Unitary Process Discrimination with Error Margin DEX-SMI Workshop - PowerPoint PPT Presentation

Unitary Process Discrimination with Error Margin DEX-SMI Workshop on Quantum Statistical Inference March 2-4, 2009, National Institute of Informatics (NII), Tokyo A. Hayashi (Fukui) T. Hashimoto (Fukui), M. Horibe (Fukui) and M. Hayashi (Tohoku)

Unitary process discrimination with error margin unitary process input output U U U ... 1 2 3 | φ> U | φ>=|φ > i i State discrimination {| φ i � ≡ U i | φ �} Inconclusive result (”I don’t know”) is allowed Maximize the success probability P success Margin on error probability P error < m m = 1 : minimum-error discrimination m = 0 : unambiguous discrimination Two solvable cases: { U 1 , U 2 } (two unitary processes) { T g } g ∈ G ( T g is a projective representation of a finite group G )

Two unitary processes { U 1 , U 2 } First, fix input | φ � Two-state discriminaiton ρ 1 = | φ 1 �� φ 1 | , | φ 1 � ≡ U 1 | φ � ρ 2 = | φ 2 �� φ 2 | , | φ 2 � ≡ U 2 | φ � We assume the same occurrence probabilities POVM { E µ } µ =1 , 2 , 3 E 1 for ρ 1 , E 2 for ρ 2 , E 3 ≡ E ? for “I don’t know”, Phys. Rev. A78, 012333(2008) by A. Hayashi, T. Hashimoto and M. Horibe Joint probabilities The state is ρ a and measurement outcome is µ : P ρ a , E µ = tr [ E µ ρ a ]

Strong error-margin conditions Success probability to be optimized � � p ◦ = 1 P ρ 1 , E 1 + P ρ 2 , E 2 2 Margin m on conditional error probabilities tr [ E 1 ρ 2 ] = tr [ E 1 ρ 1 ] + tr [ E 1 ρ 2 ] ≤ m P ρ 2 | E 1 tr [ E 2 ρ 1 ] P ρ 1 | E 2 = tr [ E 2 ρ 1 ] + tr [ E 2 ρ 2 ] ≤ m m = 1 ⇐ ⇒ Minimum error discrimination � � p ◦ = 1 � 1 − |� φ 1 | φ 2 �| 2 1 + 2 m = 0 ⇐ ⇒ Unambiguous discrimination p ◦ = 1 − |� φ 1 | φ 2 �|

Optimization problem Optimization maximize: p ◦ = 1 � � tr [ E 1 ρ 1 ] + tr [ E 2 ρ 2 ] , 2 subject to: E 1 ≥ 0 , E 2 ≥ 0 , E 1 + E 2 ≤ 1 , � � tr [ E 1 ρ 2 ] ≤ m tr [ E 1 ρ 1 ] + tr [ E 1 ρ 2 ] , � � tr [ E 2 ρ 1 ] ≤ m tr [ E 2 ρ 1 ] + tr [ E 2 ρ 2 ] . Equal occurrence probabilities Semidefinite programming

Bloch vector representation In the 2-dim subspace = < | φ 1 � , | φ 2 � > Bloch vector representation ρ a = 1 + n a · σ E µ = α µ + β µ · σ , 2 Optimization in terms of parameters { α µ , β µ } maximize: p ◦ = 1 � � α 1 + β 1 · n 1 + α 2 + β 2 · n 2 , 2 subject to: α 1 ≥ | β 1 | , α 2 ≥ | β 2 | , α 1 + α 2 + | β 1 + β 2 | ≤ 1 , � � α 1 + β 1 · n 2 ≤ m 2 α 1 + β 1 · ( n 1 + n 2 ) , � � α 2 + β 2 · n 1 ≤ m 2 α 2 + β 2 · ( n 1 + n 2 ) .

