,j=13‍(ρO)ij|gi〉〈gj|, so we can introduce an isomorphism between sets of mixed states S(Q) and S(H) by equalizing matrix elements (ρI)ij=(ρO)ij=ρij for all i,j=1,2,3.As we already mentioned,ρO is a state in antisymmetrized subspace of the tensor product space H3⊗H3. Let us observe this state in context of the whole space H3⊗H3. It is easy to verify that its matrix representation is as follows: (6)ρO′=120000000000ρ11ρ12-ρ110ρ13-ρ12-ρ1300ρ21ρ22-ρ210ρ23-ρ22-ρ2300-ρ11-ρ12ρ110-ρ13ρ12ρ1300000000000ρ31ρ32-ρ310ρ33-ρ32-ρ3300-ρ21-ρ22ρ210-ρ23ρ22ρ2300-ρ31-ρ32ρ310-ρ33ρ32ρ330000000000.Now using the partial trace operation, introduced in the previous section, we are able to obtain single three-dimensional quasifermionic states: (7)ρ1=ρ2=12ρ11+ρ22ρ23-ρ13ρ32ρ11+ρ33ρ12-ρ31ρ21ρ22+ρ33.The map M:ρI↦ρ1,2 is a completely positive trace-preserving (CPTP) map. It is obvious because the map ρI↦ρO is an isomorphism and expanding ρO to ρO′ is simply adding extra dimensions to Hilbert space which do not participate in constructing of the state ρO′; that is, the spectrum of ρO′ is equal to the spectrum of ρO and hence of ρI. The partial trace is obviously a CPTP map, so the whole map M is CPTP. Despite such trivial construction of reduced states ρ1,2 in general their spectrum is different from ρI because their construction was out of quasifermionic antisymmetric subspace H⊂H3⊗H3.


## 4. Correlations between Different Degrees of Freedom of Qutrit

The well-known Peres-Horodecki criterion or PPT criterion is a necessary condition for a mixed state to be separable. It states that if the density operatorρ of a bipartite system is separable then partial transpose of one of the subsystems is positive. The partial transpose of the second subsystem of ρO′ in matrix form is the following: (8)ρO′PT=120000-ρ11-ρ210-ρ12-ρ220ρ11ρ210000-ρ13-ρ230ρ12ρ220ρ13ρ23000000ρ110-ρ31ρ120-ρ32-ρ110ρ31000ρ130-ρ33-ρ120ρ32-ρ130ρ33000000ρ21ρ310ρ22ρ320-ρ21-ρ310000ρ23ρ330-ρ22-ρ320-ρ23-ρ330000.It can be easily verified that ρO′PT has negative values for various ρI states; that is, it is an entangled state. Actually, with the aid of entanglement monotone called negativity it is possible to show that this state is always entangled.The entanglement monotoneE is nonnegative linear functional over state ρ with two main properties:(1)
The monotone is a convex functional:E(∑ipiρi)≤∑ipiE(ρi).(2)
The functional is an entanglement monotone, that is, it does not increase on average under local quantum operations and classical communication (LOCC).Most of the entanglement monotones, like distillability monotones and monotones of formation, are immensely difficult to compute. The more easily countable measure of entanglement is called negativity monotone. It is defined as follows:(9)Eρ=ρPT1-12=∑iλi-λi2,where X1=Tr⁡(X†X) is a trace norm. In case of Hermitian operators it is just a sum of modules of negative eigenvalues.Now it is possible to show that in case of initial pure statesρI=|ψ〉〈ψ| their negativity is E=1/2 and in case of maximally chaotic state ρI=1/30001/30001/3E=1/3. More formally, one can use another more convenient entanglement monotone for pure states |ψ〉∈HA⊗HB: the entropy of the subsystem S=SA=SB=-TrρAln⁡ρA. In our case if the initial state ρI=|ψ〉〈ψ| and |ψ〉=a1|e1〉+a2|e2〉+a3|e3〉 then the density matrices of quasifermion subspaces are as follows: (10)ρ1=ρ2=12a12+a22a2a3∗-a1a3∗a2∗a3a12+a32a1a2∗-a1∗a3a1∗a2a22+a32.A simple technical calculation shows that nonzero eigenvalues of this matrix are λ1,2=1/2. This means that the entanglement measure of ρO′ is S=ln⁡2 for all initial pure states ρI. Thus the negativities E of all pure states ρO′ are also equal to each other. En elementary calculation for an arbitrary pure state shows that E=1/2.The previous considerations about pure states negativities prove that for two different mixed states with equal coefficientsρI1=p1|ψ1〉〈ψ1|+p2|ψ2〉〈ψ2|+p3|ψ3〉〈ψ3| and ρI2=p1|ϕ1〉〈ϕ1|+p2|ϕ2〉〈ϕ2|+p3|ϕ3〉〈ϕ3| negativities are equal to each other. Thus to analyze all mixed states it is enough to consider a general density matrix: (11)ρI=p1000p2000p3.The partial transpose of ρO′ then will be the following: (12)ρO′PT=120000-p1000-p20p1000000000p2000000000p100000-p10000000-p300000p3000000000p2000000000p30-p2000-p30000.Its negative eigenvalues are the negative solutions of the third-order equation: (13)λ3-p12+p22+p32λ+2p1p2p3=0.Careful analysis shows that for every distribution pi there is only one negative solution. The infimum of negativity is situated at maximally chaotic state p1=p2=p3=1/3 and is equal to E=1/3.


## 5. Conclusions

In this paper the simplest nontrivial case of quantum correlations inside qutrit was analyzed. It was demonstrated that for arbitrary mixed qutrit state the negativity is nonzero for corresponding virtual bipartite system of antisymmetrized qutrits. Thus the correlations between the degrees of freedom of qutrit emerge. These considerations could be generalized for qudits with more degrees of freedom in a straightforward manner.



---

*Source: 7213197-2016-10-24.xml*