Bloch sphere model for two-qubit pure states
Chu-Ryang Wie1
State University of New York at Buffalo, Department of Electrical Engineering, 230B Davis Hall, Buffalo, NY 14260 (Dated: March 28, 2014)
The two-qubit pure state is explicitly parameterized by three unit 2-spheres and a phase factor. For separable states, two of the three unit spheres are the Bloch spheres of each qubit with
coordinates (A,A) and (B,B). The third sphere parameterizes the degree and phase of concurrence, an entanglement measure. This sphere may be considered a ‘variable’ complex imaginary unit t where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states. We argue that the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuth
angles, A and B, of the two quasi-Bloch spheres. On these three unit spheres and a phase factor (B), the two-qubit pure states and unitary gates can be geometrically represented. Accomplished by means of Hopf fibration, the complex amplitudes (, , , ) of a two-qubit pure state and the Bloch sphere parameters are related by a single quaternionic relation:
θA cos j 2 BBkk cos sin eBB j e j θ A tA 22 sin e 2
described by a unit 7-sphere S7 for pure states and by a four-dimensional special unitary group SU(4) for a mixed I. INTRODUCTION state density matrix and two-qubit operators. There have been various attempts to parameterize the two qubit state The single qubit Bloch sphere provides a useful means of space, including an explicit parameterization of SU(4) [4], visualizing the state and a neat description of many single and construction of special unitary group on two qubit qubit operations, enabling physical intuitions and serving Hilbert space using geometric algebra of a six- as an excellent testbed for ideas about quantum dimensional real Euclidean vector space [5]. Havel and computation and information [1]. However there is no Doran presented, as a special case to their discussion of simple generalization of Bloch sphere for multiple qubits, SU(4) mixed state model, an explicit parameterization of not even for two qubits. Much that is weird and a two-qubit pure state with seven angle parameters which wonderful about quantum mechanics can be appreciated was written as a convex combination of tensor products of by considering the properties of the quantum states of two one-qubit states [5]. Mosseri and Dandoloff applied qubits [2]. Any multi-qubit quantum logic gates can be Hopf fibration and attempted to develop a Bloch sphere built out of a two-qubit operation and a number of single- model for two-qubit pure state [6,7]. They proposed a 3- bit operations [1-3]. All of the above points give a strong dimensional ball with equal-concurrence concentric reason to develop a practical Bloch sphere-like tool for spherical shells as a possible model, which obviously has the two-qubit states and gates. In spite of a considerable far too few degrees of freedom to specify the two-qubit amount of efforts spent to develop a feasible two-qubit states. However, their approach with Hof fibration has a Bloch sphere model [4-7], there still is no Bloch sphere close connection to our work in this paper. model that is simple enough to serve as useful a tool as the single qubit Bloch sphere does for state representation In this paper, we limit to the two-qubit pure states and and manipulation. attempt to find a Bloch sphere model that is simple and practical. We parameterize the S7 two-qubit pure state For complete description, two qubit states require seven space by 7 angle parameters which are grouped into the parameters for pure states and fifteen parameters for two Bloch coordinates ( , ) and ( , ), the general mixed states. The two qubit state space is usually A A B B entanglement and its relative phase (, ), and a phase
1 Electronic address: [email protected] 2
angle (B). These seven angle parameters are related to Throughout this paper, k will be assumed to be the the two-qubit quantum state amplitudes, , , and via imaginary unit of ordinary complex numbers. A a single equation. The rest of this article is organized as quaternion forms non-commutative algebra and may be follows: In section II, we briefly review the Hopf map written in terms of two complex numbers as in Eqs.(2) with the non-commutative quaternion algebra. The Hopf and (3), or in terms of four real numbers, a, b, c and d as fibration maps the S7 two-qubit pure state space to an S4 base space. This is done by mapping to each point of the q a bi cj dkabcd, , , , (5) S4 base space the entire S3 fiber space. This Hopf map enables us to model a quantum state in the S7 space by a A pure imaginary unit quaternion squares to -1 like the product of a function in the S4 base space and a function three imaginary units in Eq.(4) and thus, a pure unit in the S3 fiber space. In section III, we define the three quaternion t provides the same relation between an unit 2-spheres and a phase factor to serve as the Bloch exponential function and trigonometric (sine and cosine) sphere model for two-qubit pure states. In section IV, we functions as the complex imaginary unit does, and it can present a streamlined procedure for, given the four be parameterized in terms of two real numbers. That is, complex state amplitudes (, , and ), how to find the seven angle parameters of the Bloch spheres, and vice t2 1; et cos tsin ; . (6) versa. In this section we also show some concrete t i sin cos j sin sin k cos ; , , examples of Bloch coordinates of the maximally entangled Bell states and several separable states, and The normalization condition examples of the two-qubit quantum gates, CNOT, CZ and
SWAP gates. In section V, we discuss the entanglement 2 2 2 2 2 2 2 2 q0AABB q 1 q 0 q 1 1 (7) sphere parameters (, ), phase angle parameter (B), and the mixed-state single-qubit Bloch ball after partial trace and its relation to the quasi-Bloch sphere. In section VI, identifies the state space of a two-qubit pure state as a unit we give a summary conclusion. 7-sphere, S7, which is embedded in 8 .