Optimal success probability  � � A m 1 − |� φ 1 | φ 2 �| , (0 ≤ m ≤ m c ) ,    p ◦ = � �  1 � 1 − |� φ 1 | φ 2 �| 2 1 + , ( m c ≤ m ≤ 1) ,   2 where � � m c = 1 � 1 − |� φ 1 | φ 2 �| 2 1 − , 2 and A m is an increasing function of error margin and defined to be 1 − m � � � A m = 1 + 2 m (1 − m ) . (1 − 2 m ) 2

Success probability p ◦ ( m ) 1 p : success probability 0.8 p m 0.6 0.4 ο 0.2 p u 0 0 0.1 0.2 0.3 0.4 0.5 m c m : error margin p m : minimum error discrimination p u : unambiguous discrimination

Strong and weak error-margin conditions I Strong conditions P ρ 2 | E 1 ≤ m , P ρ 1 | E 2 ≤ m � � = 1 − |� φ 1 | φ 2 �| p ◦ A m , (0 ≤ m ≤ m c ) 1 − m � � � A m ≡ 1 + 2 m (1 − m ) (1 − 2 m ) 2 Weak conditions p × = P E 1 ,ρ 2 + P E 2 ,ρ 1 ≤ m � √ m + � 2 � p ◦ = 1 − |� φ 1 | φ 2 �| , (0 ≤ m ≤ m c ) POVM E 1 , E 2 , E 3 : rank 0 or 1

Strong and weak error-margin conditions II 1 p : success probability 0.8 p m 0.6 Weak 0.4 Strong ο 0.2 p u 0 0 0.1 0.2 0.3 0.4 0.5 m c m : error margin p m : minimum error discrimination p u : unambiguous discrimination

Optimal discrimination by LOCC Local Operations and Classical Communication (LOCC) Bipartite state ? ? Local operations Alice Bob Classical communication

Two Orthogonal pure states Two orthogonal pure states can be perfectly discriminated by LOCC:(Local Operations and Classical Communication) (Walgate et al. , 2000) Example ： 1 1 | φ 1 � = √ ( | 0 �| 0 � + | 1 �| 1 � ) = √ ( | + �| + � + | − �| − � ) 2 2 1 1 √ √ | φ 2 � = ( | 0 �| 0 � − | 1 �| 1 � ) = ( | + �| − � + | − �| + � ) 2 2 1 where , | ± � = √ ( | 0 � ± | 1 � ) 2 In general, if � φ 1 | φ 2 � = 0 � | φ 1 � = � � | i � : Orthonormal basis i | i �| ξ i � i | i �| η i � , where | φ 2 � = � � ξ i | η i � = 0

Local discrimination of nonorthogonal states Local discrimination Two non-orthogonal pure states (generally entangled) can be optimally discriminated by LOCC. Discrimination with minimum error: Virmani et al . (2001) � 1 − |� φ 1 | φ 2 �| 2 � m c = 1 Error margin : m c ≤ m ≤ 1, 2 Unambiguous discrimination: Chen et al . (2001,2002), Ji et al . (2005) Error margin : m = 0 We can show For any error margin, two pure states can be optimally discriminated by LOCC.

Three-element POVM of rank 0 or 1 Optimal POVM of discrimination with error margin { E 1 , E 2 , E 3 } : each element is of rank 0 or 1 Theorem Let V be a two-dimensional subspace of a multipartite tensor-product space H , and P be the projector onto the subspace V . Then, for any three-element POVM { E 1 , E 2 , E 3 } of V with every element being of rank 0 or 1, there exists a one-way LOCC POVM { E L 1 , E L 2 , E L 3 } of H such that E µ = PE L µ P ( µ = 1 , 2 , 3) . Proj. Op. P { E ,E ,E } rank 0 or 1 1 2 3 L 2-dim. subspace V E = P E P µ µ L L L { E ,E ,E } : LOCC 1 2 3 Multi partite H

Unitary process { U 1 , U 2 } discrimination I Discrimination between states {| φ 1 � , | φ 2 �} P max ( m , | φ 1 � , | φ 2 � ) = f ( m , |� φ 1 | φ 2 �| ) � √ m + √ 1 − s √ � 2  � � 0 ≤ m < 1 1 − s 2 1 − ,  2 f ( m , s ) = √ √ � � � � 1 1 1 − s 2 1 − s 2 1 + , 1 − ≤ m ≤ 1  2 2 f ( m , s ) is decreasing with respect to s Discrimination between processes { U 1 , U 2 } P pure max ( m ) = max | φ � P max ( m , | φ 1 � , | φ 2 � ) = f ( m , µ ) � � φ | U + � � µ ≡ min 1 U 2 | φ � � | φ � Remark: The optimal success probability is attained by a pure-state input, since P pure max ( m ) is concave with respect to m