The Hopf map from the two-qubit state space S7 to the base S4 is a composition of two maps: first from S7 to 4 II. THE HOPF FIBRATION OF TWO-QUBIT (+), then to S4 by an inverse stereographic projection [7, STATES AND PARAMETERIZATION 8]. An explicit treatment of conversion from the complex For two-qubit pure states, amplitudes, and , to the quaternion ‘amplitudes’ q and q is useful in relating the target space of the Hopf Ψ 00 01 10 11 (1) 0 1 AB map to each qubit space (and thus identifying the Bloch parameters of each space). We do this as follows: where, A. In the basis of qubit-A ,,, and 2 2 2 2 1, Ψ 0 0 1 1 0 1 AB ABBABB our Bloch sphere representation starts by introducing quaternion ‘amplitudes’ by defining them as follows: j q Ψ 0 jj 1 0 A (8) A AA j q1A q01AA j, q j (2) or where, after the Hopf fibration, qubit-A is identified with the base space S4 and the qubit-B with the fiber space S3. (3) q01BB j, q j The above conversion process keeps the qubit-A basis vectors intact while identifying the qubit-B basis vectors where, the quaternion imaginary units i, j and k satisfy the with the quaternion units: 0 1, 1 j . This following multiplication rules and identities: BB correspondence is applied again when recovering the i2 j 2 k 2 ijk 1, i jk kj , qubit-B state at the end of calculation. Let us define a (4) density matrix-like quantity as follows: j ki ik, k ij ji .
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q j 0 A Ψ Ψ jj , Ψ Ψ qq , BBB AAAAA01 j q1A (9) 0 j (12) j B , jj , j 0 j 22 B where the over-bar indicates the quaternion or complex 22 conjugation which reverses the sign of imaginary units: i i, j j and kk . As will be shown below, Where B is the reduced density matrix of qubit-B: all local parameters of the qubit-B are cancelled out in . It is interesting to note that BA tr ΨΨAB AB this dyad ΨΨ; and the only parameters remaining AA whether we choose qubit-A basis or qubit-B basis, the in A are the Bloch coordinates of the base qubit (qubit- resulting quasi density matrix consists of the reduced A) and two nonlocal parameters (entanglement and density matrix of the base qubit plus an identical term phase). These four parameters (of the Hopf base) are representing the concurrence. Also the quasi-density determined from the products of the complex amplitudes matrix possesses the pure-state density matrix , , and . From Eq.(9) we obtain 2 properties: , tr( )=1, and det( )=0. A two- qubit SWAP operation will send the quasi state Ψ to 0 j A , where AA Ψ , or the quasi-density matrix to , and vice j 0 B B 22 versa. The trajectory of a SWAP gate on the Bloch (10) spheres will be shown later. A 22 C. Hopf fibration
Here, is the concurrence, an entanglement 2 Before we parameterize the bipartite pure state space S7, measure [9], and A is the reduced density matrix for we briefly review the Hopf fibration. The Hopf fibration qubit-A given by . It is worth consists of a composition of two maps: the Hopf map h1 A tr B ΨΨ AB AB which maps S7 to 4 , followed by an inverse noting that the off-diagonal element of represents the A stereographic projection h which maps to coherence of qubit-A after qubit-B is traced out, and 2 S4\(1,0,0,0,0). Eq.(10) shows that the other off-diagonal element of represents the entanglement of the composite state. q0 1 h1: qq 0 1 QQQiQjQk 1 2 3 4 q B. In the basis of qubit-B 1 4 (13) S 7 We write the bipartite state in the qubit-B basis by 22 q, q , Q ; q q 1; Q ; i 1,2,3,4 identifying the qubit-A basis vectors with the quaternion 0 1 0 1 i units 1 and j as follows:
h2: Q 1 , Q 2 , Q 3 , Q 4 x 0 , x 1 , x 2 , x 3 , x 4 Ψ 0 1 0 0 1 1 44 AB AABAAB S (14) 4 xx2 1, , j q ii Ψ jj 0 1 0B (11) i0 B BB j q1B The stereographic projection gives where, after the Hopf fibration, qubit-B will be in the base 1 x1 x 2x3 x 4 (15) 4 3 q0 q 1 Q Q 1, Q 2 , Q 3 , Q 4 , , , space S and qubit-A in the fiber space S . 1111xxxx0000 which maps S 4 \ 1,0,0,0,0 to , or vice versa. This is The quasi density matrix is illustrated in Fig.1.
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1 h3. Its fiber S is a unit circle which can be represented by a unit complex number: ek.
D. Parameterization of the S4 Hopf base and the S3 Hopf fiber
The above discussion of Hopf fibration of the S7 two- qubit pure state space suggests a parameterization in terms of two groups of parameters: the S4 base space parameters and the S3 fiber space parameters. We first parameterize the S4 Hopf base as follows:
4 4 Figure 1 Stereographic projection from S to where S 4 the coordinates of S4 are related to the coordinates of S 7 θ through Eq.(15). 3 cos S q0 2 q (19) Therefore, a pair of quaternions, q0 and q1, are mapped to f 4 q1 θ t the coordinate (x0, x1, x2, x3, x4) on the 4-sphere S which sin e is embedded in 5 with the North pole excluded. This 2 S3 t i sin cos j sin sin k cos 74 , (20) Hopf fibration reads h: S S where h h21 h . An tx i t y j t z k ; , , , inspection shows that all the pairs of quaternions (q0, q1), which differ only by a right multiplication of an arbitrary 3 where qf is the unit quaternion, representing the S Hopf unit quaternion qf, will map to the same point Q in . fiber. By substituting Eqs.(19) and (20) into Eq.(15), we qq ()q10 qff q q Q 10 where find 22 q1 qq1 f x0 cos , x 1 sin cos , x 2 sin sin sin cos 2 qf q f qff q q f 1 (16) ccos btxy , x3 sin sin sin sin c sin bt , Therefore, x sin sin cos bcos bt , 4 z (21) qq0 f b sin sin , c sin sin sin bsin hQ: for any unit quaternion qf (17) 1 qq1 f Here, the S4 Hopf base is parameterized in terms of two This unit quaternion q is the S3 fiber in the Hopf map f angles and and a pure-imaginary unit quaternion t S3 h: S74 S . It is worth repeating here that the Hopf map which is parameterized by another two angles and . 3 7 4 5 h maps an entire subspace S of S to a single point of the The Cartesian coordinate of S embedded in is (x0, x1, 4 base space, the 4-sphere S . The unit quaternion qf is x2, x3, x4), as shown in Fig.1 and given in Eq.(21) in terms S1 32 of these four angle parameters. We may regard the pure- itself fibrated by another Hopf map h3: SS [8], imaginary unit quaternion t as a ‘variable’ complex which maps an entire circle S1, a subspace of S3, to a 2 imaginary unit for the complex plane onto which a 2- single point on S . sphere S2 with (, ) spherical coordinate is mapped by a stereographic projection. Hence, the Hopf base S4 SS32 consists of a unit 2-sphere S2 (parameters: , ) sitting 2 2 h3 : qf q f kq f (18 ) atop another S (parameters: , ). The latter S represents the ‘variable’ imaginary unit t. Also, a pure- where k is an imaginary unit. The image of the map h3, imaginary quaternion can be viewed as a vector in a 3- dimensional space with the orthogonal axes defined by qff kq , is another pure-imaginary unit quaternion which imaginary units i, j, k. The angles, and , define the is obtained by rotating k around an axis of rotation given - Bloch coordinates of the qubit in the base space. The by the imaginary part of qf by an angle of rotation, 2cos 1 pure unit quaternion t embodies the nonlocal properties (Re(qf)), given by the real part of qf [8]. A pure unit 2 such as concurrence and its phase. This will be discussed quaternion can be parameterized by a unit 2-sphere S and 4 further in the next section. The interpretation of the S will form the base (or the image space) of the Hopf map
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2 4 5 base as being a 2-sphere S sitting on top of another 2- The Hopf base (S ) coordinates (x1, x2, x3, x4, x0) in R are 2 2 sphere S is also suggested by the following relation: separated into two S Cartesian coordinates, (x1, b, x0) and (tx, ty, tz), according to: 2 2 2 2 2 2 2 2 2 2 2 xxxxxxxbttt22222222222 1 x1 + x2 + x3 + x4 + x0 = x1 + b (tx + ty + tz ) + x0 = 1 0 1 2 3 4 0 1 x y z For the S4 Hopf base, the first S2 is qubit-A Bloch sphere 42 SSS 2 2 with x1-b-x0 axis, and the second S parameterizes the (22) global parameter t with the degree of entanglement and its 4 This equation suggests that the S Hopf base can be phase on the t -t -t (or equivalently, i-j-k) axis. The 2 x y z parameterized in terms of two independent S spheres. qubit-B Bloch sphere is identical to a single qubit Bloch 2 The first S is in the x1-b-x0 coordinate frame (the base sphere coordinate as it arises from the Hopf fiber, a unit 2 3 qubit Bloch sphere), and the second S is in the tx-ty-tz (or quaternion qB represented by 3-sphere S which is further i-j-k) coordinate frame (of the nonlocal parameter t). 2 mapped by h3 to the S base (Bloch sphere, coordinates B 1 and B) and the S fiber (a phase factor, exp(kB)). See The unit quaternion qf representing the Hopf fiber enters Eqs.(23) and (24). Therefore, the S7 pure two-qubit states Eq.(19) through a right multiplication to the base are represented by three unit 2-spheres (S2) and one unit quaternions, and because the quaternion multiplication is circle S1. non-commutative, this relative position is important. The unit quaternion qf is parameterized as 2 Qubit-A quasi-Bloch Sphere S (A, A): S 2 1 3 S x sin cos , b sin sin , x cos . S 10AAAAA q(cosff sin ekkff j ) e ; , , (23) For general, entangled states, the qubit-A quasi-Bloch f22 f f f coordinate is (A, A). For separable states, the qubit-A According to the Hopf map defined in Eq.(18), a given 2 Bloch coordinate is also (A, A). [c=bsin; x2=c cos; point (f, f) on the base space S has a preimage which is 3 x3=c sin; x4=b cos.] a great circle on the 3-sphere S , and this great circle is parameterized by a unit complex number exp(kf). This 2 Entanglement Sphere S (, ): tx=sincos, great circle is also called the Hopf circle and is the 1- t =sinsin, t =cos; t=t i+t j+t k. sphere S1 of Eq.(23). In order to correctly parameterize y z x y z 3 1 For separable states (zero concurrence, c=bsin=0), the great circles of S , the Hopf fiber S , exp(kf), must either the qubit-A Bloch coordinate is at the great circle in appear at the right end of the expression in Eq(23), and the x x -plane (i.e., b=0), or the entanglement sphere once again, this relative position is important because 0 1 coordinate is at its North pole (=0, t=k) or at South pole quaternion multiplication is not commutative. The Hopf
k (=, t=-k). The phase factor influences . fiber e f not only provides the global phase of the 2 bipartite state but it affects both the relative phase of Qubit-B quasi-Bloch Sphere S (B, B): qubit-B state and the azimuth angle of t. This will be xBBBBBBBB sin cos , y sin sin , z cos shown explicitly below. For general states, the qubit-B quasi-Bloch coordinate is
k B In summary, the angle parameters are: Hopf base S4 (, , (B, B). The phase factor e influences B. But (B, 3 B) is used in plotting the Bloch sphere coordinate in this , ) and Hopf fiber S (f, f, f) which is further mapped 2 1 paper. to S base (f, f) and S fiber (f).