P ( m ) max Discrimination between processes { U 1 , U 2 } II � � p 1 2 1 + 1 � � 2 1 0.8 0.6 PSfrag repla emen ts 1 � � 0.4 m � � p 1 2 m = 1 � 1 � � 2 0.2 0 0 0.2 0.4 0.6 0.8 1 m : error margin � � � φ | U + � µ ≡ min | φ � 1 U 2 | φ � �

Minimum fidelity µ � d � � � � = � � � φ | U + q a e i θ a � � ≡ min 1 U 2 | φ � min µ � � � � | φ � q a ≥ 0 , P a q a =1 � a =1 � where { e i θ 1 , e i θ 2 , . . . , e i θ d } are eigenvalues of U + 1 U 2 µ µ=0 µ>0

Unitary processes { T g } g ∈ G : T g is a unitary projective repersentation of a finite group G Unitary projective representation T g T h = c g , h T gh ( T + g = T − 1 g , | c g , h | = 1 , g , h ∈ G ) where { c g , h } is a factor set. State discrimination � � 1 P φ g = | G | , | φ g � = T g | φ � g ∈ G where | G | is the order of G .

Covariant measurement ✓ ✏ Covariant POVM Optimal POVM { E g , E ? } can be assumed covariant: T g E ? T + T g E h T + g = E ? , g = E gh ✒ ✑ ✓ ✏ Optimization with covariant POVM maximize: P ◦ = � φ | E 1 | φ � � T g E 1 T + subject to: E 1 ≥ 0 , g ≤ 1 g ∈ G weak error-margin condition � � φ | T + P × = h E 1 T h | φ � ≤ m h ( � =1) ✒ ✑

If { T g } is irreducible (I) By Schur’s lemma g = | G | � T g E 1 T + d tr [ E 1 ] · 1 ( d = dimension) g ∈ G g ∈ G T g E 1 T + Completeness of POVM : � g ≤ 1 P ◦ = tr [ E 1 ρ ] ≤ tr [ E 1 ] ≤ d | G | T + � � Error-margin condition : � g E 1 T g ρ ≤ m g ( � =1) tr m P ◦ = tr [ E 1 ρ ] ≤ tr [ E 1 ] ≤ | G | d − 1

If { T g } is irreducible (II) Maximal success probability P ( m ) max m � (0 ≤ m ≤ m c ) d − 1 , | G | P max ( m ) = d d | G | , ( m c ≤ m ≤ 1) j G j PSfrag repla emen ts 1 − d m c = | G | d 0 m = 1 � 1 m j G j

General case For any E 1 ≥ 0 min( m r , d r ) d r � T g E 1 T + � g ≥ E 1 , κ ≡ κ | G | r g ∈ G | G | = order of G , r = irreducible representation d r = dimension of r , m r = multiplicity of r (Proof: orthogonality of representation matrices and Schwarz inequality) d r d r Note: � | G | = 1 (Plancherel measure) r g ∈ G T g E 1 T + Completeness of POVM : � g ≤ 1 P ◦ = tr [ E 1 ρ ] ≤ tr [ E 1 ] ≤ κ T + � � Error-margin condition : � g E 1 T g ρ ≤ m g ( � =1) tr κ P ◦ = tr [ E 1 ρ ] ≤ tr [ E 1 ] ≤ 1 − κ m

Maximal success probability κ � (0 ≤ m ≤ m c ) 1 − κ m , P max ( m ) = ◦ ( m c ≤ m ≤ 1) κ, = 1 − κ m c min( m r , d r ) d r � κ = | G | r Note : With a sufficient large ancilla, d 2 d 2 � � r r κ → | G | → | G | = 1 (Plancherel measure) r r ( m r ≥ 1) Optimal input and POVM min( m r , d r ) � 1 d r � � | φ � = √ κ | G || r , a , a � r a =1 E 1 = P max ( m ) | φ �� φ | ◦