k III. TWO-QUBIT BLOCH SPHERE MODEL The Phase S1: e B In the remainder of this paper we discuss parameterization 1 k B S ( e ) is the Hopf fiber of qB and it must appear at the with respect to the qubit-A basis. The Hopf map, right-most end of the expression for quaternion qB, as S3 h: S74 S and the parameterization Eq.(19), and the shown in Eq.(25) below. This phase factor may be S1 considered the global phase only after properly adjusting 32 Hopf map h3: SS and the parameterization Eq.(23), (i.e., shifting by 2B) the azimuths and B. are combined as q cosBB sin ekkBB j e (25) θ B cos A 22 j 2 BBkkBB ΨA cos sin e j e j θ k A tA 22 2 1 B sin e In summary, the three S spheres and a S unit circle e 2 provide a complete description of the S7 two-qubit pure (24) state. This is shown in Fig.2.
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2 1 k Figure 2: Three S spheres and one S phase factor ( e B ). The single qubit states |A> and |B> do not represent a physical state but are only shown here as a placeholder for the coordinate. The entanglement sphere parameterizes the pure unit quaternion t, where affects the degree of concurrence and is its phase angle (see Eq.(28)). The Bloch coordinates and the state amplitudes are related by Eq.(24). Phase factor ek B affects both global phase and relative k phase. For nonzero B, the two phase angles need to be translated -2B and BB-2B and then, the factor e B can be regarded as the global phase factor. The concurrence is given by c = sinAsinAsin = b sin.
1x x x k 1 0 1 4 (29) There are four important points to be mentioned here: 1) A 2 x x k1 x The azimuth of the entanglement sphere is the phase 1 4 0 angle of concurrence. 2) For B≠0, the phase factor influences the azimuth angles of both the qubit-B sphere Here, A is the reduced density matrix of qubit-A, and |c| and the entanglement sphere, and therefore care must be is the concurrence [9]. Consider, for example, these taken before it can be discarded as the global phase factor. maximally entangled states (MES) with a varying relative k k 3) The qubit-A hemispheric points (b) and the phase: 00 e 11 and 01 e 10 for 0,2 . All entanglement sphere antipodal points (t) are identified 2 2 according to (b, t) = (-b, -t). This is a two-fold ambiguity. of their Bloch coordinates are the same as when =0 And 4) this two-qubit Bloch sphere model, Eq.(24), can (which are two of the Bell states, see section IV) except not be used to find the Bloch coordinates for the separable for the angle : = /2 + for the former and =3/2 + states of the form Ψ 1 ψ . Instead, this state ABA B for the latter. Hence, parameterizes the relative phase of may be represented as two single-qubit states. Also, if the the entangled states. We will discuss this further in section V. qubit-B quasi-Bloch coordinate is B=, then the angles
B and B are interchangeable. These points are discussed in more detail below. (2) The phase factor ek B is the global phase only if it is located at the left-most end of expression. This can be (1) Azimuth of entanglement sphere is the phase angle seen by rewriting Eq.(24) as below: of concurrence. j ΨA Writing the parameter t as j k t cos k sin e i (26) (30) θ it follows that cos A k B 2 BBkBB2 1 k 0 j ecos sin e j 2 (27) θ ce A kkBABt 22 AA sin e e e 2 j 0 2 where, by comparing to Eq.(10), we find 1 k It is clear that the Hopf circle S ( e B ) is not simply the global phase, but it is also part of the relative phase of k 2 qubit-B. It also similarly affects the azimuth of the ce2 , c concurrence , (28) entanglement sphere. A unit quaternion is a spinor and a
kkBB c sinAA sin sin rotation operator [8, 10]. For example, e te is another pure unit quaternion which is t rotated clockwise and around the k-axis by angle 2B:
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ekkBB te (31) A. Parameter determination procedure i sin cos 2 BB j sin sin 2 k cos
Hence, for B≠0, the azimuth angles of the entanglement It is straightforward to obtain the amplitudes , and from the seven angle parameters (we used imaginary unit sphere () and qubit-B sphere (B) need be rotated k B i in the formulas of Eq.(34)): clockwise by 2B before ignoring the phase factor e as the global phase. ABAB iiBBB () -2B; and B B-2B (32) cos cos e;; cos sin e Note that this same phase factor must be factored out of 2 2 2 2 the state amplitudes for Eqs.(24) and (30) to B (34) cosAA i sin cos cos remain consistent after the global phase factor is 2 i sinA e B ; discarded. 2 B i B i sinA sin sin e 2 (3) The b hemispheric points of qubit-A sphere and the t antipodal points of the entanglement sphere are B cosAA i sin cos sin 2 i identified according to: (b,t)=(-b,-t); or (b,-t)=(-b,t). sinA e BB This introduces a two-fold ambiguity in the combined 2 i sin sin cos B ei B coordinates of the qubit-A sphere and the entanglement A 2 sphere. This is because only the bt product appears in the base parameters: In these expressions, it is clear that factoring out the 1x x bt = 1 01 and global phase requires the azimuth angles and B to A be shifted accordingly: -2B and BB-2B. Note 2 x10 bt1 x here that since no quaternion is involved in Eq.(34), the 1 x 1 0 (33) imaginary unit k was replaced by i. We present in this ΨA qB x bt section a streamlined procedure on how to obtain the 21 x0 1 Bloch coordinates (the seven angles) from the complex The Bloch sphere coordinates with the same bt product amplitudes , and . (while all other coordinates are being the same)
correspond to the same state. On a different note, Eq.(33) can be used as a shortcut to Eq.(24) when determining the The four angle parameters (A, A, , ) and their Cartesian coordinates (x , x , x , x , x ) of the Hopf base Bloch sphere coordinates. The column vector in the quasi 0 1 2 3 4 state can be taken straight from the first column of the are determined from the quasi density matrix which is given below in Eq.(39). Once the base parameters have quasi density matrix. One can determine the values of x0, been found, the three parameters (B, B, B) of the Hopf x1, b and t by comparing A of Eq.(33) with Eq.(10), and fiber qB are found from the quasi state Ψ , Eqs.(35) and noting that t is a pure-imaginary unit quaternion. A A detailed step-by-step parameter determination procedure (25). From Eqs.(24) and (25),
of the seven angle parameters is given in the next section θA (section IV). cos j 2 (35) Ψ A qB j θ A tA (4) This Bloch sphere parameterization of Eq.(24) is not sin e valid for a separable state given by Ψ 1 ψ . 2 ABA B the quasi density matrix is For this state, qubit-A is at the South pole, x0= -1 or A=. θA In this case, the qubit-B can be plotted on its Bloch sphere cos 2 θθt using the single qubit state ψ . This separable state Ψ Ψq q cosAA , sin e A B ABAA θ B A tA 22 appears to be the only exception to our model, Eq.(24). sin e Another minor issue occurs when the qubit-B is at the 2 1 South pole: z = -1 or =. In this case, the terms, kB (36) B B e ItnA σ 2 and k B , are interchangeable in q with the appropriate e B Where, change of sign. The final state, however, is consistent after factoring out the global phase factor.
IV. PROCEDURE TO OBTAIN (A, A), (, ), (B, B), B FROM , AND EXAMPLES
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θ qB qB 1, σ t x , y t , z , cos A step-4) 2 j ΨABBBBBqq , , nAAAAAAsin cos , sin sin , cos x10 , b , x θ j A tA sin e t sin cos i sin sin j cos k ; 2
Solving for qB, we get θAB , θ , 0, ; ABB , , , 0, 2π (37) θθ The Pauli matrices are defined as q cosAA j sin etA j B 22 0 1 0t 1 0 (38) x , yt , z 1 0t 0 0 1 The qubit-B parameters are found by equating this qB to The Hopf fiber qB (parameters: B, B, B) is completely eliminated from . Hence, depends only on t and the kk A BBBB qBBBB cos sin e j e , , qubit-A parameters. Here, t is the imaginary unit in the 22 Pauli matrix y(t). The quasi-density matrix is written in full as Through these four steps, the seven angle parameters are 1 1x x bt obtained from the four complex amplitudes of the state. 01 A At this point, if B0, then 2B may be subtracted from 2 x10 bt1 x (39) and B (where they become the new and B) while the 1 1x0 x 1 x 4 k x 3 x 2 k j k global phase factor e B is factored out of the complex 2 x x k x x k j1 x 1 4 3 2 0 amplitudes . Then the global phase can be reset 22 j (that is, set B=0) or ignored. All these four steps can be done quickly using the shortcut Eq.(33). j 22 B. Examples of two-qubit states on Bloch sphere where, x cos , x sin cos , b sin sin , c bsin , 01AAAAA 1. Maximally entangled states, Bell states As a first example, we plot the four maximally entangled x2 ccos , x 3 csin , x 4 bcos , bt xi 2 xj 3 xk 4 4 states (MES), namely the Bell states defined as: All angle parameters of the S Hopf base, (A, A) and (, 00 11 01 10 ), and the equivalent Cartesian coordinates (x0, x1, x2, x3, , , 00 01 (40) x4), are obtained. The procedure is outlined step by step 22 below: 00 11 01 10 , 1022 11 4 1. S Hopf base parameters:(A,A),(,)(x0, x1, x2, x3, x4) In the qubit-A basis, converting qubit-B basis states to quaternion units, we get the following quasi states: 2 2 2 2 step-1) x0 cosAA 111 j 00 , 01 , 22j 1 2 AA11xx00 (41) sinA 1 x0 , cos , sin 2 2 2 2 111 j , step-2) 10 11 x14 kx 2 A , 22j 1 x sin cos two possiblevalues The quasi density matrices are: 1 AAA b sinAA sin 111 jj 1 00 , 01 , 22 jj 1 1 (42) x4 bcos c bsin 111 jj 1 10 , 11 step-3) x32 kx 2 22 jj1 1 x ccos , x c sin 23 For this example we use the shortcut Eq.(33). From the t i sin cos j sin sin k cos density matrix we find:
00 : x0=0, x1=0, b=1, t=j 2. S3 Hopf fiber parameters: ( , ) and B B B : x =0, x =0, b=1, t=-j Once the base parameters are determined, the parameters, 01 0 1 ( , ) and , are obtained from the quasi state . : x =0, x =0, b=1, t=-j B B B ΨA 10 0 1
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: qB=1 B=0, B=0 11 : x0=0, x1=0, b=1, t=j 10 From the quasi state in Eq.(33), equating it to Eq.(41), we : q =j =, =0, =0 11 B B B B find qB, which we then equate to Eq.(25), to find The Bloch coordinates and Bloch spheres are summarized : qB=1 B=0, B=0 00 in the Table I and Table II, respectively. : q =j =, =0, =0 01 B B B B
Table I The Bloch coordinates of maximally entangled Bell states. Since (b, t) and (-b, -t) give the same state, the coordinate for |cd> alternate to the table is cd = (x1, b, x0) & (tx, ty, tz): 00 = (0,-1,0) & (0,-1,0); 01 = (0,-1,0) & (0, 1, 0); 10 = (0,-1,0) & (0, 1, 0); 11 = (0,-1,0) & (0, -1, 0).
Bloch sphere |00> |01> |10> |10> Coordinates (x0, b, x1) (0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0) (tx, ty, tz) (0, 1, 0) (0, -1, 0) (0, -1, 0) (0, 1, 0) (xB, yB, zB) (0, 0, 1) (0, 0, -1) (0, 0, 1) (0, 0, -1) B 0 0 0 0
Table II Bell State Bloch Spheres. For all cases, B=0. The first three Bloch sphere rows are for the Bell states. In the first two rows, filled (red) and hollow (blue) dots are the alternate coordinates (b,t) and (-b,-t), respectively. The last row is the qubit-A Bloch sphere obtained by applying CNOT gate to the Bell states. The CNOT gate did not affect the qubit-B Bloch coordinate.
|00> |01> |10> |11> Qubit-A Bloch Sphere
Entanglement Sphere
Qubit-B Sphere before and after CNOT
CNOT CNOT|00> CNOT|01> CNOT|10> CNOT|11> Qubit-A after CNOT
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Applying the CNOT gate to the Bell state, a product state θθ cosAA cos results. The qubit-B state remained unchanged, while 22 (43) qubit-A coordinate changed to x = 1, b=0 (which gives 1 θθ AAtkAA zero concurrence, c = b sin = 0). The entanglement sin e sin e sphere can be ignored for any state with b=0. The product 22 state can be easily written down from the Bloch SS42 coordinates. For the entangled states, the states can be In this case, the quasi state recovered from the Bloch coordinates using Eq.(24). θA cos k 2 k2 (44) ΨeB cosBB sin eBB j A θ A kA 22 2. Separable states sin e 2 For separable states the concurrence is zero, which is corresponds to the composite state: satisfied if b=sin sin =0 or if sin=0: the former θA B A A cos cos k 22 (45) condition (b=0) is satisfied if the qubit-A is on the great Ψe B ψ ψ ABθ A B circle on the x x -plane (b=0). In this case, the A kA B kBB2 0 1 sin e sin e entanglement sphere is undefined because t appears in the 22 quasi-state and quasi-density matrix only as a product The global phase factor ek B may be ignored (i.e., set to with b (see Eq.(33)). In the second case (sin=0), the unity) after the azimuth of qubit-B Bloch sphere has been entanglement sphere coordinate is either at the North pole translated: BB-2B. It is clear that when t = k (or –k), (=0) or at the South pole (=). In both cases, since the the entire coordinate of qubit-A Bloch sphere is valid for concurrence is zero (c=0), the angle is undefined (see the single qubit state. The separable state Bloch spheres step-3). From the two-fold ambiguity in (b, t), choosing are shown in Fig.3. Some specific examples are shown in t=k (i.e., =0, North pole), the qubit-A Bloch sphere the Table III. coordinate agrees with the single qubit state of the original product state.
4 For a separable state where t equals k, the Hopf base S is reduced to S2:
Figure 3: The separable state Bloch spheres of Eq.(45). The entanglement sphere is necessary when the qubit-A Bloch coordinate is at b≠0, and in which case, the entanglement sphere coordinate must be either the North pole (filled gray dot) or the South pole (open gray dot).
Table III Separable state examples: ΨAB ψ A ψ B . In all cases, B = 0. In the coordinates of qubit-A sphere and entanglement sphere, the two pairs of corresponding shape (or color) coordinates give the same bt product in Eq.(33). The North pole (filled red dot) of entanglement sphere (i.e., t = k) gives qubit-A coordinate which agrees with the single- qubit state |A>.
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Qubit-A Sphere Entanglement Sphere Qubit-B Sphere
C. Trajectory of CNOT, CZ and SWAP gates Then, the phase is fixed at /2, =/2, and the rotation angle increases from =0 to = around a given axis We plot the Bloch sphere trajectory of two-qubit gates. (marked ‘2’ in the figure). Let us consider a controlled-unitary, C(U), gate in two steps. Here, U is a single-qubit unitary which can be kk written as '1 e Rn 0 e , 0 ; 2 (47) k k n σ kk 2 2 '2' e Rnn kR , 0 U e Rn e e (46) We consider the trajectories of CNOT, CZ and SWAP where is a phase angle, R () is the rotation operator n gates following these two steps. that rotates a single-qubit state by an angle around the axis n, and is the Pauli matrix, = (x,y(k),z). The 1. CNOT and CZ gates Pauli X and Z gates can be written as 00 11 Let us take as an example. The CNOT k 10 2 X e Rx tobeused inCNOT and SWAP 2 gate and CZ gate trajectories are shown in Figures 4 and k 5, respectively. For the CZ gate in Figure 5, the entire 2 Z e Rz tobeused inCZ process amounts to a change in the nonlocal phase angle by (from =3/2 to /2 counterclockwise). We take a two step process: First, the phase angle increases from 0 to /2, = 0/2 (marked ‘1’ in the figure) while the rotation angle is fixed at zero, =0.
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01 Figure 4: Evolution of CNOT gate on the Bell state |10>: . The alternate (b, t) =(-b,-t) CNOT 10 0 2 trajectory is not shown. B=0. See Eq.(47).
Figure 5: Evolution of the CZ gate on the Bell state |10>: CZ|10>=|00>. The alternate (b, t) =(-b,-t) is not shown. B=0. See Eq.(47).
2. SWAP gate figure, followed by an X-rotation 0 which is marked ‘2’ in the figure. As an example, in Fig.6 we We consider the SWAP operation as show the SWAP gate operating on a separable state 01 k 11 . The resulting state is another separable Ψ AB 2 k 2 k SWAP X e Rxx e R 10 k 11 0 state . 2 , 0 2
Similar to the CNOT and CZ gates above, we first let the phase evolve 0 which is marked as ‘1’ in the 2
00kk 01 0 1 00k 01 00 k 10 0 k 1 Figure 6: The action of a SWAP gate on 0 : SWAP 0. 22 2 2 2
The hollow red circle in the entanglement sphere is the initial state at =/2, =0. B=0.
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V. DISCUSSION quasi-Bloch sphere would have its B coordinate depend The two quasi-Bloch spheres seem to be a natural choice on 1-2. The same argument applies to the a-c and b-d for their name and coordinates because for separable pairs, leading us to expect the qubit-A Bloch coordinate states, these spheres reduce to the respective single qubit A to depend on 1+2. Moreover, the a-d and the b-c Bloch sphere (with the azimuth of qubit-B Bloch sphere pairs are entangled pairs with a relative phase factor adjusted to B-2B). The phase angle B is not simply the exp(2k1) and exp(2k2), respectively. In section III, it global phase, but it also affects the azimuth angles B and was mentioned for the two MES states that = 21+/2 . After properly adjusting these two azimuth angles (that for a=1/2=d (b=0=c), and =22+ 3/2 for b=1/2=c is, after subtracting 2B from each and replacing the (a=0=d). Another example may be a=b=d=1/3, c=0, azimuth with these new values, Eq.(32)), the phase factor which is a partially entangled state (concurrence = 2/3). exp(kB) can be considered a pure global phase factor and For this state we obtain =21+/2, B=1-2, and cosA ignored. = (1/2)cos(1+2). For a separable state example, we
find B=1-2 for a=1/2=b (c=0=d); A=1+2 for The entanglement sphere and its coordinates deserve b=1/2=d (a=0=c); and A=1+2 and B=1-2 for more discussion. This sphere seems to capture the a=b=c=d=1/2. In all cases, any non-zero global phase nonlocal parameters. The maximally entangled states angle had been factored out to the left end of exist on the equator of this sphere (= /2) together with B expression as in Eq.(30) with the angles and properly the possible two coordinates b=±1 (East and West) on the B shifted. These examples suggest that is a ‘nonlocal’ equator of qubit-A Bloch sphere. If = 0 or , then the relative phase angle of the composite state while and state is separable and the Bloch spheres may be A considered the single-qubit Bloch sphere of each qubit. B are the ‘local’ relative phase angles. However, this A product state can also happen if the qubit-A coordinate interpretation needs more investigation. is on the great circle corresponding to b=0 (prime meridian of the qubit-A sphere). Let us discuss the single-qubit Bloch sphere (or ball) of qubit-A after a partial trace over qubit-B. Once qubit-B is traced out, the quasi-state vector is no longer a well The azimuth of the entanglement sphere deserves much ΨA more discussion. This appears to parameterize a defined quantity. The partial trace over qubit-B maps the
‘nonlocal relative phase’ angle of the composite state. As two-quit density matrix AB=|AB><AB| to the reduced was mentioned in section III (see Eqs.(27) and (28) and density matrix A. Similarly, let us define the partial trace discussion there) this angle is the phase angle of over qubit-B as a map on the quasi-density matrix , concurrence. It was mentioned that this angle is equal to A tr : . the relative phase angle of MES states offset by some B A A fixed constant. Furthermore, the azimuth angles A and 1 1x01 x bt trBA: B seem to parameterize the ‘local’ relative phase angle in 2 x bt1 x the sense that they capture the relative phase angle 10 (49) between the ‘separable pairs’ of the basis components in 1 1x x x k 0 1 4 the composite state. To be more specific, let us consider A 2 x1 x 4 k1 x 0 the following state: Let us note first that A has the properties of a pure-state 2 k k1 k 2 k 2 k 1 density matrix: = , tr( )=1, and det( )=0. ΨAB e ae00 be 01 ce 10 de 11 A (48) Two things are immediately clear in this definition: (1) The partial trace over qubit-B is equivalent to where a, b, c, d are a non-negative real number, projecting the unit quaternion t onto the k-axis in the normalized; and , 1 and 2 are real numbers. This is a entanglement sphere: bt btzk = bcosk = x4k (see state with a definite phase factor for the concurrence Fig.7(b)). And (where, = (ad-bc)ek2) while the component basis (2) indicates that the qubit-A quasi-Bloch sphere is the states can vary with a relative phase from each other. Riemann sphere with x1-b-x0 axes. This unit sphere Suppose we consider a pair of component states at a time. intersects at its equator a complex plane which has x1-real The a-b pair is for a product state with |0> for qubit-A axis (x) and b-imaginary axis (y) with imaginary unit t, with qubit-B having a relative phase factor exp[k(1-2)] and a point is given by z=x+ty on this complex plane. between its two basis states. For the c-d pair, qubit-A is Then, in terms of the stereographic projection, the point in |1> and qubit-B has a relative phase factor exp[-k(1- (A, A) on the Bloch sphere is projected to a point z on 2)] between its two basis states. Therefore, the qubit-B
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x bt t the complex plane given by z1 cotA e A (see VI. SUMMARY 12 x0 A two-qubit pure-state Bloch sphere model was proposed Fig.7(a)). One the other hand, after tracing out qubit-B, A indicates that the Bloch ball coordinate axes for the and some examples of two-qubit states and gates were mixed-state single qubit-A are x -x -x where the complex presented. The model consists of three unit 2-spheres 1 4 0 (Bloch spheres of each qubit and a nonlocal, plane has x1-real axis and x4-imaginary axis with imaginary unit k. Then, the qubit-A Bloch coordinate is entanglement sphere) and a phase factor. This model can 2 2 2 2 consistently represent both separable and entangled two- that of a Bloch ball given by x0 +x1 +x4 =1-c where c is the concurrence of the two-qubit state before taking the qubit pure states except one notable exception, Ψ 1 ψ , in which case, the states can be partial trace. In this single-qubit mixed-state Bloch ball, a ABA B constant concurrence corresponds to a sphere of radius represented directly, without following the procedure (1-c2). Using the angular parameters in this paper, the presented here. The model is presented in terms of qubit- Bloch coordinate would be (sinAcosA, sinAsinAcos, A basis. This three sphere model seems to be able to cosA). For a fixed angle , this gives a flattened sphere, represent all two qubit pure states and two qubit gates flattened along the imaginary-axis by factor |cos| relative geometrically, enabling visualization and intuition. The to the unit 2-sphere. On the x0x1-plane, it has a unit main result is given by Eq.(24), which has a shortcut in radius (see Fig.7(c)). These relationships are illustrated in Eq.(33). A detailed step-by-step procedure of parameter Figure 7. Finally, Figure 8 summarizes the maximally determination was given as the step-1 through step-4 in entangled states and the separable states on the qubit-A section-IV. sphere and the entanglement sphere.
Figure 7 An illustration of Eq.(49). (a) The quasi-density matrix A and quasi-Bloch sphere of qubit-A. (b) The complex planes. Partial trace over qubit-B projects the t imaginary-axis (b-axis) to the k imaginary-axis (x4-axis), shrinking Bloch sphere(ball) in this direction by a factor |cos|. The k imaginary-axis (c-axis) is the concurrence axis k(-/2) (where k=e j , see Eq.(27)). (c) After tracing out qubit-B, the reduced density matrix A is plotted for a constant (at =/4) and for a constant concurrence (at c=1/2).
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Figure 8 The maximally entangled states and the separable states are summarized on the qubit-A sphere and the entanglement sphere. This summary is consistent with the examples given in the Tables I, II and III.
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